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- About
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- Abstract
- Introduction
- Field measurements
- Simulation methodology
- Results and discussion
- Conclusions
- Code and data availability
- Appendix A: Induction correction for the measured reference wind speed for the power curve measurements of the 4R-V29 wind turbine
- Appendix B: Nomenclature
- Author contributions
- Competing interests
- Acknowledgements
- Review statement
- References

**Research article**
20 May 2019

**Research article** | 20 May 2019

Power curve and wake analyses of the Vestas multi-rotor demonstrator

^{1}Technical University of Denmark, DTU Wind Energy, RisøCampus, Frederiksborgvej 399, 4000 Roskilde, Denmark^{2}Technical University of Denmark, DTU Wind Energy, Lyngby Campus, Anker Engelunds Vej 1, 2800 Lyngby, Denmark^{3}Vestas Wind System A/S, Hedeager 42, 8200 Aarhus, Denmark

^{1}Technical University of Denmark, DTU Wind Energy, RisøCampus, Frederiksborgvej 399, 4000 Roskilde, Denmark^{2}Technical University of Denmark, DTU Wind Energy, Lyngby Campus, Anker Engelunds Vej 1, 2800 Lyngby, Denmark^{3}Vestas Wind System A/S, Hedeager 42, 8200 Aarhus, Denmark

**Correspondence**: Maarten Paul van der Laan (plaa@dtu.dk)

**Correspondence**: Maarten Paul van der Laan (plaa@dtu.dk)

Abstract

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Numerical simulations of the Vestas multi-rotor demonstrator (4R-V29) are
compared with field measurements of power performance and remote sensing
measurements of the wake deficit from a short-range WindScanner lidar system.
The simulations predict a gain of 0 %–2 % in power due to the rotor
interaction at below rated wind speeds. The power curve measurements also
show that the rotor interaction increases the power performance below the
rated wind speed by 1.8 %, which can result in a 1.5 % increase in
the annual energy production. The wake measurements and numerical simulations
show four distinct wake deficits in the near wake, which merge into a
single-wake structure further downstream. Numerical simulations also show
that the wake recovery distance of a simplified 4R-V29 wind turbine is
1.03–1.44 *D*_{eq} shorter than for an equivalent single-rotor wind
turbine with a rotor diameter *D*_{eq}. In addition, the numerical
simulations show that the added wake turbulence of the simplified 4R-V29 wind
turbine is lower in the far wake compared with the equivalent single-rotor
wind turbine. The faster wake recovery and lower far-wake turbulence of such
a multi-rotor wind turbine has the potential to reduce the wind turbine
spacing within a wind farm while providing the same production output.

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van der Laan, M. P., Andersen, S. J., Ramos García, N., Angelou, N., Pirrung, G. R., Ott, S., Sjöholm, M., Sørensen, K. H., Vianna Neto, J. X., Kelly, M., Mikkelsen, T. K., and Larsen, G. C.: Power curve and wake analyses of the Vestas multi-rotor demonstrator, Wind Energ. Sci., 4, 251–271, https://doi.org/10.5194/wes-4-251-2019, 2019.

1 Introduction

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Over the past few decades, the rated power of wind turbines has been increased by upscaling the traditional concept of a horizontal axis wind turbine with a single three-bladed rotor. It is expected that this trend will continue for offshore wind turbines, although the problems that arise from realizing large wind turbine blades (> 100 m) are not trivial to solve (Jensen et al., 2017). An alternative way to increase the power output of a wind turbine is the multi-rotor concept, where a single wind turbine is equipped with multiple rotors. From a cost point of view, it can be cheaper to produce a multi-megawatt wind turbine with several rotors consisting of relatively small blades that are already mass produced compared with a single-rotor wind turbine with newly designed large blades (Jamieson et al., 2014). In addition, small blades are easier to transport than large blades, which makes a multi-rotor concept interesting for locations where infrastructure is a limiting factor. However, multi-rotor wind turbines also have disadvantages; for example, a more complex tower is required and the number of components is higher compared with single-rotor wind turbines.

The multi-rotor concept is an old idea that dates back to the start of 19th century. Between 1900 and 1910, a Danish water management wind mill, was upgraded to a twin-rotor wind mill (Holst, 1923). Around the 1930s, Honnef (1932) introduced the multi-rotor concept for an electricity generating wind turbine, as discussed by Hau (2013). In the late 20th century, the Dutch company Lagerwey built and operated several multi-rotor wind turbine concepts based on two, four and six two-bladed rotors (Jamieson, 2011). In April 2016, Vestas Wind Energy Systems A/S built a multi-rotor wind turbine demonstrator at the Risø Campus of the Technical University of Denmark. This multi-rotor wind turbine, hereafter referred to as the 4R-V29 wind turbine, consists of four V29-225 kW rotors, which are arranged as a bottom and top pair. The 4R-V29 wind turbine operated for almost 3 years and was decommissioned in December 2018. In the present article, we investigate the power performance and wake interaction of the 4R-V29 wind turbine using field measurements and numerical simulations.

The tip clearances between the rotors in multi-rotor wind turbines are typically much smaller than a single-rotor diameter, and several authors have shown that the operating rotors strongly interact with each other. Chasapogiannis et al. (2014) and Jamieson et al. (2014) performed numerical simulations of closely spaced rotors positioned in a honeycomb layout with a tip clearance of 5 % of the (single) rotor diameter. Chasapogiannis et al. (2014) simulated seven 2 MW rotors using computational fluid dynamics and vortex models, and they calculated an increase in power and thrust of about 3 % and 1.5 %, respectively, compared with seven non-interacting single rotors. In addition, Chasapogiannis et al. (2014) found that the seven individual single wakes merge into a single-wake structure at a downstream distance equal to or further than two rotor diameters. Jamieson et al. (2014) simulated a 20 MW multi-rotor wind turbine consisting of 45 444 kW rotors. They argued that wind turbine loads are reduced compared with a single-rotor 20 MW wind turbine because of load-averaging effects when using 45 small rotors; furthermore, they reported that the power performance is increased due to rotor interactions, and the fact that smaller wind turbines can respond faster to wind speed variations. More recently, Jensen et al. (2017) reported an 8 % power increase for the same multi-rotor wind turbine using a smaller tip clearance of 2.5 % or the rotor diameter. Nishino and Draper (2015) employed Reynolds-averaged Navier–Stokes (RANS) simulations of a horizontal array of actuator discs (AD) with a tip clearance of 50 % of the rotor diameter and an optimal thrust coefficient. They found an increase in the wind farm power coefficient, based on the axial induction of the ADs (up to 5 %), when increasing the number of ADs from one to nine. Nishino and Draper (2015) also simulated an infinite array of ADs, but the domain blockage ratio for this case was too high (2 %) to obtain a valid result, as also discussed in Sect. 3.2.2 of the present article.

Ghaisas et al. (2018) employed large-eddy simulations (LES) and two engineering wake models to show that a general multi-rotor wind turbine consisting of four rotors has a faster wake recovery and lower turbulent kinetic energy in the wake compared with a single rotor with an equivalent rotor area. They argued that the faster wake recovery is a result of a larger entrainment, as the ratio of the rotor perimeter and the rotor swept area is twice as high for their multi-rotor turbine compared with a single-rotor turbine. In the same work, it was also shown that different tip clearances in the range of zero to two rotor diameters hardly effect the wake recovery of the multi-rotor wind turbine, whereas the turbulent kinetic energy in the far wake varies, although it is always less than the turbulent kinetic energy in the far wake of a single rotor. Finally, it was shown that the power deficit and the added turbulent kinetic energy in the wake of a row of five multi-rotor wind turbines is less than for a row of five single-rotor wind turbines. These results suggest that a wind farm of multi-rotors has lower power losses and fatigue loads due to wakes than a wind farm of single-rotor wind turbines. In the present article, we attempt to confirm the results of Ghaisas et al. (2018) for the 4R-V29 wind turbine using different models and levels of ambient turbulence.

2 Field measurements

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Figure 1 depicts the 4R-V29 wind turbine located at the
Risø Campus of the Technical University of Denmark and a corresponding
sketch including dimensions and rotor definitions. The hub height of the
bottom rotor pair (*R*_{1} and *R*_{2}) and the top rotor pair (*R*_{3} and
*R*_{4}) are 29.04 and 59.50 m, respectively, which gives an average hub
height of 44.27 m. The horizontal distance between the nacelles for both
pairs is 31.02 m. The rotors are equipped with 13 m (V27) blades, where the
blade root is extended by 1.6 m, resulting in a rotor diameter of 29.2 m.
The rotor tilt angles and the cone angles (the angle between the individual
rotor plane and its blade axis) are all zero. To increase the horizontal
distance between the rotors (tip clearance), the 4R-V29 has a toe-out angle
of 3^{∘}, as depicted in the top view sketch in Fig. 1.
This means that the left rotors (*R*_{1} and *R*_{3}) and the right rotors
(*R*_{2} and *R*_{4}) are yawed by +3 and −3^{∘}, respectively. (A
positive yaw angle is a clockwise rotation as seen from above.) The
horizontal and vertical tip clearances are 1.86 and 1.26 m, or 6.4 % and
4.3 % of the single-rotor diameter, respectively, which is close to the
5 % used in simulations performed by
Chasapogiannis et al. (2014) and Jamieson et al. (2014). It is possible to yaw the
bottom and top pairs independently of each other, which could be beneficial
in atmospheric conditions where a strong wind veer is present (i.e., a stable
atmospheric boundary layer).

Power curve measurements of the 4R-V29 wind turbine were carried out to
quantify the effect of the rotor interaction on the power performance. For
this purpose, a test cycle of three stages was run repetitively, as
illustrated in Fig. 2. During stages 1 and 3, only rotors
*R*_{1} and *R*_{3} were in operation, respectively, whereas the other rotors were in
idle. During Stage 2, all rotors were in operation. We used two single-rotor
operation stages to account for the effect of the shear. Each stage was run
for 15 min and was post-processed to 10 min data samples via the removal of
start up and shutdown periods between the stages. By toggling the stages at
every 15 min, we minimized differences in environmental conditions between
the three data sets (one data set per stage).

The reference wind speed is measured using a commercial dual-mode
continuous-wave lidar, ZephIR 300, manufactured by ZephIR (UK)
(Medley et al., 2014). The lidar is mounted on the top platform of the 4R-V29
wind turbine at height of 60 m, as depicted in Fig. 3a. It measures the
upstream wind speed at 146 m (5 D) and 300 m (≈10.3 D), at a
height of 44.3 m, as shown in Fig. 3b. We chose to use the lidar
measurements of the reference wind speed at 146 m because the lidar
measurements at 300 m have a lower data availability and a higher volume
averaging. In order to capture the wind speed at a hub height of 44.3 m, the
instrument is configured with a tilt angle of −7^{∘}, such that the
center of the scan is directed towards the desired measurement height, as
illustrated in Fig. 3. A horizontal pair of measurements
at this height are used to determine the wind speed and yaw misalignment,
using a pair-derived algorithm. The lidar measurements are corrected in real
time for tilt variations due to the tower deflection. A sample, measured
every 1 s (1 Hz), is corrected for the difference in the induction zone for
when only one or all four rotors are in operation, as discussed in
Appendix A. The corrected data samples are averaged over
10 min and then binned in wind speed intervals of 0.5 m s^{−1}. It
should be noted that applying the induction correction after the 10 min
averaging did not make a difference for the final power curve.

The total number of available measurement cycles is depicted in Fig. 4b and corresponds to 549 10 min data
samples or approximately 91.5 h for each stage between wind speeds of 4 and
14 m s^{−1}. The total amount of data per stage is about half of the
minimal requirement as defined in the international standard
(IEC, 2005), where a power curve database should include at
least 180 h of data and a minimal of 30 min per binned wind speed. In
addition, there is not much data available above the rated wind speed. As a result, the
standard error of the mean power in a bin is high, as shown by the error bars
in the power curves of Fig. 4a. These two power curves represent the sum of power from rotors *R*_{1} and *R*_{3}
of stages 1 and 3, and the sum of power from rotors *R*_{1} and *R*_{3} of
Stage 2, both multiplied by a factor of 2. The relative difference between the
power curves is discussed in Sect. 4.2. The power
curve measurements are filtered for events where a rotor (that is planned to
operate) is not in full operation. During the power curve measurements, the
neighboring V27 and Nordtank (NTK) wind turbines were not in operation (see
Fig. 5). To avoid the influence of the other neighboring wind
turbines and flow disturbance from a motorway, the power measurements are
filtered for a wind direction sector 180–330^{∘}, which represents an
inflow from the fjord (see Fig. 5). It should be mentioned that
the wind turbine test site at the Risø Campus is not flat. The influence
of this is minimized by adjusting the lidar configured height to match the
height difference upstream, although this could slightly influence the
power curve measurements. In addition, the power curve measurements are not
filtered for turbulence intensity and atmospheric stability, as the
amount of data remaining after filtering would be too small. However, the measurements
are filtered for normalized mean fit residuals below 4 %, which removes
data samples with a high complexity of incoming flow.

The wake of the 4R-V29 turbine has been measured by three ground based
short-range WindScanners (Mikkelsen et al., 2017; Yazicioglu et al., 2016) during two
separate measurement campaigns, referred as the near-wake and the far-wake campaigns.
The measurement setup is shown in Fig. 5. The three WindScanners
measure the wake deficit by synchronously altering the line-of-sight azimuth
and elevation of each individual unit. In the near-wake campaign, the
WindScanners scanned three cross planes located at 0.5 D, 1 D and 2 D
downstream. In addition, a horizontal line at the lower hub height 1 D
downstream was rapidly scanned at about 1 Hz. Each cross plane/line is
scanned for 10 min, before moving on to the next, which means that every
40 min a complete set of three cross planes is available. The data are stored
in 1 min files and the 10 min scans are post-processed for minutes
without scan plane transitions, rendering 8 min means. The far-wake campaign
consists of only one cross plane scanned at 5.5 D downstream. It is not
possible to scan further downstream due to the presence of a highway and surrounding
trees located 170–200 m downstream of the 4R-V29 wind turbine for a wind
direction of 280^{∘}. The WindScanners are positioned in between the
near- and far-wake scanning distances. The selected WindScanner positions
allow near- and far-wake measurements to be monitored by turning the
“pointing direction” toward and away from the 4R-V29 wind turbine, respectively.
This configuration allows for the estimation of the two components of the
horizontal wind vector by assuming that the vertical component is equal to
zero. During the wake measurements, the neighboring Nordtank (NTK) and V27
wind turbines were not in operation.

Figure 6 summarizes the atmospheric conditions
during the near- and far-wake measurements, as measured at the met mast
depicted in Fig. 5. The met mast is equipped with pairs of cup
and sonic anemometers located at five heights: 18, 31, 44, 57 and 70 m. The wind speed and wind
direction are taken from a cup and a sonic anemometer, respectively, both
located at a height of 44 m, which is close to the average hub height of the
4R-V29 wind turbine. The turbulence intensity and the atmospheric stability
in terms of a Monin–Obukhov length *L*_{MO} are measured by sonic
anemometers located at heights of 44 and 18 m, respectively.

A near-wake case is selected from three consecutive post-processed scans
measured between 21:36 and 22:03 GMT+1 on 28 October 2016. A far-wake case
is taken from one post-processed scan measured between 21:45 and
21:53 GMT+1 on 1 November 2016. During these periods, the atmospheric
stability is near-neutral (*L*_{MO}=340 m) and neutral
(*L*_{MO}=661 m). The wind direction in both cases is close to
280^{∘}, and the yaw offset with respect to the upper platform is 3.4
and 8.2^{∘} for the near- and far-wake cases, respectively. The
atmospheric conditions of the two cases are listed in
Table 1, and are used as input for the numerical
simulations. Note that the simulations only consider neutral atmospheric
stability.

The wind speed and total turbulence intensity profiles measured at the met
mast during the near- and far-wake case recordings are depicted in
Fig. 7. The wind speed and turbulence intensity at
44 m (*U*_{ref} and *I*_{ref}) are used to determine the neutral
logarithmic inflow profiles defined by *z*_{0} and *u*_{*} following
Eq. (2). The results are listed in Table 1.
The far-wake profile deviates from a logarithmic profile at a height of
18 m, which could be related to the upstream fjord-land roughness changes,
as shown in Fig. 5, although this deviation is not observed in
the near-wake case inflow profile.

Spectra of 35 Hz wind velocity data measured by the sonic anemometer at
44 m are used to fit Mann turbulence spectra (Mann, 1994) utilizing three
parameters: *α**ϵ*^{⅔}, *L* and Γ. When these
parameters are used to generate a Mann turbulence box, which is employed as
inflow turbulence for the MIRAS-FLEX5 and EllipSys3D LES-AL-FLEX5 simulations
(Sect. 3), the resulting turbulence intensity in the Mann
turbulence box is lower than the measured value at the sonic anemometer,
which is not fully understood. The problem is circumvented by using an
*α**ϵ*^{⅔} that is about twice as large as original fitted
value. The final values of *α**ϵ*^{⅔},
*L* and Γ are listed in Table 1. Note that the
stream-wise dimension of the Mann turbulence box is chosen to fit an entire
measurement case (40 min) using ${\mathrm{2}}^{\mathrm{14}}\times {\mathrm{2}}^{\mathrm{7}}\times {\mathrm{2}}^{\mathrm{7}}$ points in the
stream-wise and cross direction, respectively, with a spacing of 2 m in all
directions.

3 Simulation methodology

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Four different simulations tools are employed to model the 4R-V29 wind turbine: Fuga, EllipSys3D RANS-AD, MIRAS-FLEX5 and EllipSys3D LES-AL-FLEX5. The simulation methodology for each model, ranked from the lowest to highest model fidelity, is described in the following sections. Note that a high model fidelity corresponds to an intended high accuracy at the price of a high computational cost, although good model performance is not guaranteed. All simulations that are used to model the 4R-V29 wind turbine only assume a neutral atmospheric surface layer inflow. In addition, only flat terrain with a homogeneous roughness length is modeled; hence, the effects of the fjord-land roughness change and sloping terrain are neglected.

Fuga is a fast linearized RANS model developed by Ott et al. (2011). Fuga models
a single wind turbine wake as a linear perturbation of an atmospheric surface
layer. In the present setup, a thrust force is modeled that is distributed
uniformly over the rotor-swept area. The forces are smeared out using a
two-dimensional Gaussian filter with standard deviations of *D*∕4 and *D*∕16
in the stream-wise and cross directions. The turbulence is defined using the
eddy viscosity of an atmospheric surface layer, which means that a wind
turbine wake does not affect the turbulent mixing. The resulting equations
are transformed to wave-number space in the horizontal directions to obtain a
set of mixed spectral ordinary differential equations. As these equations are
very stiff, a novel numerical solving method was developed by Ott et al. (2011).
The linearity of the model allows for the superposition of single wakes, and
is also applicable in multi-rotor configurations.

EllipSys3D is an incompressible finite volume flow solver, initially
developed by Sørensen (1994) and Michelsen (1992), which incorporates both
RANS and LES models, and has different methods of representing a wind
turbine. In this section, the RANS-AD method is discussed. The Navier–Stokes
equations are solved using the SIMPLE algorithm (Patankar and Spalding, 1972), and the
convective terms are discretized using a QUICK scheme (Leonard, 1979). The wind
turbine rotors are represented by actuator discs (ADs) based on airfoil data
as presented in Réthoré et al. (2014). The RANS-AD model can only model stiff blades.
The tip correction of Pirrung and van der Laan (2018) is applied (with a constant of
*c*_{2}=29), which is an improvement of the tip correction of Shen et al. (2005).
This modified tip correction models the induced drag due to the tip vortex,
which leads to a stronger tip loss effect on the in-plane forces than on the
out-of-plane forces. The RANS-AD model can be employed to model two different
flow cases, a uniform inflow and a neutral atmospheric surface layer, which
are described in the following sections (Sects. 3.2.1
and 3.2.2). The uniform inflow case is used to validate the AD
model of a single V29 rotor with the results of two blade element moment
codes. The neutral atmospheric surface layer flow case is used to simulate
the 4R-V29 wind turbine.

For the uniform inflow case, the numerical setup is fully described in
Pirrung and van der Laan (2018). The uniform grid spacing around the AD is set to *D*∕20,
which is fine enough to estimate
*C*_{T} and *C*_{P} within a discretization error of 0.3 % following a
previously performed grid refinement (Pirrung and van der Laan, 2018).

For the atmospheric surface layer flow case, the $k-\mathit{\epsilon}-{f}_{\text{P}}$
model from van der Laan et al. (2015) is employed, which is a modified *k*−*ε*
model developed to simulate wind turbine wakes in atmospheric turbulence. A
typical numerical domain for ADs in flat terrain and corresponding boundary
conditions are employed as described in van der Laan et al. (2015). In the present work,
a finer spacing of *D*∕20 is applied (in previous work from van der Laan et al. (2015) a
spacing of *D*∕10 was used), and a larger uniformly spaced wake domain is
used: 15 D × 5 D × 4 D (stream-wise, lateral and
vertical directions), where *D* is a single-rotor diameter, and the 4R-V29
wind turbine is placed at 3 D downstream from the start of the wake domain.
In addition, a larger outer domain is used –
116 D × 105 D × 50 D – such that the blockage effects
are negligible (blockage ratio: $\mathit{\pi}/(\mathrm{105}\times \mathrm{50})=\mathrm{0.06}\phantom{\rule{0.125em}{0ex}}\mathit{\%}$). In the RANS
simulations, we observed that a blockage ratio of 1 % for the 4R-V29 wind
turbine is not small enough when
comparing the simulated power of the 4R-V29 wind turbine with a single V29
rotor using the same domain. This is because the blockage ratio of the single
V29 rotor simulation is 4 times lower than the 4R-V29 wind turbine
simulation, and one would include a false gain in power for the 4R-V29 wind
turbine that is caused by the difference in the blockage ratio between the
V29 and 4R-V29 wind turbine RANS simulations.

The inflow conditions represent a neutral atmospheric surface layer that is in balance with the domain (without the ADs):

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}U={\displaystyle \frac{{u}_{*}}{\mathit{\kappa}}}\mathrm{ln}\left({\displaystyle \frac{z+{z}_{\mathrm{0}}}{{z}_{\mathrm{0}}}}\right),\\ \text{(1)}& {\displaystyle}& {\displaystyle}k={\displaystyle \frac{{u}_{*}^{\mathrm{2}}}{\sqrt{{C}_{\mathit{\mu}}}}},{\displaystyle}& {\displaystyle}\mathit{\epsilon}={\displaystyle \frac{{u}_{*}^{\mathrm{3}}}{\mathit{\kappa}\left(z+{z}_{\mathrm{0}}\right)}},\end{array}$$

where *U* is the stream-wise velocity, *u*_{*} is the friction velocity,
*κ*=0.4 is the Von Kármán constant, *z* is the height, *z*_{0} is the
roughness length, *k* is the turbulent kinetic energy, *C*_{μ}=0.03 the eddy
viscosity coefficient and *ε* is the turbulent dissipation. The
friction velocity and the roughness height can be set using a reference
velocity *U*_{ref} and a reference (total) turbulence intensity
${I}_{\mathrm{ref}}=\sqrt{\frac{\mathrm{2}}{\mathrm{3}}k}/{U}_{\mathrm{ref}}$, for a reference height
*z*_{ref}:

$$\begin{array}{ll}\text{(2)}& {\displaystyle}& {\displaystyle}{u}_{*}={U}_{\mathrm{ref}}{I}_{\mathrm{ref}}{\displaystyle \frac{{C}_{\mathit{\mu}}^{\mathrm{1}/\mathrm{4}}}{\sqrt{\mathrm{2}/\mathrm{3}}}},{\displaystyle}& {\displaystyle}{z}_{\mathrm{0}}={\displaystyle \frac{{z}_{\mathrm{ref}}}{\mathrm{exp}\left(\frac{\mathit{\kappa}\sqrt{\mathrm{2}/\mathrm{3}}}{{I}_{\mathrm{ref}}{C}_{\mathit{\mu}}^{\mathrm{1}/\mathrm{4}}}\right)-\mathrm{1}}}\end{array}$$

The shear exponent from the power law ($U={U}_{\mathrm{ref}}(z/{z}_{\mathrm{ref}}{)}^{\mathit{\alpha}}$) can be expressed by setting the shear at the reference height ($\partial U/\partial z{|}_{{z}_{\mathrm{ref}}}$) from the power law equal to that from the logarithmic profile and substituting Eq. (2):

$$\begin{array}{ll}\text{(3)}& {\displaystyle}\mathit{\alpha}& {\displaystyle}={\displaystyle \frac{{u}_{*}}{\mathit{\kappa}{U}_{\mathrm{ref}}}}{\displaystyle \frac{{z}_{\mathrm{ref}}}{({z}_{\mathrm{ref}}+{z}_{\mathrm{0}})}}{\displaystyle}& {\displaystyle}={I}_{\mathrm{ref}}{\displaystyle \frac{{C}_{\mathit{\mu}}^{\mathrm{1}/\mathrm{4}}}{\mathit{\kappa}\sqrt{\frac{\mathrm{2}}{\mathrm{3}}}}}\left[\mathrm{1}-\mathrm{exp}\left(-{\displaystyle \frac{\mathit{\kappa}\sqrt{\frac{\mathrm{2}}{\mathrm{3}}}}{{I}_{\mathrm{ref}}{C}_{\mathit{\mu}}^{\mathrm{1}/\mathrm{4}}}}\right)\right]\\ {\displaystyle}& {\displaystyle}=\mathrm{1.274}{I}_{\mathrm{ref}}+\mathcal{O}\left({I}_{\mathrm{ref}}^{\mathrm{2}}\right)\end{array}$$

Note that the power law is not used in the simulations; however, the relation in Eq. (3) is employed to discuss the simulations in Sect. 4.1.

The in-house solver MIRAS (Method for Interactive Rotor Aerodynamic Simulations) is a multi-fidelity computational vortex model for predicting the aerodynamic behavior of wind turbines and the corresponding wakes. It has been developed at the Technical University of Denmark over the last decade, and it has been extensively validated for small to large wind turbine rotors by Ramos-García et al. (2014a, b, 2017). The turbine aeroelastic behavior is modeled by using the MIRAS-FLEX5 aeroelastic coupling developed by Sessarego et al. (2017). FLEX5 is an aeroelastic tool developed by Øye (1996), which gives loads and deflections.

In the present study, a lifting line technique is employed as the blade aerodynamic model. The blade bound circulation is modeled by a vortex line, located at the blade quarter-chord and subdivided into vortex segments. The vorticity is released into the flow by a row of vortex filaments following the chord direction (shed vorticity, which accounts for the released vorticity due to the time variation of the bound vortex) and a row of filaments perpendicular to the chord direction (trailing vorticity, which accounts for the vorticity released due to circulation gradients along the span-wise direction of the blade).

A hybrid vortex method is used for the wake modeling, where the near wake is modeled with vortex filaments, and further downstream the filaments' circulation is transformed into a vorticity distribution on a uniform Cartesian auxiliary mesh, where the interaction is efficiently calculated using fast Fourier transform-based method developed by Hejlesen (2016). Effects of domain blockage are removed by solving the Poisson equation using a regularized Green's function solution with free-space boundary conditions in all directions except the ground, which is modeled using a slip wall. In order to avoid the periodicity of the Green's function convolution, the free-space boundary conditions are practically obtained by zero-padding the domain, as introduced by Hockney and Eastwood (1988). The ground condition is modeled by solving an extended problem, accounting for the vorticity field mirrored about the ground plane.

The prescribed velocity–vorticity boundary layer model (P2VBL) presented in Ramos-García et al. (2018) is employed to model the wind shear. This model corrects the unphysical upward deflection of the wake observed in simpler prescribed velocity shear approaches.

The Mann model (Mann, 1998) is used to generate a synthetic turbulent velocity field on a uniform mesh, commonly known as a turbulence box. The velocity field is transformed into a vortex-particle cloud, which is gradually released into the computational domain at a plane 2 D upstream of the wind turbine. All components of the Mann model velocity fluctuations are scaled by a factor 1.2 in order to reproduce the measured turbulence intensity at the hub height (as listed in Table 1). The same scaling factor is necessary in LES-AL-FLEX5 simulations, as discussed in Sect. 3.4. It is not fully understood why this scaling factor is necessary in order to reproduce the original inflow turbulence intensity, and this should be investigated further in future work.

The mesh used has an extent of ${L}_{x}\times {L}_{y}\times {L}_{z}=\mathrm{17.1}\phantom{\rule{0.125em}{0ex}}\text{D}\times \mathrm{6.2}\phantom{\rule{0.125em}{0ex}}\text{D}\times \mathrm{6.2}\phantom{\rule{0.125em}{0ex}}\text{D}$, where *L*_{x},
*L*_{y} and *L*_{z} are the stream-wise, lateral and vertical domain lengths,
respectively. A constant spacing of 0.7 m, approximately 20 cells per blade,
is used in all three directions, resulting in a mesh with $\mathrm{714}\times \mathrm{258}\times \mathrm{258}$ cells. This results in a total of about 48 million cells with a
similar number of vortex particles. Due to aeroelastic
constraints, the time step is fixed to 0.01s. A total number of 130 000 time
steps were simulated for all cases. The analysis performed in the following
sections uses the data recorded for the last 120 000 time steps. The
turbulent box used in all computations is much larger than the actual
simulated domain, $\mathrm{1122}\phantom{\rule{0.125em}{0ex}}\text{D}\times \mathrm{9}\phantom{\rule{0.125em}{0ex}}\text{D}\times \mathrm{9}\phantom{\rule{0.125em}{0ex}}\text{D}$, in
order to include large structures in the simulation. Moreover, the
discretization of the box is coarser, with a constant spacing of 2 m, which
is around 3 times larger than the computational cells. In this way, the
smaller turbulent structures are generated by the solver.

The structure of EllipSys3D is similar to that described in Sect. 3.2. For the LES cases the convective terms are discretized via a combination of the third-order QUICK scheme and the fourth-order central difference scheme in order to suppress unphysical numerical wiggles and diffusion. The pressure correction equation is solved using the PISO algorithm.

LES applies a spatial filter on the Navier–Stokes equations, which results in a filtered velocity field. The large scales are solved directly by the Navier–Stokes equations, whereas scales smaller than the filter scale are modeled using a sub-grid-scale (SGS) model, which provides the turbulence closure. The SGS model is a mixed-scale model based on an eddy-viscosity approach as described by Ta Phuoc et al. (1994).

The turbines are modeled using the actuator line (AL) technique as described by Sørensen and Shen (2002), which applies body forces along rotating lines within the numerical domain of the flow solver – here EllipSys3D. The body forces are computed using FLEX5. Therefore, the actuator lines are directly controlled by FLEX5, which means that the actuator lines are both rotating and deflecting within the flow. Additional details of the aeroelastic coupling can be found in Sørensen et al. (2015). The aeroelastic coupling also provides a turbine controller, which is made up of a variable speed P-controller for below rated wind speeds and a PI-pitch angle controller for above rated wind speeds, see Larsen and Hanson (2007) or Hansen et al. (2005) for details on turbine controllers.

The atmospheric boundary layer is modeled by applying body forces throughout the domain, see Mikkelsen et al. (2007). Applying body forces makes it possible to impose any vertical velocity profile, which is beneficial when aiming to model specific measurements, e.g., Hasager et al. (2017).

Turbulence has also been introduced 2 D upstream the turbines using body forces (see e.g., Gilling et al. (2009)), where the imposed turbulence is identical to the turbulence generated using the Mann model as described in Sect. 3.3. All components of the Mann model velocity fluctuations are scaled by a factor 1.2 in order to reproduce the measured turbulence intensity at the wind turbine position, at hub height (as listed in Table 1)

The computational mesh is ${L}_{x}\times {L}_{y}\times {L}_{z}=\mathrm{17.5}\phantom{\rule{0.125em}{0ex}}\text{D}\times \mathrm{7}\phantom{\rule{0.125em}{0ex}}\text{D}\times \mathrm{20}\phantom{\rule{0.125em}{0ex}}\text{D}$ in the stream-wise, lateral and vertical directions, respectively. This yields a blockage ratio of 2 %, which is less that than the 3 % recommended by Baetke et al. (1990). The mesh is equidistant in the streamwise direction and in a region containing the turbine and wake of 2–6 D in the lateral and from the ground up to 4 D in the vertical, which is then stretched towards the sides. This corresponds to each turbine blade being resolved by 36 cells in order to resolve the tip vortices (Troldborg, 2008), and the mesh contains a total of 131 million cells. Inlet and outlet boundary conditions were applied in the streamwise direction, and cyclic boundary conditions were applied in the lateral direction. The top boundary was modeled as a symmetry condition, and the ground was modeled with a no-slip condition.

The simulations were run with time steps of 0.0063 and 0.0069 s for the near- and far-wake case, respectively.

The statistics presented are based on 10 min of data, which were sampled after the initial transients propagated through the domain, similar to the results using MIRAS-FLEX5.

4 Results and discussion

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A comparison of the V29 rotor models from EllipSys3D RANS-AD and FLEX5 (used by EllipSys3D LES-AL-FLEX5 and MIRAS-FLEX5) is made with a HAWC2 model of the V29 provided by Vestas Wind System A/S. The Fuga rotor model is not compared with the other models because the chosen thrust force distribution is uniform and the total thrust force is a model input. Here, the deflections are switched off in FLEX5 and HAWC2 in order to make a fair aerodynamic comparison with EllipSys3D RANS-AD that can only model stiff blades. The near-wake model of Pirrung et al. (2016, 2017) is used in HAWC2, and a uniform inflow is employed without inflow turbulence or the presence of a wall.

The mechanical power and thrust force as function of the undisturbed wind
speed are plotted in Fig. 8 for the three models:
EllipSys3D RANS-AD, FLEX5 and HAWC2. For wind speeds between 5 and
8 m s^{−1}, all three models predict a similar power and thrust
coefficients that differ by approximately 2 %. The thrust
coefficient of EllipSys3D RANS-AD and HAWC2 only differ by around 1 % for
all wind speeds, whereas EllipSys3D RANS-AD overpredicts the power
coefficient by about 1 % below 9 m s^{−1} and by 2 %–6 % for
higher wind speeds. The largest differences between FLEX5 and HAWC2 are
observed around the shoulder of the power curve, which is presumably caused
by differences in control strategies.

The normalized tangential and thrust force distributions for three different
wind speeds (7, 12 and 18 m s^{−1}) are plotted in
Fig. 9 for HAWC2, FLEX5 and EllipSys3D RANS-AD. For a
wind speed of 7 m s^{−1} (below the rated wind speed), all three models predict similar
force distributions. For the higher wind speeds (12 and 18 m s^{−1}),
there are differences between the three models, mainly observed outboard and
towards the blade tip, which could be related to the different tip
corrections that are employed in each model.

The measured and simulated relative difference in power (Δ*C*_{P}) and
thrust force (Δ*C*_{T}) of the 4R-V29 wind turbine due to the rotor
interaction are depicted in Fig. 10. Δ*C*_{P} and Δ*C*_{T} are calculated as follows:

$$\begin{array}{ll}\text{(4)}& {\displaystyle}& {\displaystyle}\mathrm{\Delta}{C}_{P}={\displaystyle \frac{\left({P}_{{R}_{\mathrm{1}}}^{{s}_{\mathrm{2}}}+{P}_{{R}_{\mathrm{3}}}^{{s}_{\mathrm{2}}}\right)-\left({P}_{{R}_{\mathrm{1}}}^{{s}_{\mathrm{1}}}+{P}_{{R}_{\mathrm{3}}}^{{s}_{\mathrm{3}}}\right)}{{P}_{{R}_{\mathrm{1}}}^{{s}_{\mathrm{1}}}+{P}_{{R}_{\mathrm{3}}}^{{s}_{\mathrm{3}}}}}{\displaystyle}& {\displaystyle}\mathrm{\Delta}{C}_{T}={\displaystyle \frac{\left({T}_{{R}_{\mathrm{1}}}^{{s}_{\mathrm{2}}}+{T}_{{R}_{\mathrm{3}}}^{{s}_{\mathrm{2}}}\right)-\left({T}_{{R}_{\mathrm{1}}}^{{s}_{\mathrm{1}}}+{T}_{{R}_{\mathrm{3}}}^{{s}_{\mathrm{3}}}\right)}{{T}_{{R}_{\mathrm{1}}}^{{s}_{\mathrm{1}}}+{T}_{{R}_{\mathrm{3}}}^{{s}_{\mathrm{3}}}}},\end{array}$$

where *s*_{1}, *s*_{2} and *s*_{3} correspond to the three stages of the test cycle
as illustrated in Fig. 2, and *P* and *T* are the power and
thrust force for a rotor *R*_{i}, respectively. The measurements in Fig. 10a show that the rotor interaction
increases the power production of the 4R-V29 wind turbine for the wind speed
bins below the rated wind speed between 7.5 and 11 m s^{−1}. The standard error of the
mean Δ*C*_{P} is too large to make the same statement below
7.5 m s^{−1}. Above the rated wind speed, the effect of the rotor interaction on the
mean power is smaller than below the rated wind speed, and high uncertainties of the mean
power for 11.5 and 13 m s^{−1} are observed. The weighted average of
Δ*C*_{P} (using the number of observations per bin) for a wind speed
between 5 and 11 m s^{−1} is 1.8±0.2 *%*, which supports the
observed bias towards a power gain below the rated wind speed. The rotor interaction of the
4R-V29 wind turbine increases the annual energy production by
1.5±0.2 *%* if we assume a Weibull distribution for the wind speed with shape
and scale parameters of 2 and 7.5 m s^{−1}, respectively (corresponding
to a mean wind speed of about 6.7 m s^{−1}), and we assume a zero power
gain below 5 m s^{−1} and above 11 m s^{−1}. The 0.2 %
uncertainty represents the standard error of the mean and does not represent
measurement uncertainties directly, which could be a lot higher than
0.2 %. However, the analysis is focused on the relative differences
between the test cycles as illustrated in
Fig. 4. In addition, we have removed
uncertainties due to measurement biases as much as possible (e.g., induction
correction), as discussed in Sect. 2.2.

The RANS-AD simulations in Fig. 10 are
performed for three different turbulence intensities (5 %, 10 % and
20 %), and a larger power and thrust force below the rated wind speed is predicted when
all four rotors are in operation for the two lowest turbulence intensities
(5 % and 10 %). The largest gain in power (2 %) is found for the
lowest turbulence intensity, where the shear is also the lowest. For a large
turbulence intensity, the effect of the rotor interaction is almost zero
below the rated wind speed. The loss in power above rated power is not interesting because
it is possible to adapt the pitch angle such that the rated power is reached.
Note that the V29 rotor starts to pitch out between 10 and 11 m s^{−1}.
Figure 10b shows that Δ*C*_{T} from the RANS-AD simulations follow the trends of the Δ*C*_{P}. This
indicates that the axial induction of the 4R-V29 wind turbine is increased
due to the rotor interaction. The measured power gain including the standard
error of the mean is of the same order as the RANS-AD simulations, except for
the wind speed bins of 8.5, 11, 12 and 14 m s^{−1}, where the measured
power gain is underpredicted by the simulations. The lower measured power
gain for wind speeds below 7.5 m s^{−1} compared with wind speeds
between 7.5 and 9.5 m s^{−1} could also be related to the fact that a high
turbulence intensity is more common at low wind speeds, and the RANS-AD
simulations show that the power gain decreases with increasing turbulence
intensity.

Two results of MIRAS-FLEX5 for respective wind speeds of 7 and 10.6 m s^{−1} using
the Mann inflow turbulence of the far-wake case, which has a turbulence intensity
of about 10 %, are also depicted in
Fig. 10. Each result represents the mean of
two consecutive 10 min averages, and the error bar represents the standard
error of the mean. The power gain predicted by MIRAS-FLEX5 for respective wind speeds
of 7 and 10.6 m s^{−1} is 0.3 % higher and 0.1 % lower,
respectively, compared with the results from RANS-AD (for a turbulence
intensity of 10 %); however, the trend regarding wind speed is the same. The
gain in the thrust coefficient from MIRAS-FLEX5 is 0.7 % higher and 0.1 %
lower than RANS-AD for 7 and 10.6 m s^{−1}, respectively. The higher
gains for 7 m s^{−1} from MIRAS-FLEX5 are not caused by a difference in
domain blockage when operating one or four rotors as the effects of domain
blockage are avoided, as discussed in Sect. 3.3.

Δ*C*_{P} and Δ*C*_{T} for a bottom rotor (*R*_{1}) and a top rotor
(*R*_{3}) calculated by the RANS-AD simulations for three different turbulence
intensities are plotted in Fig. 11.
The measurements in Fig. 11 also
depict Δ*C*_{P} for one bottom (*R*_{1}) and one top rotor (*R*_{3}). The
RANS-AD simulations indicate that the difference in Δ*C*_{P} and Δ*C*_{T} within a horizontal pair (*R*_{1} compared to *R*_{2} and *R*_{3} compared to
*R*_{4}) is negligible (results of *R*_{2} and *R*_{4} are not shown in
Fig. 11 to improve readability),
whereas the difference between a vertical pair is clearly visible. The bottom
rotors produce more Δ*C*_{P} and Δ*C*_{T} than the top rotors, and
the difference between the bottom and the top pair increases with turbulence
intensity, which is probably due to associated increased shear. For the largest
turbulence intensity (20 %) and shear (*α*=0.25), only the bottom
rotors produces more power, which could be caused by the difference in thrust
force between the top and bottom rotors. In other words, the high thrust
force of the top rotors creates a blockage effect that pushes more wind
downwards into the rotor plane of the bottom rotors. Two results of
MIRAS-FLEX5, corresponding to respective wind speeds of 7 and 10.6 m s^{−1} and a
turbulence intensity of about 10 %, confirm that the bottom rotors
produce more Δ*C*_{P} and Δ*C*_{T} than the top rotors. In
addition, the difference between MIRAS-FLEX5 and RANS-AD is largest for
the bottom rotor for 7 m s^{−1} in terms of Δ*C*_{T} (1 %), where
MIRAS-FLEX5 also shows the largest standard error of the mean because the
lower rotor experiences a lower inflow wind speed and a higher turbulence
level compared with the top rotor. The measurements also indicate that the bottom
rotor is mainly responsible for the power gain, although the standard error
of the mean of the bottom and top rotor overlap for most of the wind speed
bins. In addition, one could argue that the sloping terrain, as illustrated
in Fig. 5, may have influenced the difference between the top and
the bottom pair, as sloping terrain can lead to a speedup close to the
ground that enhances the wind resource for the lower rotor pair. The terrain
effects could be included and studied in future work.

Results of the near-wake test case are discussed in Sect. 4.3.1, whereas Sect. 4.3.2 presents results of the far-wake test case including the near-wake to far-wake development.

Contours of the stream-wise velocity at three downstream distances, measured by the short-range WindScanner and simulated by four models (LES-AL-FLEX5, MIRAS-FLEX5, RANS-AD and Fuga) are depicted in Fig. 12. The measurements and simulations show four distinct wakes, which are most visible at $x/D=\mathrm{0.5}$. At this distance, the measurements and Fuga show a stronger deficit at the bottom rotors compared with the top rotors, which is also visible in the RANS-AD results with smaller differences between the top and bottom rotors. The mixing in the surface layer linearly increases with height in RANS-AD and Fuga, which increases the mixing of the top rotors compared with the bottom rotors. In the higher fidelity models – LES-AL-FLEX5 and MIRAS-FLEX5 – the inflow turbulence is modeled by Mann turbulence that has a uniform turbulent mixing in the vertical direction. This could explain why LES-AL-FLEX5 and MIRAS-FLEX5 do not show a clear difference in wake deficit between the bottom and top rotors. Note that all models include a sheared inflow, which can also cause a difference in the wake deficit between the top and bottom rotors. At $x/D=\mathrm{2}$, the measurements show much lower velocities compared with all four models.

Profiles of the stream-wise velocity normalized by the inflow at three heights, corresponding to the bottom rotor hub height (29.04 m), the center reference height (44.27 m) and the top rotor hub height (59.5 m) are plotted in Fig. 13. Results of the WindScanner and the four models are shown, taken at three downstream distances. It is clear that measured velocity inside and outside of the wake, at the bottom rotor hub height and at the center height are lower than predicted by all four models. This suggests that the actual shear and reference wind speed at the 4R-V29 wind turbine could have been different to values measured at the reference met mast. Unfortunately, it is not possible to determine the free-stream conditions from the WindScanner data because of the limited horizontal extent of the scanned planes. In addition, the atmospheric conditions of the near-wake case measured at the reference met mast was near-neutral (see Table 1), which could have increased the measured wake deficit.

The measurements and all of models, except Fuga, show the buildup of a traditional double bell-shaped near-wake profile at the center height in the downstream direction, as depicted in Fig. 13. Fuga is based on a linearized RANS approach, which means that it is designed to describe the far wake properly; however, it cannot predict the nonlinear near wake accurately, especially for a high thrust coefficient, as shown by Ebenhoch et al. (2017). Nevertheless, the other models yield very similar results.

Profiles of the turbulence intensity *I* ($I=\sqrt{\mathrm{2}/\mathrm{3}k}/{U}_{\mathrm{ref}}$) are
plotted in Fig. 14 using the same definition as in
Fig. 13. Only the results of LES-AL-FLEX5, MIRAS-FLEX5 and
RANS-AD are shown, as the WindScanner cannot measure *I*, and Fuga cannot
model *I* in the wake because it uses a turbulence closure that is unaffected
by the wake. Figure 14 shows that RANS-AD has
smaller peaks in *I* to LES-AL-FLEX; this is due to the fact that an AD model
simulates a ring root and tip vortex, whereas an AL model resolves a (smeared)
root and tip vortex per blade.

The results of the far-wake case are plotted in Figs. 15,
16 and 17, which follow the same
definition as in Figs. 12, 13 and
14, respectively. In addition, six downstream distances
are depicted to show the full downstream development of the 4R-V29 wind
turbine wake. Only measurements of the stream-wise velocity at $x/D=\mathrm{5.5}$ are
available. The four individual wakes merge into a single structure between
$x/D=\mathrm{2}$ and $x/D=\mathrm{3}$ as shown in Figs. 15
and 16. The middle column of Fig. 16
depicts how a bell-shaped near-wake structure forms at the center height up
to and including $x/D=\mathrm{3}$, whereas the single wakes at the bottom and top hub
heights cannot be distinguished from each other at this distance.
Further downstream, at $x/D=\mathrm{5.5}$, the fifth row of plots in
Fig. 15 shows that all models capture the measured
single-wake structure at $x/D=\mathrm{5.5}$, although the wake of Fuga has moved downwards
compared with the measurements and other models. The magnitude of the wake
deficit at $x/D=\mathrm{5.5}$ is underpredicted by all models, as seen in
Fig. 16, where the measured wake at the bottom hub height
is also skewed. The measured wake skewness could be a terrain effect or a
results of the 8.2^{∘} yaw misalignment, as discussed in
Sect. 2.3. In addition, the close proximity of the
highway and surrounding trees, as discussed in
Sect. 2.3, could have influenced the measurements of
the far wake. Furthermore, we would like to point out that it is challenging
to compare the models with a single 8 min averaged result from the
WindScanner.

The inflow Mann turbulence that is used in LES-AL-FLEX5 and MIRAS-FLEX5
results in a turbulent kinetic energy profile that has a higher value near
the ground and a lower value above the center height compared with the
reference turbulent kinetic energy at the center height. The turbulent
kinetic energy profile in the RANS-AD simulations is constant with height.
Hence, the comparison of the RANS-AD simulations with the LES-AL-FLEX5 and
MIRAS-FLEX5 simulations in terms of turbulence intensity
(Fig. 17) at *z*=29.04 m and *z*=59.5 m is not entirely
fair. At the center height (*z*=29.04 m), where the ambient turbulence
intensity levels between the models are similar, the turbulence intensity in
the far wake is higher in the RANS-AD simulations compared with LES-AL-FLEX5
(about 0.02 at $x/D=\mathrm{12}$, $y/D=\mathrm{0}$), which was also previously observed by
van der Laan et al. (2015) for single AD simulations. The largest difference in
turbulence intensity between the LES-AL-FLEX5 and MIRAS-FLEX5 simulations are
found in the near wake for the lowest rotor pair (*z*=29.04 m).

The presented near- and far-wake cases show that the models follow the measured trends, but there are not enough measured data to validate the simulations. More wake measurements of the 4R-V29 wind turbine are required in order to perform a model validation.

The wake recovery of a multi-rotor wind turbine is very important for placing
several multi-rotors together in wind farms. Therefore, the aim here is to
quantify the wake recovery of a multi-rotor wind turbine operating in an
atmospheric surface layer with respect to an equivalent single-rotor wind
turbine that has the same rotor area, force distributions, tip speed ratio
(TSR) and total thrust force. In order to do so, a simplification of the 4R-V29 wind
turbine is used so that a fair comparison with a equivalent single-rotor
wind turbine can be made. The simplified 4R-V29 wind turbine has a zero
toe-out angle, and the force distributions are defined by prescribed
normalized blade force distributions (calculated by Réthoré et al., 2014,
employing a detached eddy simulation of the NREL-5MW rotor for a wind speed
of 8 m s^{−1}). The blade force distributions are scaled by the hub
height velocity *U*_{H}, *R*, *C*_{T}, *C*_{P} and the rotational speed
(RPM) as discussed by van der Laan et al. (2015). The resulting AD force distributions
are uniform over the azimuth, and the effect of shear on the AD force
distributions are neglected. The dimensions and scaling parameters of the
simplified 4R-V29 wind turbine and an equivalent single-rotor wind turbine
referred as V58, are summarized in Table 2. The
inflow is an atmospheric surface layer, with *U*_{ref}=7 m s^{−1}
and three different *I*_{ref} at *z*_{ref}=44.27 m: 5 %, 10 % and 20 %.
The hub height wind speed for the bottom and top
rotor pairs is different for the simplified 4R-V29 wind turbine due to the
shear. In order to model the same total thrust force for the V58 wind
turbine, the thrust coefficient of the V58 is adjusted. The rotational speed
is set to assure a TSR of 7.6 for all rotors.

Figure 18 depicts the wake recovery in terms of
stream-wise velocity and added turbulence intensity of the simplified 4R-V29
multi-rotor wind turbine and the equivalent V58 single-rotor wind turbine as
a function of the stream-wise distance *x* normalized by the single-rotor
diameter (*D*_{eq}=58.4 m) for three turbulence intensities (5 %,
10 % and 20 %). The wake recovery is calculated as rotor-integrated
values normalized by the same integral without an AD. Note that four
integrals are calculated for the multi-rotor and summed up for each
downstream distance. Figure 18a, c and e show that the wake recovery distance
in terms of stream-wise velocity of a simplified 4R-V29 multi-rotor wind
turbine is about 1.03–1.44 *D*_{eq} shorter than the wake recovery
distance of a V58 single-rotor wind turbine, which is a remarkable
difference. The largest difference is found for the lowest ambient turbulence
intensities (5 %). This suggests that the horizontal area of a wind farm
consisting of 4R-V29 wind turbines positioned in a regular rectangular layout
can be reduced compared with a wind farm consisting of V58 wind turbines. The
area could be reduced by $\mathrm{1}-(\mathrm{1}-\mathrm{1.44}/s{)}^{\mathrm{2}}$ and $\mathrm{1}-(\mathrm{1}-\mathrm{1.03}/s{)}^{\mathrm{2}}$ (for
*I*_{ref}=5 % and *I*_{ref}=20 %, respectively), with
*s* as the horizontal and vertical inter-turbine spacing in *D*_{eq}.
For example, for *s*=8 *D*_{eq} the RANS predicted reduction in wind
farm area would be 24 %–32 %; this significant reduction in the area
required could also reduce cost and potentially increase the power production
by increasing the number of installed turbines in a given area. This result
is a rough extrapolation that should be verified by wind farm simulations of
multi-rotor wind turbines.

Figure 18b, d and fshow that the added wake turbulence is larger for the
multi-rotor wind turbine in the near wake for *I*_{ref}=5 % and
*I*_{ref}=10 % for $x/{D}_{\mathrm{eq}}<\mathrm{3}$ and $x/{D}_{\mathrm{eq}}<\mathrm{2}$,
respectively, but is smaller in the far wake with respect to the added wake
turbulence of single-rotor wind turbine. It is not possible to shift the
added wake turbulence of the multi-rotor wind turbine downstream to match the
added wake turbulence of the single-rotor wind turbine in the same manner as
the wake recovery. The lower wake turbulence in the far wake has the
potential to reduce blade fatigue loads that are caused by wake turbulence.

The increased wake recovery of a multi-rotor wind turbine could be related to the fact that the total thrust force is more distributed compared with a single-rotor wind turbine. Ghaisas et al. (2018) also obtained a faster wake recovery for a multi-rotor wind turbine, and argued that it is caused by a larger entrainment because the ratio of the rotor perimeter and the rotor swept area is twice as high for the multi-rotor wind turbine with four rotors.

5 Conclusions

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Numerical simulations and field measurements of the Vestas multi-rotor wind turbine (4R-V29) have been performed. The simulations show an increased thrust force and axial induction of the 4R-V29 wind turbine compared with a single rotor. In addition, the simulations calculate a 0 %–2 % enhancement of the power performance of the 4R-V29 multi-rotor wind turbine below the rated wind speed due to the interaction of the rotors. The largest gain in power is obtained for a low turbulence intensity that is associated with a low shear. The relative power gain is largest for the bottom rotor pair. Power curve measurements of the 4R-V29 wind turbine also show that rotor interaction increases the power performance below the rated wind speed by 1.8 %, which can result in a 1.5 % increase in the annual energy production.

Two flow cases based on short-range WindScanner wake measurements of the 4R-V29 wind turbine are used to compare the multi-rotor wake deficit simulated by four numerical models. In the near wake, four distinct wake deficits are visible that merge into a single structure at a downstream distance of 2–3 D. More wake measurements are required to validate the numerical models.

The wake recovery of a simplified 4R-V29 wind turbine is quantified by
comparison with the wake recovery of an equivalent single-rotor V58 wind
turbine. RANS simulations show that the wake recovery distance in terms of
the stream-wise velocity of the simplified 4R-V29 wind turbine is
1.03–1.44 *D*_{eq} shorter than a the wake recovery distance of the
equivalent single-rotor wind turbine with a rotor diameter *D*_{eq}.
In addition, it is found that the added wake turbulence of the simplified
4R-V29 wind turbine is smaller than the equivalent single-rotor V58 wind
turbine in the far wake. The fast wake recovery of a multi-rotor wind turbine
could potentially lead to closer spaced wind turbines in multi-rotor wind
farms and needs to be further investigated.

Code and data availability

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Code and data availability.

The numerical results are generated using proprietary software, although the data presented can be made available upon request from the corresponding author.

Appendix A: Induction correction for the measured reference wind speed for the power curve measurements of the 4R-V29 wind turbine

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The measured effect of rotor interaction on the power production is
quantified using the test cycle in Fig. 2, where the
combined power curves of two single-rotor operation stages (stages 1 and 3)
are compared with the power curve of a stage where all four rotors are in
operation (Stage 2). The reference wind speed in these power curve
measurements is taken at 5 D (146 m) upstream, as discussed in
Sect. 2.2. As the induction zone in stages 1
and 3 is smaller than in Stage 2, a lower reference wind speed is measured when
all four rotors are in operation. Hence, the power curve of Stage 2 will be
shifted towards the left, and an artificial bias towards a power gain due to
the rotor interaction would be measured. To avoid this, the reference wind
speed is corrected by a factor *f*_{cor} when all four rotors are in
operation (Stage 2):

$$\begin{array}{}\text{(A1)}& {\displaystyle}{f}_{\mathrm{cor}}={\displaystyle \frac{\frac{\mathrm{1}}{\mathrm{2}}\left({U}_{\mathrm{ref}}^{\mathrm{Stage},\mathrm{1}}+{U}_{\mathrm{ref}}^{\mathrm{Stage},\mathrm{3}}\right)}{{U}_{\mathrm{ref}}^{\mathrm{Stage},\mathrm{2}}}}\end{array}$$

for each undisturbed wind speed with an interval of 1 m s^{−1}. The
induction correction factor can only be calculated if the undisturbed wind
speed is known. Therefore, the RANS simulations in Sect. 3.2.2
are used to calculate *f*_{cor}, and the results are shown in
Fig. A1 for a reference turbulence intensity of 10 %.
*f*_{cor} follows the thrust coefficient curve, and below the rated
wind speed, where the thrust coefficient is the highest, the measured
reference wind speed for Stage 2 is 0.7 % lower than the reference wind
speed in stages 1 and 3.

*f*_{cor} is also calculated using a simple induction model from
Troldborg and Meyer Forsting (2017), which has been developed to model the induction of a
single rotor in a uniform inflow. The simple induction model is only a
function of the thrust coefficient, rotor radius and spatial coordinates. The
thrust coefficient of the RANS simulations is used as input. The induction
zone for Stage 2 is calculated by superposition of the induction of the four
individual rotors. Figure A1 shows that the induction of the
4R-V29 wind turbine at $x=-\mathrm{5}$ D is underestimated by the simple induction
model compared with the RANS simulations and should not be used to correct of
the reference wind speed in Stage 2. We chose to use the RANS results to
correct the reference wind speed, as Meyer Forsting et al. (2017) have shown
that RANS-AD simulations compare well with lidar measurements of the
induction zone when measurement uncertainty is included in the validation
method.

The influence of the ambient turbulence intensity at a height of 44.27 m on
*f*_{cor} in the RANS simulations is also investigated for three
different turbulence intensities (5 %, 10 % and 20 %). The
results are same for a turbulence intensity of 5 % and 10 %, whereas
the *f*_{cor} is slightly higher for a turbulence intensity of
20 % (*f*_{cor}=1.0073 below the rated wind speed). As the power curve
measurements are filtered for a wind direction from the fjord, we expect that
the ambient turbulence intensity is lower than 20 % and that a
*f*_{cor} based on a turbulence intensity of 10 % is justified.

Appendix B: Nomenclature

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D |
Rotor diameter of each single rotor of the 4R-V29 |

wind turbine. | |

D_{eq} |
Rotor diameter of an equivalent single rotor wind |

turbine (D_{eq}=2 D). |

Author contributions

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Author contributions.

MPVDL performed the EllipSys3D RANS-AD and Fuga simulations, produced all figures and drafted the article. SO, MPVDL and MK investigated the meteorology of the wake measurements. SJA and NRG performed the EllipSys3D LES-AD-FLEX5 and the MIRAS-FLEX5 simulations, respectively. GRP contributed to the validation of the numerical single-rotor models. NA, MS and TKM planned, executed and post-processed the WindScanner wake measurements. KHS and JXVN executed and post-processed the power curve measurements. GCL planned and managed the research related to this article. All authors jointly finalized the paper.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

This is work is sponsored by Vestas Wind System A/S.

Review statement

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Review statement.

This paper was edited by Johan Meyers and reviewed by Peter Jamieson, Dominic von Terzi and one anonymous referee.

References

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Baetke, F., Werner, H., and Wengle, H.: Numerical simulation of turbulent flow over surface-mounted obstacles with sharp edges and corners, J. Wind Eng. Ind. Aerod., 35, 129–147, https://doi.org/10.1016/0167-6105(90)90213-V, 1990. a

Chasapogiannis, P., Prospathopoulos, J. M., Voutsinas, S. G., and Chaviaropoulos, T. K.: Analysis of the aerodynamic performance of the multi-rotor concept, J. Phys. Conf. Ser., 524, 1–11, https://doi.org/10.1088/1742-6596/524/1/012084, 2014. a, b, c, d

Ebenhoch, R., Muro, B., Dahlberg, J.-Å., Berkesten Hägglund, P., and Segalini, A.: A linearized numerical model of wind-farm flows, Wind Energy, 20, 859–875, https://doi.org/10.1002/we.2067, 2017. a

Ghaisas, N. S., Ghate, A. S. ., and Lele, S. K.: Large-eddy simulation study of multi-rotor wind turbines, J. Phys. Conf. Ser., 1037, 1–10, https://doi.org/10.1088/1742-6596/1037/7/072021, 2018. a, b, c

Gilling, L., Sørensen, N., and Rethore, P.: Imposing Resolved Turbulence by an Actuator in a Detached Eddy Simulation of an Airfoil, in: EWEC 2009 Proceedings online, European Wind Energy Association (EWEA), Marseille, France, https://doi.org/10.1002/we.449, 2009. a

Hansen, M. H., Hansen, A., Larsen, T. J., Oye, S., Sorensen, P., and Fuglsang, P.: Control design for a pitch-regulated, variable speed wind turbine, Denmark, Forskningscenter Risœ, Risœ-R, Danmarks Tekniske Universitet, Risø Nationallaboratoriet for Bæredygtig Energi, 2005. a

Hasager, C., Nygaard, N., Volker, P., Karagali, I., Andersen, S., and Badger, J.: Wind Farm Wake: The 2016 Horns Rev Photo Case, Energies, 10, https://doi.org/10.3390/en10030317, 2017. a

Hau, E.: Wind Turbines: Fundamentals, Technologies, Application, Economics, 3rd edn., Springer, 2013. a

Hejlesen, M. M.: A high order regularisation method for solving the Poisson equation and selected applications using vortex methods, PhD thesis, Technical University of Denmark, 2016. a

Hockney, R. W. and Eastwood, J. W.: Computer Simulation Using Particles, Institute of Physics Publishing, Bristol, PA, USA, 2nd edn., 1988. a

Holst, H.: Opfindelsernes Bog, Nordisk Forlag, 1923. a

Honnef, H.: Windkraftwerke, Vieweg, 1932. a

IEC 61400-12-1 Ed. 1: Wind turbines – Part 12-1: Power performance measurements of electricity producing wind turbines, International Electrotechnical Commission, 2005. a

Jamieson, P.: Innovation in Wind Turbine Design, Wiley, 2011. a

Jamieson, P., Chaviaropoulos, T., Voutsinas, S., Branney, M., Sieros, G., and Chasapogiannis, P.: The structural design and preliminary aerodynamic evaluation of a multi-rotor system as a solution for offshore systems of 20 MW or more unit capacity, in: PO.ID 203 EWEC & Excibition Barcelona, EWEA, 2014. a, b, c, d

Jensen, P. H., Chaviaropoulos, T., and Natarajan, A.: LCOE reduction for the next generation offshore wind turbines, Tech. rep., INNWIND.EU, 2017. a, b

Larsen, T. and Hanson, T.: A method to avoid negative damped low frequent tower vibrations for a floating, pitch controlled wind turbine, J. Phys. Conf. Ser., 75, 012073, https://doi.org/10.1088/1742-6596/75/1/012073, 2007. a

Leonard, B. P.: A stable and accurate convective modelling procedure based on quadratic upstream interpolation, Computer Methods in Applied Mechanics and Engineering, 19, 59–98, 1979. a

Mann, J.: The spatial structure of neutral atmospheric surface-layer turbulence, J. Fluid Mech., 273, 141–168, 1994. a

Mann, J.: Wind Field Simulation, Probab. Eng. Mech., 13, 269–282, 1998. a

Medley, J., Barker, W., Harris, M., Pitter, M., Slinger, C., Mikkelsen, T., and Sjöholm, M.: Evaluation of wind flow with a nacelle-mounted, continuous wave wind lidar, Proceedings of Ewea 2014, Barcelona, Spain, 2014. a

Meyer Forsting, A. R., Troldborg, N., Murcia Leon, J. P., Sathe, A., Angelou, N., and Vignaroli, A.: Validation of a CFD model with a synchronized triple-lidar system in the wind turbine induction zone, Wind Energy, 20, 1481–1498, https://doi.org/10.1002/we.2103, 2017. a

Michelsen, J. A.: Basis3D – a platform for development of multiblock PDE solvers, Tech. Rep. AFM 92-05, Technical University of Denmark, Lyngby, Denmark, 1992. a

Mikkelsen, R., Sørensen, J., and Troldborg, N.: Prescribed wind shear modelling with the actuator line technique, in: EWEC: European Wind Energy Conf., Milan, Italy, 2007. a

Mikkelsen, T. K., Sjöholm, M., Angelou, N., and Mann, J.: 3D WindScanner lidar measurements of wind and turbulence around wind turbines, buildings and bridges, IOP Conference Series: Materials Science and Engineering, 276, 012004, https://doi.org/10.1088/1757-899X/276/1/012004, 2017. a

Nishino, T. and Draper, S.: Local blockage effect for wind turbines, J. Phys. Conf. Ser., 625, 012010, https://doi.org/10.1088/1742-6596/625/1/012010, 2015. a, b

Ott, S., Berg, J., and Nielsen, M.: Linearised CFD Models for Wakes, Tech. Rep. Risø-R-1772, Risø, 2011. a, b

Øye, S.: Flex4 simulation of wind turbine dynamics, Danmarks Tekniske Universitet, Lyngby, Denmark, 71–76, 1996. a

Patankar, S. V. and Spalding, D. B.: A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows, Int. J. Heat Mass Tran., 15, 1787–1806, 1972. a

Pirrung, G., Riziotis, V., Madsen, H., Hansen, M., and Kim, T.: Comparison of a coupled near- and far-wake model with a free-wake vortex code, Wind Energ. Sci., 2, 15–33, https://doi.org/10.5194/wes-2-15-2017, 2017. a

Pirrung, G. R. and van der Laan, M. P.: A simple improvement of a tip loss model for actuator disc and actuator line simulations, Wind Energ. Sci. Discuss., https://doi.org/10.5194/wes-2018-59, 2018. a, b, c

Pirrung, G. R., Madsen, H. A., Kim, T., and Heinz, J.: A coupled near and far wake model for wind turbine aerodynamics, Wind Energy, 19, 2053–2069, https://doi.org/10.1002/we.1969, 2016. a

Ramos-García, N., Shen, W. Z., and Sørensen, J. N.: Validation of a three-dimensional viscous-inviscid interactive solver for wind turbine rotors, Renew. Energ., 70, 78–92, https://doi.org/10.1016/j.renene.2014.04.001, 2014a. a

Ramos-García, N., Shen, W. Z., and Sørensen, J. N.: Three-dimensional viscous-inviscid coupling method for wind turbine computations, Wind Energy, 19, 67–93, https://doi.org/10.1002/we.1821, 2014b. a

Ramos-García, N., M. Mølholm, J. N. S., and Walther, J. H.: Hybrid vortex simulations of wind turbines using a three-dimensional viscous-inviscid panel method, Wind Energy, 20, 1187–1889, 2017. a

Ramos-García, N., Juul, H. S., Sørensen, J. N., and Walther, J. H.: Vortex simulations of wind turbines operating in atmospheric conditions using a prescribed velocity-vorticity boundary layer model, Wind Energy, 21, 1216-1231, https://doi.org/10.1002/we.2225, 2018. a

Réthoré, P.-E., van der Laan, M. P., Troldborg, N., Zahle, F., and Sørensen, N. N.: Verification and validation of an actuator disc model, Wind Energy, 17, 919–937, 2014. a, b

Sessarego, M., Ramos-García, N., Sørensen, J., and Shen, W.: Development of an aeroelastic code based on three-dimensional viscous-inviscid method for wind turbine computations, Wind Energy, 20, 1145–1170, https://doi.org/10.1002/we.2085, 2017. a

Shen, W. Z., Mikkelsen, R., Sørensen, J. N., and Bak, C.: Tip loss corrections for wind turbine computations, Wind Energy, 8, 457–475, 2005. a

Sørensen, J. N. and Shen, W. Z.: Numerical modelling of Wind Turbine Wakes, J. Fluid. Eng., 124, 393–399, https://doi.org/10.1115/1.1471361, 2002. a

Sørensen, J. N., Mikkelsen, R., Henningson, D. S., Ivanell, S., Sarmast, S., and Andersen, S. J.: Simulation of wind turbine wakes using the actuator line technique, Philos. T. R. Soc. A, 373, 20140071, https://doi.org/10.1098/rsta.2014.0071, 2015. a

Sørensen, N. N.: General purpose flow solver applied to flow over hills, PhD thesis, Risø National Laboratory, Roskilde, Denmark, 1994. a

Ta Phuoc, L., Lardat, R., Coutanceau, M., and Pineau, G.: Rechereche et analyse de modeles de turbulence de sous maille adaptes aux ecoulements instationnaires decolles, Technical Report LIMSI Report 93074, LIMSI, France, 1994. a

Troldborg, N.: Actuator line modeling of wind turbine wakes, PhD thesis, DTU Mechanical Engineering, Technical University of Denmark, Lyngby, Denmark, 2008.

Troldborg, N. and Meyer Forsting, A. R.: A simple model of the wind turbine induction zone derived from numerical simulations, Wind Energy, 20, 2011–2020, https://doi.org/10.1002/we.2137, 2017. a

van der Laan, M. P., Sørensen, N. N., Réthoré, P.-E., Mann, J.,
Kelly, M. C., Troldborg, N., Schepers, J. G., and Machefaux, E.: An improved
*k*-*ε* model applied to a wind turbine wake in atmospheric
turbulence, Wind Energy, 18, 889–907, https://doi.org/10.1002/we.1736, 2015. a, b, c, d, e

Yazicioglu, H., Angelou, N., Mikkelsen, T. K., and Trujillo, J.-J.: Characterization of wind velocities in the wake of a full scale wind turbine using three ground-based synchronized WindScanners, J. Phys. Conf. Ser., 753, 032032, https://doi.org/10.1088/1742-6596/753/3/032032, 2016. a

Short summary

Over the past few decades, single-rotor wind turbines have increased in size with the blades being extended toward lengths of 100 m. An alternative upscaling of turbines can be achieved by using multi-rotor wind turbines. In this article, measurements and numerical simulations of a utility-scale four-rotor wind turbine show that rotor interaction leads to increased energy production and faster wake recovery; these findings may allow for the design of wind farms with improved energy production.

Over the past few decades, single-rotor wind turbines have increased in size with the blades...

Wind Energy Science

The interactive open-access journal of the European Academy of Wind Energy