1. business and industrial
2. paper industry

Numerical computations of wind turbine wakes
and wake interaction
by
Karl Nilsson
May 2015
Technical Reports from
Royal Institute of Technology
Department of Mechanics
SE-100 44 Stockholm, Sweden
Akademisk avhandling som med tillst˚
and av Kungliga Tekniska H¨ogskolan i
Stockholm framl¨
agges till offentlig granskning f¨or avl¨aggande av teknologie
doktorexamen torsdagen den 4 juni 2015 kl 10.15 i sal F3, Kungliga Tekniska
H¨
ogskolan, Lindstedtsv¨
agen 26, Stockholm.
©Karl Nilsson 2015
Universitetsservice US-AB, Stockholm 2015
Numeriska ber¨
akningar av vakar bakom vindkraft och vakinteraktion
Karl Nilsson
Linn´e Flow Centre, KTH Mekanik, SE-100 44 Stockholm, Sverige
Sammanfattning
N¨
ar vindkraftverk placeras i parker reduceras den totala produktionen p˚
a grund
av vakeffekter. I parken ¨
ar turbulensniv˚
aer generellt h¨oga vilket leder till ¨okade
laster p˚
a verk inne i parken. N¨
ar man planerar f¨or nya vindkraftparker ¨ar
det d¨
arf¨
or n¨
odv¨
andigt att ha kunskap om f¨
oruts¨attningarna hos fl¨odet f¨or att
kunna uppskatta produktionsf¨
orluster samt f¨or att kunna optimera valet av
vindkraftverk. Numeriska ber¨
akningar och en ¨okande datorkapacitet g¨or det
m¨
ojligt att studera fl¨
odet bakom vindkraftverk i detalj vilket har varit huvudfokus i detta doktorandarbete.
Den numeriska modellen anv¨
ands f¨
or att uppskatta produktionen i den
befintliga vindkraftsparken Lillgrund. Simuleringsresultatet j¨amf¨ors med data
fr˚
an parken vilka ¨
overensst¨
ammer v¨
al. Dock, s˚
a identifieras n˚
agra os¨akerheter
i resultatet p˚
a grund av den rotormodell som anv¨ants. En av os¨akerheterna ¨ar
att en generisk rotor ¨
ar anv¨
and i Lillgrundfallet. Felet som beg˚
as genom denna
generalisering analyseras genom att simulera tre olika rotorkonfigurationer.
Generellt s˚
a observeras att valet av konfiguration inte ¨ar kritiskt om syftet
a¨r att studera medelv¨
arden av vakparametrar. Ytterligare en os¨akerhet introduceras i Lillgrundfallet genom att vindkraftverken simuleras utan en styrrutin
f¨
or att optimera effektuttaget. En styrrutin har d¨arf¨or anpassats och implementerats i den numeriska modellen och anv¨ants i simuleringar. Generellt,
s˚
a medf¨
or styrrutinen att hastigheter i vakfl¨odet ¨okar, turbulensniv˚
an och de
axiella krafter som verkar p˚
a vindkraftverket minskar. Effektproduktionen observeras d¨
aremot p˚
averkas mindre av ifall en styrrutin anv¨ands eller ej.
Effekterna av den f¨
orenklade teknik f¨
or att modellera det atmosf¨ariska
gr¨
ansskiktet analyseras i syfte att verifiera teknikens f¨orm˚
aga att anv¨andas i
mycket l˚
anga dom¨
aner. Desto l¨
angre ner i dom¨anen som analyseras, desto
mindre beror turbulensen i fl¨
odet av den initiala turbulensen och mer av vindprofilen som anv¨
ands.
En f¨
orfinad variant av den numeriska modellen valideras genom att anv¨anda
m¨
atningar av fl¨
odet bakom en experimentell rotor. J¨amf¨orelsen baseras p˚
a position, storlek och styrka hos spetsvirvlarna bakom verket samt hastigheter
i vakfl¨
odet. Generellt, s˚
a ¨
overensst¨
ammer m¨atningar och simuleringar bra
f¨
orutom hos storleken p˚
a spetsvirvlarna vilken ¨ar kraftigt ¨overskattad i simuleringarna.
Keywords: Vindkraftverk, Vindvakar, Vakinteraktion, Actuator disc-metoden,
Actuator line-metoden, CFD, LES, EllipSys3D
iii
Numerical computations of wind turbine wakes and wake
interaction
Karl Nilsson
Linn´e Flow Centre, KTH Mechanics, SE-100 44 Stockholm, Sweden
Abstract
When wind turbines are placed in farms, wake effects reduce the overall power
production. Also, turbine loads are significantly increased since turbulence
levels are high within the wake flow. Therefore, when planning for a wind farm,
it is imperative to have an understanding of the flow conditions in the farm in
order to estimate the power losses and to optimize the durability of the turbines
to be selected for the farm. The possibilities granted by numerical modeling
and the development of computational capabilities give an opportunity to study
these flow conditions in detail, which has been the key focus of this doctoral
work.
The actuator disc method is used for predicting the power production of
the Lillgrund wind farm. The results of the simulations are compared to measurements from the actual wind farm, which are found to be in very good
agreement. However, some uncertainties are identified in the modeling of the
turbine. One of the uncertainties is that a generic rotor is used in the Lillgrund
case. In order to quantify the errors resulting from this generalization three
different rotor configurations are simulated in various flow conditions. Generally, it can be stated that the choice of rotor configuration is not crucial if
the intention of the simulations is to compute the mean wake characteristics
subject to turbulent inflow. Another uncertainty is that the turbines in the
Lillgrund case were simulated without a power controller. Therefore, a power
controller is implemented and used in simulations. Generally, the controller
reduces the thrust of the turbines, reduces turbulence intensity and increases
velocity levels in the wake flow. However, the use of a controller was observed
to have a small impact on the power production.
The effects of using the technique of imposing pregenerated turbulence and
a prescribed boundary layer in the simulation are analyzed in order to verify
its applicability in very long domains. It is observed that close to the plane
of imposed turbulence, the conditions are mainly dependent on the imposed
turbulence while far downstream the turbulence, regardless of its initial characteristics, is in near equilibrium with the prescribed wind shear.
The actuator line method is validated using measurements of the near wake
behind the MEXICO rotor. The analysis is performed by comparing position,
size and circulation of the tip vortices, as well as velocity distributions in the
wake flow. The simulations and measurements are generally found to be in
good agreement apart from the tip vortex size, which is greatly overestimated
in the simulations.
Keywords: Wind turbine, Wakes, Wake interaction, Actuator disc method,
Actuator line method, CFD, LES, EllipSys3D
iv
Preface
In the first part of this thesis a brief introduction, the methodologies, a summary of the results and the overall conclusions of the project are presented.
The second part contains six papers which are adjusted to comply with the
present thesis format for consistency.
List of appended papers
Paper 1. K. Nilsson, S. Ivanell, K.S. Hansen, R. Mikkelsen, J.N.
Sørensen, S.-P. Breton & D. Henningson 2015 Large-eddy simulations
of the Lillgrund wind farm. Wind Energy 18 449-467
Paper 2. K. Nilsson, S.-P. Breton, J.N. Sørensen & S. Ivanell 2014
Airfoil data sensitivity analysis for actuator disc simulations used in wind turbine applications. Journal of Physics: Conference series 524 012135
Paper 3. K. Nilsson, S.-P. Breton & S. Ivanell 2015 Evaluation of
the effects of using a power controller in LES/ACD simulations. Technical report
Paper 4. S.-P. Breton, K. Nilsson, H. Olivares-Espinosa, C. Masson, L. Dufresne & S. Ivanell 2014 Study of the influence of imposed
turbulence on the asymptotic wake deficit in a very long line of wind turbines.
Renewable Energy 70 153-163
Paper 5. K. Nilsson, O. Eriksson, N. Svensson, S.-P. Breton & S.
Ivanell 2015 Large-eddy simulations of the evolution of imposed turbulence
in prescribed boundary layers in a very long domain. To be submitted to Renewable Energy
Paper 6. K. Nilsson, W.Z. Shen, J.N. Sørensen, S.-P. Breton &
S. Ivanell 2015 Validation of the actuator line method using near wake measurements of the MEXICO rotor. Wind Energy 18 499-514
v
Other papers
The following papers are not included in this thesis due to an overlap in content
or that their content goes beyond the scope of this thesis.
Paper 1. S.-P. Breton, K. Nilsson, S. Ivanell, H. Olivares-Espinosa,
C. Masson & L. Dufresne 2014 Comparative CFD study of the effect of
the presence of downstream turbines on upstream ones using a rotational speed
control system. Journal of Physics: Conference series 555 012014
Paper 2. O. Eriksson, K. Nilsson, S.-P. Breton & S. Ivanell 2014
Analysis of long distance wakes behind a row of turbines - a parameter study.
Journal of Physics: Conference series 524 012152
´n, K. Nilsson, S.-P. Breton & S.
Paper 3. S. Martinen, I. Carle
Ivanell 2014 Analysis of the effect of curtailment on power and fatigue loads
of two aligned wind turbines using an actuator disc approach. Journal of
Physics: Conference series 524 012182
vi
Division of work among authors
The main advisor for the project is Prof. Dan S. Henningson (DH) and the
co-advisor is Associate Prof. Stefan Ivanell (SI).
Paper 1
This work consists of large-eddy simulations of the Lillgrund wind farm using
the actuator disc method. The principal aim of this work is to validate the
method by comparing simulated power with measurements from the actual
farm. The simulations and analysis of the simulation results were performed
by Karl Nilsson (KN). The measurement data was extracted and processed by
Kurt S. Hansen (KH). The manuscript was written by KN with input from
SI, Simon-Philippe Breton (SPB), Robert Mikkelsen (RM), Jens N. Sørensen
(JN), DH and KH.
Paper 2
This work investigates the sensitivity of the rotor configuration used in actuator
disc simulations. The simulations and the analysis of the results were performed
by KN. The manuscript was written by KN with input from SPB, JN and SI.
Paper 3
This work analyzes the effects of using a power controller in actuator disc
simulations. The simulations and the analysis of the results were performed by
KN. The manuscript was written by KN with input from SPB and SI.
Paper 4
This work investigates the wake development in a very long line of turbines
simulated using the actuator disc method. The aim is to determine if and
when an asymptotic wake state is reached. The simulations, the analysis of the
results and the writing of the manuscript were performed jointly by KN and
SPB with input from SI, Hugo Olivares-Espinosa (HOE), Christian Masson
(CM) and Louis Dufresne (LD).
Paper 5
This work analyzes the effects of using different prescribed boundary layer
shapes and synthetic turbulence boxes in a very long domain both in the absence and presence of wind turbines. The simulations and the analysis of the
results were performed by KN with input from Ola Eriksson (OE) and SI. Nina
Svensson (NS) contributed with a wind profile determined from Weather Research & Forecasting (WRF) simulations. The manuscript was written by KN
with input from SI, SPB, OE and NS.
Paper 6
This work consists of a validation study of near wake features simulated with the
actutator line method. For this purpose, the simulated wake flow is compared
to data from the MEXICO experiment. The simulations were performed by
Wen Z. Shen (WZS). The analysis of the simulation and experimental results
was performed by KN. The manuscript was written by KN with input from
JN, SPB, SI and WZS.
vii
viii
Contents
Sammanfattning
iii
Abstract
iv
Preface
v
Part I
xi
Chapter 1. Introduction
1.1. Wind turbine wakes and wake interaction
1.2. Earlier work in the field
1.3. Objectives of the work
1
2
6
8
Chapter 2. Numerical model
2.1. Flow solver, EllipSys3D
2.2. Turbine modeling
2.3. Prescribed boundary layer and imposed turbulence
2.4. Power control strategy
9
9
10
14
16
Chapter 3. Actuator disc simulations
3.1. Validation
3.2. Effects of rotor configuration
3.3. Effects of using a power controller
3.4. Effects of imposed turbulence and wind shear
19
19
22
25
27
Chapter 4.
Actuator line simulations
31
Chapter 5.
Conclusions
35
Acknowledgements
37
Bibliography
39
Part II
43
Paper 1.
Large-eddy simulations of the Lillgrund wind farm
47
Paper 2.
Airfoil data sensitivity analysis for actuator disc
simulations used in wind turbine applications
77
ix
Paper 3.
Evaluation of the effects of using a power controller in
LES/ACD simulations
99
Paper 4.
Study of the influence of imposed turbulence on the
asymptotic wake deficit in a very long line of wind
turbines
117
Paper 5.
Large-eddy simulations of the evolution of imposed
turbulence in prescribed boundary layers in a very
long domain
143
Paper 6.
Validation of the actuator line method using near wake
measurements of the MEXICO rotor
165
Part I
Overview and summary
CHAPTER 1
Introduction
The power in the wind has been harvested for thousands of years by sailboats
and windmills and has been an integral part of the evolution of modern society.
The first wind turbines used for electricity generation were constructed in the
late 19th century by pioneers in U.S.A. and Denmark (Brush and LaCour).
In the wake of the oil crisis of the 1970’s, several countries, including the UK,
Germany, Denmark and Sweden, started governmentally funded programs to
increase wind power development (Burton et al. 2001). From then until now
turbines have evolved drastically in terms of both size and nominal power.
They have gone from having a diameter of around 20 meters and a nominal
power of less than 100 kW in the 80’s, to the Vestas V164 which has a diameter
of 164 meters and a nominal power of 8 MW of today. The general concept of
using a three-bladed horizontal-axis turbine has remained the same but some
technological advancements have been made. For example, depending on the
type of generator, some turbines use a gear box, some do not (direct drive) and
some use a hybrid concept with a combination of the two.
During the oil crisis the main driver for wind power development was the
high oil price. One of the advantages of using wind power for electricity generation is that it relies on a freely available wind as fuel source. This in combination with the wind turbine’s very low emissions of CO2 , which is very beneficial
from a climate change perspective, have led to political incentives in a number of countries in order to promote wind power and other types of renewable
electricity sources. As a result, wind power has shown an annual growth rate
of approximately 25% over the last decade and in the middle of 2014 the total installed wind power world wide was around 318,000 MW (WWEA 2014).
One drawback of wind power is that its production depends on the wind speed
which is varying at all times. At low wind speeds the production is close to
zero and for high speeds, normally in the order of 25m/s, the turbines shut
down for safety reasons. Also, the proximity between wind turbines and people’s residential and recreational areas has, in some places, led to conflicts.
Since wind turbines can be seen and heard over long distances, some people
have found them disturbing. A possible mitigation to the problem of having
varying wind speeds, and which also distances wind turbines from people, is to
place turbines in offshore locations where speeds are relatively higher and more
stable. However, the costs associated with offshore wind power are normally
significantly higher compared to onshore sites due to increased complexity in
1
2
1. INTRODUCTION
grid connection, construction and maintenance. In order to reduce these costs
offshore turbines are often placed in farms.
However, once the turbines are placed in farms their power production is
reduced and fatigue loads are increased by wake effects (discussed in Section
1.1). It is therefore imperative, at the planning stage, to have an understanding
of the wake power losses within a farm, which normally can be in the order of
10% to 20% of yearly production (Barthelmie et al. 2009). The Lillgrund wind
farm, which is used as a reference site throughout this work, has losses in the
order of 30%. These losses are related to the farm’s special layout which will be
addressed in Section 3.1. Additionally, information about the flow conditions
inside a farm is important in order to be able to optimize the durability of the
turbines selected for the farm.
The possibilities granted by numerical models, and the development of
computational resources give a unique opportunity to study and increase the
understanding of flow conditions in large wind farms. To give a perspective of
the development of computational resources, the simulations performed within
this project began in 2010 using the Neolith cluster at the National Supercomputer Centre (NSC) at Link¨
oping University in Sweden. In 2012, a migration
to the new Triolith cluster occurred which resulted in a computational speedup of 3-4 times and a 5 fold increase in prioritized simulation time allowing our
research group to perform larger and more demanding simulations.
1.1. Wind turbine wakes and wake interaction
According to the well-known Betz limit (Betz 1920) no more than approximately 59.3% (16/27) of the kinetic energy in the wind can be extracted and
transformed into rotational energy by the rotor. This means that the turbine extracts momentum from the flow leading to lower kinetic energy in the
wake flow, and thus a lower mean velocity, compared to the undisturbed flow
upstream of the turbine.
Furthermore, as the turbine blades sweep the air, each blade induces a
vortex sheet which transforms into tip and root vortices behind the turbine.
Since the blades are rotating, these vortices create helical structures as seen
in Figure 1.1. These structures will eventually break down into smaller scale
turbulence due to different instability mechanisms such as the vortex pairing
of the tip vortices (Sarmast et al. 2014) as depicted in Figure 1.1. This figure
is generated by a large-eddy simulation (LES) in which the turbine is modeled
using the so called actuator line (ACL) method.
In this work the turbine rotors are modeled either using the ACL method
(Sørensen & Shen 2002) or the actuator disc (ACD) method (Mikkelsen 2003;
Sørensen & Myken 1992) which will be described in detail in Section 2.2.
1.1.1. Wake regions
The wake behind a wind turbine is characterized by two regions; the near and
the far wake. The near wake is the region in the vicinity of the turbine where the
1.1. WIND TURBINE WAKES AND WAKE INTERACTION
3
Figure 1.1. Wake structure generated by the LES/ACL approach visualized with iso-contours of vorticity (reproduced
with courtesy of S. Sarmast).
wake features are directly linked to the rotor geometry and its aerodynamics as
well as to the inflow conditions. The near wake is characterized by the helical
vortex structures described above which gradually undergoes a transition into
smaller scale turbulence. The near wake is followed by the far wake in which
the influence of the rotor characteristics is less important. The far wake is more
influenced by the surrounding environment, such as wakes from other turbines
and the topography. In the far wake the cross-sectional profiles of velocity and
turbulence intensity have self-similar distributions (Vermeer et al. 2003).
The wake structure is visualized in Figure 1.2 which depicts iso-contours
of vorticity in the plane-cut, intersecting the center of the turbines, of the flow
field of two ACD simulations. The simulations are performed using a freestream
velocity, U0 = 8m/s, and two different ambient turbulence intensity levels,
T Iamb = 4.5% and 8.9%. In ACD simulations, the helical vortex structure is not
present and instead a shear layer is generated. Despite this simplification, the
general idea of a near and a far wake region is still valid. In Figure 1.2, it is seen
that the shear layer is more unstable and breaks down earlier into smaller scale
turbulence when T Iamb = 8.9% compared to when T Iamb = 4.5%. Transversal
profiles of the mean axial velocity component intersecting the center of the
turbines are depicted in Figure 1.3. It is observed that initially the velocity
profiles are largely influenced by the turbine characteristics. This influence
is observed to gradually decline as the wake progresses further downstream
and somewhere in the interval of 6R < z < 8R for T Iamb = 4.5% and of
4R < z < 6R for T Iamb = 8.9% the profiles’ shapes become Gaussian in which
the rotor characteristics are not visible.
1.1.2. Wake interaction
The simplest case of wake interaction is when the wake from one wind turbine
interacts with a second turbine standing in the wake of the first one. Since the
4
1. INTRODUCTION
a)
b)
Figure 1.2. Iso-contours of vorticity for a) T Iamb = 4.5%
and for b) T Iamb = 8.9%. Dashed lines represent the downstream positions used for determination of the velocity profiles
in Figure 1.3.
z = 0R
z = 4R
z = 2R
z = 6R
z = 8R
z = 10R
11.5
11
x/R
10.5
10
9.5
9
8.5
0.5
1
0.5
1
0.5
1
0.5
1
0.5
1
0.5
1
W/U0
Figure 1.3. Velocity profiles at different downstream positions in the wake flow. Solid lines denote T Iamb = 4.5% and
dashed lines T Iamb = 8.9%
wake flow, as discussed above, is characterized by a relatively low mean velocity and high turbulence level, the turbine operating in the wake of the other
turbine, experiences a decreased power production and increased fatigue loads
compared to stand-alone turbines. The way the interaction occurs depends
1.1. WIND TURBINE WAKES AND WAKE INTERACTION
5
on how the turbines are positioned relative to each other and, also, on the
atmospheric conditions (e.g. wind direction, wind speed, turbulence intensity,
atmospheric stability and wind veer). In wind farms, a more complex scenario
arises since a multiple of wakes interacts. In Figure 1.4, the wake interaction
in one inflow angle of the Lillgrund wind farm is depicted.
Figure 1.4. Wake interaction in the Lillgrund wind farm
generated using the LES/ACD approach visualized with isocontours of vorticity.
1.1.3. Wake modeling
Like stated above, in this work the focus is on the ACD and ACL methods.
However, wind turbine wake simuations can be performed using models with
a large variety of complexity. There are simple wake models assuming linearly
expanding wakes (Jensen 1983) and there are more sophisticated models solving
the thin shear layer approximation using an eddy viscosity approach (Ainslie
1988). There are vortex models, which in an efficient and accurate manner
solve the wake flow behind a single turbine (Okulov & Sørensen 2010; Segalini
& Alfredsson 2013). Furthermore, there are models based on the linearized
Navier-Stokes equations such as the Fuga model (Ott et al. 2011) and there
are wind turbine models which, combined with CFD simulations, compute the
entire wake flow. The simplest model in this case is the constantly loaded
ACD. The ACD method used in the present work is more complex since the
disc loadings are computed with the use of airfoil data (Mikkelsen 2003). The
ACL method (Sørensen & Shen 2002) also used in the present work, is similar
to the ACD method in its use of airfoil data but it also models each blade of the
6
1. INTRODUCTION
turbine causing computational demands to increase. Ultimately, the turbine
can be modeled using the actual geometry of the blades which, compared to
the ACL method, is far more computationally demanding. For an extensive list
of different wake models and turbine models, the reader is referred to Crespo
et al. (1999) and Vermeer et al. (2003).
In Barthelmie et al. (2009), the power production of the Horns Rev wind
farm was estimated using wake models with different levels of complexity. The
different estimated values were benchmarked and compared with measured
production. The results showed that the less complex wake models tended to
overestimate the production while more complex models showed the opposite
trend.
1.2. Earlier work in the field
1.2.1. Experiments
The wake characteristics behind a model scale wind turbine were analyzed
experimentally in a pioneer work by Alfredsson & Dahlberg (1979). Some of
the results were determined using smoke visualization showing both the helical
tip vortex structure and the mechanism of the pairing of the tip vortices. Later,
Alfredsson & Dahlberg (1981) added a second turbine which that was traversed
in the spanwise direction in the wake flow of another turbine, effectively showing
the effects of wake interaction on power production. Several years later, detailed
measurements of the near wake flow were performed in the MEXICO project
(Snel et al. 2007). The rotor in this case (the MEXICO rotor) measured 4.5m
in diameter and the results were based on load and pressure data measurements
combined with Particle Image Velocimetry (PIV) measurements. The analysis
of the measured data was performed within the MexNext Annex (Schepers
et al. 2012), giving a unique possibility to validate numerical methods.
1.2.2. Single turbine simulations
R´ethor´e (2009) studied modified versions of the k − ε model associated with
RANS simulations using an ACD method by comparing them with LES and
measurements. This work showed that all of the simulations using the former
model had problems capturing the full extent of the wake flow. Generally,
these simulations captured either the wake flow near the rotor or the flow
further downstream, depending on the modification of the k − ε model used.
Port´e-Agel et al. (2011) studied the velocity and the turbulent statistics behind a single turbine which was parameterized by using an ACD method with
constant loading (this method is referred to as ACD-NR), an ACD method
which considers turbine-induced flow rotation (ACD) and an ACL method.
The results were compared to wind tunnel measurements and field measurements where the ACD and ACL methods yielded good predictions of the studied parameters while the results of the ACD-NR showed significant deviations
from the studied parameters. Troldborg et al. (2010) performed simulations
(LES/ACL) on a single turbine in uniform inflow conditions. Ivanell et al.
1.2. EARLIER WORK IN THE FIELD
7
(2009) and Sarmast (2014) analyzed the near wake features behind a single
turbine drawing conclusions on the stability of the tip vortices. In a recent
study by Sarmast et al. (2015), the vortex model described in Segalini & Alfredsson (2013) was compared to LES/ACL simulations and the results showed
a very good agreement.
1.2.3. Farm simulations
The numerical method used in the present work has previously been used by
Ivanell et al. (2008b), Ivanell (2009), Troldborg et al. (2011, 2014) and Keck
et al. (2014). The work by Ivanell et al. (2008b) studied a farm of 9 turbines
and special attention was paid to the dependency of the inflow angle of the wind
and the impact of a turbulent inflow. The work of Ivanell (2009) studied 20
turbines in the Horns Rev wind farm. By using periodic boundary conditions,
an infinitely wide wind farm was modeled and a good approximation of the
layout of the farm, consisting of 80 turbines, was achieved. The results from
this work exhibited a good, yet not perfect, match with the measured data
from the farm. Troldborg et al. (2011) performed simulations of the wake
interaction between two turbines when varying the inflow to mimic laminar,
offshore and onshore conditions. The correlation between the imposed wind
profile and the atmospheric turbulence was investigated in Troldborg et al.
(2014) and it was shown that imposed turbulence was sustained to a large
extent throughout the computational domain if a linear wind profile equal to
that used in the Mann model (Mann 1994, 1998) was used. In Keck et al.
(2014) a wind profile defined by the power law was used and it was shown
that the turbulence approached equilibrium with the wind shear profile when
progressing downstream. Other work in the field has been performed by, among
others, Ammara et al. (2002), Calaf et al. (2010), Lu & Port´e-Agel (2011) and
Churchfield et al. (2012). Ammara et al. (2002) analyzed a two-row periodic
wind farm using RANS simulations combined with an ACD method. The
results showed an acceptable level of accuracy for the farm’s power performance.
Calaf et al. (2010) performed simulation of a fully developed wind turbine array
imposed in a neutral boundary layer. The wind turbines were modeled using
a constantly loaded ACD approach. The results from this work show that
the kinetic vertical fluxes were in the same order of magnitude as the power
extracted by the wind turbines. Lu & Port´e-Agel (2011) performed simulations
on a very large wind farm in a stable atmospheric boundary layer. Churchfield
et al. (2012) performed computations on two aligned turbines parameterized
using a flexible ACL model which were imposed in a turbulent atmospheric
boundary layer. The evolution of turbulence in long lines of wind turbines has
been the subject of research in Andersen et al. (2013) where an infinite row
of turbines was simulated using periodic boundary conditions at the inlet and
outlet. The turbines were in this case parameterized using the ACL model.
This study was performed without using an atmospheric boundary layer and
thus it studied the turbulence inherent to the turbines themselves. In a recent
study by Goit & Meyers (2015), a wind farm was simulated using LES combined
8
1. INTRODUCTION
with a constantly loaded ACD approach with the aim of increasing the power
production by means of optimally controlling the boundary layer above the
wind farm. Compared to a non-controlled reference case, the optimized case
resulted in an increased power yield of up to 16%.
1.3. Objectives of the work
The key focus of this doctoral work has been to predict and to analyze wake
flows after wind turbines with the aim of increasing the knowledge and understanding of how turbines operating in this wake flow behave. For this study LES
computations are performed using the computational fluid dynamics (CFD)
tool EllipSys3D (described in Section 2.1). The turbines are parameterized
using either the ACD or the ACL method (described in Section 2.2). The atmospheric conditions are modeled using the prescribed boundary layer (PBL)
technique and with pregenerated synthetic turbulence (both described in Section 2.3).
The scientific work leading to this thesis addresses the following questions:
• How well does the numerical model perform when compared to measurements?
• How sensitive are the wake flow and the turbine operational characteristics to the rotor configuration?
• How will pregenerated turbulence and the wind profile shape affect the
wake flow and operational characteristics of turbines?
CHAPTER 2
Numerical model
Resolving all length and time scales associated with wind turbines is not possible with today’s computational capabilities. Therefore, in this work the approach of LES is used in combination with the ACD and ACL methods. To
further reduce the needed computational capability, the atmospheric conditions
are modeled using pregenerated synthetic turbulence combined with the PBL
technique. In this chapter the numerical model is described in detail.
2.1. Flow solver, EllipSys3D
The simulations are performed with the EllipSys3D code, originally developed
by Michelsen (1992, 1994) and Sørensen (1995) at DTU/Risø. The EllipSys3D
code is a general purpose 3D solver based on a finite volume discretization of the
Navier-Stokes equations in general curvilinear coordinates and is parallelized
using a message passing interface (MPI). The code is formulated in primitive
variables, i.e., pressure and velocity, in a collocated storage arrangement. The
numerical method uses a blend of a third order Quadratic Upwind Interpolation
for Convective Kinematics (QUICK) scheme (10%) and a fourth order Central
Differences Scheme (CDS) (90%) for the convective terms, while it uses a second
order CDS for the remaining terms. The usage of the blend is a compromise
between avoiding non-physical numerical wiggles and limiting numerical diffusion. The pressure correction equation is based on the Semi-Implicit Method
for Pressure-Linked Equations (SIMPLE) algorithm. The numerical model has
been described, used and validated by Mikkelsen (2003), Troldborg et al. (2011,
2010) and Ivanell et al. (2008a,b) among others. The governing equations are
defined by the filtered incompressible Navier-Stokes equations given by,
1 0
∂U
1
1
1
+ U · ∇U = − ∇P + ∇ (ν + νSGS )∇U + fW
+ f0
+ f0
∂t
ρ
ρ T G ρ P BL ρ turb
(2.1)
and,
∇·U=0
(2.2)
where U is the filtered velocity, P is the filtered pressure, t is time, ρ is
the density of air, f 0 are forcing terms used for imposing turbines (WTG),
9
10
2. NUMERICAL MODEL
prescribed boundary layer (PBL) and the synthetic turbulence (turb), ν is the
kinematic viscosity and νSGS is the sub-grid scale (SGS) viscosity. The details
on how to introduce the body forces are presented in Sections 2.2 and 2.3.
The computations are carried out as LES. With the LES technique large
eddies are resolved explicitly, and eddies smaller than a certain size are filtered out and modeled by an eddy-viscosity based SGS model. For this work
the mixed scale model developed by Ta Phuoc (1994) is used because of i) its
simplicity, ii) it takes dissipation into account, and iii) it ensures a correct interaction between the smallest resolved scales and the largest unresolved scales.
For more information about the LES technique and the SGS model, the reader
is referred to Sagaut (2006) and Ta Phuoc (1994). The SGS viscosity is defined
by,
νSGS = ρCm |∇ × U(x, t)|αm qc2
m
1−α
2
(x, t)∆
1+αm
(2.3)
where Cm = 0.01 and αm = 0.5 are model constants, ∆ = ∆V ol1/3 is
the cut-off filter size, where ∆V ol is the volume of a given cell, and qc2 is the
turbulent kinetic energy which is evaluated as,
qc2 =
1
(U(x, t))0 (U(x, t))0
2
(2.4)
e defines the highest frequency part of the resolved
where (U)0 = U − U
velocity field which is found by using a test filter, denoted with the ∼ symbol,
with a cut-off size equal to 2∆.
2.2. Turbine modeling
The local velocities, forces and angles acting on and associated with a wind
turbine blade section are depicted in Figure 2.1. In this figure Uz and Uθ are
the velocities in the axial and tangential directions, respectively, and Urel is the
local relative velocity. Ω is the angular velocity of the turbine, r is the local
radius, ϕ is the local flow angle, θP is the local pitch angle and α is the local
angle of attack. L and D are the lift and drag forces per unit length and F is the
resulting force per unit length. F is projected onto θ and z axes which gives the
forces Fθ and Fz acting in the tangential and normal directions to the plane of
rotation, respectively. Fθ contributes to the aerodynamic torque, Maero , while
Fz contributes to the thrust force, T . From Figure 2.1 the following expressions
for Urel , ϕ and α are found,
Urel =
p
Uz2 + (Ωr − Uθ )2
ϕ = tan−1
Uz
Ωr − Uθ
(2.5)
(2.6)
2.2. TURBINE MODELING
11
L
F"
F
!
Fz
z
D
#
"P
$%r
!
Urel
U"
Uz
$"
Figure 2.1. Velocities, forces and angles associated with a
section of a wind turbine blade.
α = ϕ − θP
(2.7)
Furthermore, the axial and azimuthal induction factors, a and a0 are determined by,
a=
U0 − Uz
U0
a0 = −
Uθ
Ωr
(2.8)
(2.9)
where U0 is the free stream velocity.
2.2.1. Actuator disc method
Disc theory has been presented in a variety of publications and will be not, for
the sake of conciseness, be described here. The reader is referred to Hansen
(2008) and Burton et al. (2001) for disc theory and to Mikkelsen (2003) for the
connection between the theory and the current numerical application. In the
ACD method (Mikkelsen 2003; Sørensen & Myken 1992) the blades are represented by body forces. The body forces are computed using a blade element
approach combined with tabulated airfoil data (lift and drag coefficients, CL
12
2. NUMERICAL MODEL
and CD ). Therefore, the flow past the rotor is solved without resolving the
boundary layer on the blades which significantly reduces the computational
demands. The main uncertainties of the method are therefore found in its dependence of the quality of the airfoil data. The body forces are distributed on a
disc representing the blades as depicted in Figure 2.2. It is emphasized that the
discretization in this figure is very crude and that, normally, 21 points along
the rotor radius, R, and 81 points in the azimuthal direction are employed.
dA=rdrdθ
rdθ
dr
r
R
Figure 2.2. Local discretization of the ACD.
The lift and drag coefficients, CL (α, δ/c) and CD (α, δ/c), are interpolated
from tabulated data, where δ is the local thickness and c is the local chord
length of the blade. L and D are determined from,
(L, D) =
1 2
ρU cB(CL eL , CD eD )
2 rel
(2.10)
where B is the number of blades. F is projected in the θ and z axes by,
F = (Fθ , Fz ) = (Lsin(ϕ) − Dcos(ϕ), Lcos(ϕ) + Dsin(ϕ))
(2.11)
By looking at the small area of the disc of differential size, dA, in Figure
2.2, the force per unit area is given by,
f=
Fdr
Fdr
Fdr
=
=
nθ dA
nθ rdrdθ
2πrdr
(2.12)
where nθ is the number of points on the disc in the azimuthal direction.
The resulting body force is found by,
f0 =
f
dz
(2.13)
where dz is the length of a cell in the axial direction. In order to avoid
numerical problems, the body forces are smeared over some cells in the axial
2.2. TURBINE MODELING
13
direction. In this process, the single point force determined at the disc at a
certain position is distributed in one-dimensional Gaussian manner around the
point in such way that the integral amount of the distributed force is equal to
the value of the single point force. The force distribution is determined by the
convolution, denoted by ∗, of the body force and a regularization kernel, η ,
according to,
0
0
fW
T G = f ∗ η
(2.14)
2
1
η (p) = √ e−(p/)
π
(2.15)
where η is defined as,
where p is the distance between cell centered grid points and the rotor disc,
and is a constant smearing parameter, normally equal to the length of two
computational cells in the axial direction.
T and Maero are determined by,
Z Z
Z
T =
R
Z
2π
fz dA =
A
fz rdrdθ
0
Z Z
Z
Maero =
R
Z
fθ rdA =
A
(2.16)
0
0
2π
fθ r2 drdθ
(2.17)
0
where fz and fθ are forces per unit area in the axial and tangential directions, respectively. The power, P , is determined from,
P = ΩMaero
(2.18)
2.2.2. Actuator line method
Throughout the project, the focus has been on the development and understanding of flow behavior inside wind farms using the ACD method, however,
the ACL method is used in a validation study. In the ACL method the rotor
is modeled using forces distributed along three lines representing the blades
which are rotating with a certain velocity. The procedure to determine the
body forces is in principle the same for both the ACL and the ACD methods,
however, as the ACL method includes the rotation of the blades, it requires a
higher resolution and a smaller time step since the tip of the blade, with the
highest rotational velocity, cannot travel more than one grid point in one time
step (Troldborg 2008).
14
2. NUMERICAL MODEL
2.3. Prescribed boundary layer and imposed turbulence
In this section, the methodology to introduce the atmospheric conditions using
prescribed boundary layers and synthetic pregenerated turbulence is presented.
2.3.1. Prescribed boundary layer
The shear profile used in the simulations is introduced and maintained using
body forces. This technique is from now on referred to as the prescribed boundary layer (PBL) technique and is described in detail in Troldborg et al. (2014).
The body forces, fP0 BL , are first determined from the desired velocity profile
and imposed in the domain by performing a steady-state simulation in the absence of wind turbines and atmospheric turbulence. Then, still in the absence
of wind turbines and turbulence, a few unsteady time steps are performed ensuring a time-dependent solution. The results are saved and used as inputs
for cases involving turbines and turbulence. There are two main advantages
of using the PBL technique compared to traditional simulations in which the
shear profile is let to develop in a very long domain or by the use of precursor
simulations. The first advantage is that it drastically reduces the simulation
time since the precursor simulation step in a PBL simulation only needs to be
run in the order of 10-20 time steps. The second advantage is that any given
wind profile can be simulated. However, the model is restricted to simulating
only neutrally stratified atmospheric conditions.
In this work, the shear profiles are defined either using a combination of a
power law and a parabolic function,
(
W (y) =
2
U0 (c
2y +
c1 y) y ≤ ∆
U0
y
hhub
αpl
y>∆
(2.19)
a linear function,

y < yL
 WL
WL + β(y − yL ) yL ≤ y ≤ yH
W (y) =

WH
y > yH
(2.20)
or by the logarithmic law,
W (y) =
u? y
ln
κ z0
(2.21)
In Equation 2.19, αpl is the power law exponent, ∆ is a given height and
c1 and c2 parameters used to ensure a smooth transition between the power
law and the parabolic profile which is used between 0 ≤ z < ∆. c1 and c2 are
defined by,
2.3. PRESCRIBED BOUNDARY LAYER AND IMPOSED TURBULENCE
c1 =
c2 =
U0
(2 − α)
hhub
U0
hhub ∆
∆
hhub
(α − 1)
15
α−1
∆
hhub
(2.22)
α−1
(2.23)
In Equation 2.20, yL and yH , are the lowest the highest vertical positions
of the linear shear profile, respectively. WL and WH are the axial velocities at
−WL
these positions and β = WyH
. In Equation 2.21, u? is the friction velocity,
H −yL
κ is the von K´
arm´
an constant and z0 is the roughness length.
2.3.2. Imposed turbulence
The atmospheric turbulence is pregenerated using the Mann model (Mann
1994, 1998). The turbulence field, which is referred to as the Mann box, computed by this model is homogeneous, Gaussian, anisotropic, and has the same
second order statistics as neutrally stratified atmospheric turbulence and is
generated assuming a linear wind profile. By adjusting the roughness length,
z0 , the velocity, U0 , at a certain defined height, zh , in the Mann model, different shear profiles are simulated resulting in different levels of turbulence. The
turbulence is imposed in the domain using body forces. From a simple control
volume analysis and momentum balance the following expression is found,
f =m
˙ f u + ρ
du
dt
(2.24)
where m
˙ f is the mass flux, is a smearing parameter and u is a vector
containing fluctuations. The resulting body force is determined by,
f0 =
f
dz
(2.25)
Before imposing the body forces in the computations, they are smeared
in a Gaussian manner in the z-direction in order to avoid numerical problems
using,
0
fturb
= f 0 ∗ η
(2.26)
where η is the same type of smearing function as used in the ACD method
(Equation 2.15).
The Mann box consists of Nz × Nx × Ny grid points, and by assuming
the Taylor’s frozen hypothesis, each grid point in the axial direction is a time
step. Therefore, the Mann box is divided into Nz number of xy-planes. The
fluctuations in these xy-planes are introduced at a certain streamwise location
16
2. NUMERICAL MODEL
which is located within the equidistant region of the grid and upstream of
possible wind turbines and are then convected downstream by the flow solver.
2.4. Power control strategy
In order to simulate realistic conditions of variable speed turbines, a power
controller needs to be used in order to ensure that the turbines are operating
at desired conditions. Below rated power, modern turbines are designed to
operate with a constant tip speed ratio, λ = ΩR/U0 , regardless of the incoming
wind velocity ensuring that the turbine is operating optimally. Here, Ω is the
angular velocity and R is the radius of the turbine rotor
For a power control strategy to be efficient and usable for both individual
turbines and for farms of such, it needs to rely on parameters evaluated at
the turbine position. The controller strategy described and used here basically
requires i) determination of the aerodynamic torque (which is performed using
information available at the disc location), ii) tabulated information about the
generator torque and iii) a relationship between these different torques by a
governing equation.
2.4.1. Determination of different torques
The aerodynamic torque, Maero , is determined by Equation 2.17. The generator torque, Mgen , is tabulated as a function of RPM. This curve is created
using the methodology described in Jonkman et al. (2009). A principal sketch
of the generator torque curve is shown in Figure 2.3. The curve is divided into
the following regions:
•
•
•
•
•
Region
Region
Region
Region
Region
1: Start-up region
1 1/2: Linear transition between region 1 and 2
2: Desired operation region
2 1/2: Linear transition between region 2 and 3
3: Pitch control region
At the start-up region Mgen is null, which allows the system to accelerate. In
the desired operation region, the generator torque makes sure that the turbine
operates at the desired tip speed ratio. In the pitch region, the turbine is controlled to operate at rated power by pitching the blades and reducing the lift
force. Regions 1, 1 1/2, 2 and 2 1/2 are denoted as the generator torque controller (GTC) and region 3 is denoted as the pitch controller. It is emphasized
that only the GTC is described here.
2.4.2. Governing equation
The controller makes use of Newton’s second law of motion for rotating bodies,
defined by,
˙ = ID Ω˙
Maero − Mgen = (Irotor + Igen )Ω
(2.27)
2.4. POWER CONTROL STRATEGY
16
17
x 10
Region 1 1/2
14
Torque [Nm]
12
10
8
Region 3
Region 2
Region 1
6
4
Region 2 1/2
2
0
0
2
4
6
8
10
RPM
12
14
16
18
20
Figure 2.3. A principal sketch of a generator
torque curve.
where Irotor and Igen are the moments of inertia for the rotating parts which
are simplified as the drivetrain moment of inertia, ID , and Ω˙ is the resulting
angular acceleration of the system.
2.4.3. Control procedure
The following procedure is used for controlling the turbine:
1. At time t:
• Maero is determined by Ω(t).
• Mgen is determined from the look-up table using Ω(t).
• Ω˙ is determined by Equation 2.27,
Maero − Mgen
Ω˙ =
(2.28)
ID
• The pertubation of the angular velocity, ∆Ω, is determined by
integrating Ω˙ in time by,
˙
∆Ω = Ω∆t
(2.29)
2. At the proceeding time step, t + ∆t, the new angular velocity, Ω(t + ∆t),
is determined by adding ∆Ω to Ω(t),
Ω(t + ∆t) = Ω(t) + ∆Ω
(2.30)
The input to the control strategy is therefore the angular velocity, the
tangential forces acting on the disc and the tabulated generator torque, while
the output is a new angular velocity.
CHAPTER 3
Actuator disc simulations
3.1. Validation
The Lillgrund wind farm is situated offshore between Sweden and Denmark
and consists of 48 wind turbines, as depicted in Figure 3.1. Compared to other
offshore wind farms, the turbines in the Lillgrund farm are positioned very
tightly, with internal turbine spacings as small as 6.6R, where R = 46.5m is
the rotor radius. The farm efficiency of Lillgrund, as discussed in Chapter
1, is therefore lower in comparison to other wind farms but thanks to the
characteristics of the farm it is very interesting from a wake interaction point of
view. Therefore, it is used for validating the LES/ACD approach by comparing
measured with simulated power production. This section presents the main
results from Paper 1.
70
Wind turbines
Met.mast
Row 8
Row 7
N
Row 6
Row 5
60
Row 4
South-North direction, Rotor radii [-]
Row 3
Row 2
50
Row 1
40
Row A
Row B
30
Row C
20
Row D
U0,120
Row E
10
Row F
0
Row G
U0,222
-10
-10
0
Row H
10
20
30
40
50
60
West-East direction, Rotor radii [-]
Figure 3.1. Layout of the Lillgrund wind farm.
19
70
20
3. ACTUATOR DISC SIMULATIONS
The simulations are performed using flow conditions determined from measurements. The atmospheric turbulence is modeled using synthetic turbulence
created by the Mann model (Mann 1994, 1998) characterized by a turbulence
intensity of 5.7%. The PBL is defined by the power law (Equation 2.19) using
∆ = 0.4R and αpl = 0.11.
Two inflow angles (120±2.5◦ and 222±2.5◦ ) are considered in the analysis,
as denoted by the black arrows in Figure 3.1, but only the 120 ± 2.5◦ case is
presented here for the sake of conciseness. The power production is filtered for
7.5 ≤ U0 ≤ 8.5m/s and is normalized with the median production of Row 1 in
Figure 3.1.
In Figure 3.2, the average production of all full length rows (Rows B-D)
is depicted. The general agreement between the simulations and the measurements is very good. However, there are small discrepancies in the agreement,
where an overestimation of the power for the second turbine in the row is
observed in the simulation results.
P/Pmd
1
LES
Measurements
0.8
0.6
0.4
0.2
0
5
10
15
20
25
30
35
40
45
z/R
Figure 3.2. Average production of Rows B-D predicted from
computations compared with measurements. The production
is normalized with the median production of Row 1 of turbines.
In Figure 3.3 the production of Row E turbines is shown. This row is
of special interest since it contains of a hole in the middle where no turbines
are placed and where the flow is allowed to recover over a longer distance.
The production is again slightly overestimated for the second turbine in the
row and also for the turbines after the recovery hole. The production of the
turbines after the hole is observed to have a similar trend compared to the
measurements. It is worth noting that the production after the recovery hole is
almost twice as high as the production before the hole, which shows the effects
of placing the turbines so close to each other.
A simple power curtailment study is performed to determine how the total
production of the farm changes when the power extraction of the front row turbines is decreased. The power curtailment is performed by pitching the blades
3.1. VALIDATION
P/Pmd
1
21
LES
Measurements
0.8
0.6
0.4
0.2
0
5
10
15
20
25
30
35
40
45
z/R
Figure 3.3. Production of Rows E predicted from computations compared with measurements. The production is normalized with the median production of Row 1 of turbines.
2◦ , 4◦ , and 6◦ which allows more kinetic energy to pass to the downstream turbines which will, ideally, experience increased production and decreased loads.
In Figure 3.4, which is based on the average production of all full length rows
(Row B-D) normalized with the median production of Row 1 for the 0◦ pitch
case, it can be seen that only the second turbine in the row benefits from the
curtailment and that the increase in production at this turbine is lower than
the loss at the first turbine. This simple curtailment strategy does not, in
other words, increase the overall production of the farm and more advanced
strategies needs to be evaluated, see e.g. Goit & Meyers (2015).
P/Pmd
1
0◦
2◦
4◦
6◦
0.8
0.6
0.4
0.2
0
5
10
15
20
25
30
35
40
45
z/R
Figure 3.4. The effect of curtailment on the average production of Rows B-D. The production is normalized with the
median production of Row 1 of turbines for the 0◦ pitch case.
The agreement in power production when comparing simulations with measurements is very good. However, some uncertainties in the results are found.
The airfoil data and blade geometry describing the actual turbine at the site
are unavailable for this study. Therefore a downscaled version of the NREL
22
3. ACTUATOR DISC SIMULATIONS
5MW turbine (Jonkman et al. 2009) is used in its place. This turbine is fitted
against the thrust and power coefficients of the real turbine. Additionally, the
simulations are performed giving the turbines a fixed angular velocity. For the
real turbine at the site a power controller is used which forces each turbine to
adapt to the conditions it is operating in by dynamically changing its angular
velocity. The implications of the generalization of the rotor configuration and
the lack of a power controller are discussed in the following sections.
There are also uncertainties in the way the atmospheric conditions are measured due to positioning and type of instruments used in the measurements
(Barthelmie et al. 2009). In the case of Lillgrund, the measurement mast is
located very near the farm as indicated in Figure 3.1. Therefore, the mast is
in most flow directions shadowed or influenced by the farm which obviously increases the uncertainties regarding the incoming flow conditions. Additionally,
no measurements of the water temperature is gathered at the Lillgrund wind
farm site. Instead, this information is taken from a nearby located lightship.
As a result there are uncertainties in the stability classification of the site. The
simulations are carried out assuming neutral atmospheric conditions.
3.2. Effects of rotor configuration
As discussed in the previous section, the data describing the wind turbine
rotor in the Lillgrund wind farm is unavailable for this project. Therefore a
generic rotor model is used. The lack of real turbine data is believed to be a
general issue and does not only apply to the current project. In this section,
which presents the main results from Paper 2, the error resulting from the use
of a generic wind turbine is analyzed. For this purpose data sets from three
different airfoils are used, which are all intended to be representations of the
turbine used in the Lillgrund wind farm. The first turbine is the downscaled
NREL turbine (Jonkman et al. 2009) used in Paper 1 and is in this study
referred to as AF1. The second turbine is based on the DU21 profile which is
one of the profiles used in the AF1 turbine. This profile is used for the entire
span width of the rotor. The chord, c, and twist, Φ, distributions are designed
to fit the CP and CT curves of AF1. This turbine is referred to as AF2. The
third turbine, referred to as AF3, is the generic SWT-2.3-93 turbine which is
described in detail in Churchfield (2013). The chord and twist distributions of
these turbines are shown in Figures 3.5 and 3.6, respectively.
These rotor configurations are simulated in a number of different flow conditions in which a line of 8 turbines is imposed. For the sake of conciseness, the
results from only one of these flow conditions are presented here. The turbine
spacing in this case is 6.6R and the ambient turbulence intensity is 4.5%. This
flow condition is chosen since it is the best representation of the Lillgrund wind
farm in the 120◦ inflow direction. The simulations are performed without a
power controller and all turbines are set to operate with a fixed angular velocity. This is chosen since it facilitates the comparison of the rotor configurations.
3.2. EFFECTS OF ROTOR CONFIGURATION
23
0.15
AF1
AF2
AF3
c/R
0.1
0.05
0
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
r/R
Figure 3.5. Chord distributions of the three rotor configurations as functions of the radial position on the blade.
15
AF1
AF2
AF3
Φ [◦ ]
10
5
0
-5
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
r/R
Figure 3.6. Twist distributions of the three rotor configurations as functions of the radial position on the blade.
However, it makes the comparison with the measurements less accurate since
the turbines in the measurement case are controlled.
The relative mean power and thrust are depicted in Figures 3.7 and 3.8,
respectively. For the power, measurement data from the Lillgrund wind farm
is added for comparison purposes. Since the simulation is not an exact representation of the measurement case, the value of the comparison is limited.
Only small differences are found in the results of the power and the thrust for
the different rotor configurations. The main difference found is that AF2 gives
slightly lower values for both the power and the thrust for the second turbine
in the row. It is also shown that the simulated power is in very good agreement
with the measurements.
The mean axial velocity component, W , and the standard deviation of the
axial velocity component, σz , are depicted in Figures 3.9 and 3.10, respectively.
These are measured in detail down to 6R behind the first turbine and then at
3.3R behind each turbine in the remaining flow field. Generally, AF1 and
AF3 agree well for both W and for σz . AF2 renders a lower value for W for
z < 26.9R after which it is very similar to AF1 and AF3. For σz , AF2 it is
very similar to AF1 and AF3.
24
3. ACTUATOR DISC SIMULATIONS
Based on these results, it can be stated that if average quantities are analyzed, the choice of rotor configuration is not critical for the results, even
though small differences are observed, given that the rotors are similar in size
and power. This behavior can be put in relation to the fact, as discussed in
Section 1.1, that the wake flow becomes less dependent on the turbine characteristics far downstream of the turbine.
1
AF1
AF2
AF3
Measurements
P/P1
0.8
0.6
0.4
0.2
0
5
10
15
20
25
30
35
40
45
z/R
Figure 3.7. Relative mean power as function of downstream distance.
1
AF1
AF2
AF3
T /T1
0.8
0.6
0.4
0.2
0
5
10
15
20
25
30
35
40
45
z/R
Figure 3.8. Relative mean thrust as function of downstream distance.
1
AF1
AF2
AF3
W/U0
0.8
0.6
0.4
0.2
0
10
20
30
40
50
z/R
Figure 3.9. Normalized mean axial velocity component as
function of the downstream distance.
3.3. EFFECTS OF USING A POWER CONTROLLER
25
σz /U0
0.2
0.15
0.1
AF1
AF2
AF3
0.05
0
0
10
20
30
40
50
z/R
Figure 3.10. Normalized standard deviation of axial velocity
component as function of the downstream distance.
3.3. Effects of using a power controller
In the validation study the turbines are given a fixed angular velocity. Variable
speed wind turbines are controlled so that their angular velocity is directly
proportional to the incoming velocity for conditions below rated power as described in Section 2.4. In other words the turbines are controlled to operate on
a fixed tip speed ratio which maintains a constant angle of attack. Therefore,
the implication of assuming a fixed angular velocity is that the angle of attack
will vary with the incoming velocity which will have an impact on the determination of the body forces, as described in Section 2.2. This section presents the
main results from Paper 3 in which controlled and non-controlled simulations
are compared.
For this study the turbine described in Paper 1 is used for simulating 21
turbines in a row, positioned with a spacing of 6.6R, which is representative
of the spacing in the Lillgrund wind farm in the 120◦ inflow direction. Five
different simulation cases are performed. The first 3 are performed using a
controller which forces the turbines to operate at a tip speed ratio of 8.2. In
these cases, the moment of inertia, ID , is set to 0.28 · 107 kgm2 , 1.20 · 107 kgm2
and 2.12 · 107 kgm2 , respectively. These cases are referred to as IL (low inertia),
IM (medium inertia) and IH (high inertia). The last two cases are performed
without a controller and in these cases each turbine is given a fixed angular
velocity. In the first case, referred to as NC (non-controlled), the angular
velocity, Ω, is set to 8.2U0 /R for all turbines in the row. In the second case,
referred to as SC (semi-controlled), the angular velocities of the turbines are
determined from the average velocities determined in case IM.
As shown in Figures 3.11-3.14, there are only marginal difference between
the controlled cases and case SC. If a controller is used, the upstream effects
of a rotor are reduced, turbulence levels in the wake flow are reduced and
velocity levels are increased compared to case NC. What might appear slightly
counterintuitive is that the controller has the smallest impact on the power
production itself. However, this can be explained by the fact that the analyzed
turbine has a power coefficient, CP , which shows a quite flat behavior over a
26
3. ACTUATOR DISC SIMULATIONS
large span of tip speed ratios. A turbine with a different behavior of CP would,
therefore, show a different sensitivity to if a power controller is used.
1
IM
IL
IH
NC
SC
P/P1
0.8
0.6
0.4
0.2
0
20
40
60
80
100
120
z/R
Figure 3.11. Relative mean power as function of the downstream distance.
1
IM
IL
IH
NC
SC
T /T1
0.8
0.6
0.4
0.2
0
20
40
60
80
100
120
z/R
0.2
0.2
0.15
0.15
σz /U0
σz /U0
Figure 3.12. Relative mean thrust as function of the downstream distance.
IM
0.1
IL
IH
NC
SC
WTG0.05
0.1
0.05
18
20
22
24
z/R
26
28
30
IM
IL
IH
NC
SC
WTG
144
146
148
150
152
154
z/R
Figure 3.13. Normalized standard deviation of the axial velocity component for the flow at (left) the beginning and
(right) the end of the row of turbines.
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.4
0.3
W/U0
W/U0
3.4. EFFECTS OF IMPOSED TURBULENCE AND WIND SHEAR
IM
IL
IH
NC
SC
WTG
0.5
IM
IL
IH
NC
SC
WTG
0.4
0.3
0.2
27
0.2
18
20
22
24
z/R
26
28
30
144
146
148
150
152
154
z/R
Figure 3.14. Normalized mean axial velocity component at
(left) the beginning and (right) the end of the row of turbines.
3.4. Effects of imposed turbulence and wind shear
As shown in the validation study, the model is found to accurately predict the
power production of Lillgrund which is a relatively short wind farm. If comparing Lillgrund to the Horns Rev wind farm, the latter is found to be more
than two times longer in the 270◦ direction compared to the 120◦ direction of
Lillgrund. Furthermore, the flow evolution for long distances using the simplified technique to take the atmospheric conditions into account has not been
analyzed in detail which naturally poses uncertainties when simulating long
arrays of wind turbines, such as in the Horns Rev case. In this section, the
main results from Paper 4 and Paper 5 are presented which will provide new
insights in the flow development in very large domains.
First, simulations are conducted using uniform flow conditions. In the absence of wind turbines, as no wind shear is present, the initial Mann turbulence
is found to decay as it progresses downstream. This is expected since there is
no shear profile present to sustain the turbulence. The simulations are then
performed in the presence of a row of 10 wind turbines. In the presence of
wind turbines the wake conditions, defined in terms of W and σz , and the
power production, P , are found to be in a nearly asymptotic state far down
in the row of turbines and to be dependent on the level of the imposed Mann
turbulence.
Second, simulations are performed using the PBL technique. In the absence
of wind turbines, three different Mann turbulence boxes (M1, M2 and M3) of
different levels are simulated using three different wind shear profiles (WP1,
WP2 and WP3). In Figure 3.15, the downstream evolution of σi is depicted.
The general finding is that when progressing downstream, the turbulence characteristics undergo a transition from Mann turbulence to PBL turbulence and
that the PBL turbulence is found to be sustained by the wind shear profile far
downstream. Furthermore, close to the Mann turbulence plane, the characteristics of the turbulence are observed to change drastically.
28
3. ACTUATOR DISC SIMULATIONS
M1
M2
M3
0.08
WP1
0.06
0.04
0.02
0.06
WP2
σi /U0
0.08
0.04
i=z
i=x
i=y
0.08
0.06
0.04
WP3
0.02
0.02
17
80
168
293 17
80
168
293 17
80
168
293
z/R
Figure 3.15. Normalized standard deviations of the velocity
components as functions of the downstream distance.
Therefore, it is of interest to determine how the behavior of the wake
conditions and the power production depends on where you place the row
of turbines relative to the Mann turbulence plane. For this purpose three
sets of simulations are performed using the Mann turbulence boxes and wind
profiles defined above. In each of the sets of simulations a row of 10 turbines
is placed so that the distance between the first turbine in the row and the
turbulence plane is 4R, 67R and 155R. These cases are referred to as T1, T2
and T3, respectively. The resulting power is depicted in Figure 3.16, in which
measured production is added for comparison purposes. The measurements
(Hansen 2011) are defined as a mean of the 6 inner rows of turbines in the
270 ± 2.5◦ direction for U0 = 8 ± 0.5m/s and for a turbulence intensity level of
approximately 7%.
When imposing the turbines according to case T1, the levels and trends of
the power are mainly observed to be dictated by the initial turbulence and less
by the wind shear. The ambient flow outside of the wake flow, on the other
hand, will in such a situation inevitably change with downstream distance,
which will affect the production trend. When imposing turbines according
to case T2, the levels and the trends of the power are influenced both on the
imposed turbulence and the wind shear. When imposing the turbines according
to case T3, the levels and trends of the power are observed to be influenced
mainly by the turbulence dictated by the wind shear profile and less by the
initial turbulence. Attention need to be paid here, since the Mann turbulence
is fitted to atmospheric conditions which is not necessarily the case with the
29
3.4. EFFECTS OF IMPOSED TURBULENCE AND WIND SHEAR
PBL turbulence which is found far downstream where the turbulence level is
in equilibrium with the wind shear.
T1
T2
T3
1
WP1
0.8
0.6
0.4
WP2
0.6
0.4
1
0.8
WP3
P/Pref
1
0.8
0.6
0.4
M1
0
50
100
0
50
100
0
M2
50
M3
HR, 270◦
100
(z − Li,W T G1 )/R
Figure 3.16. Relative mean power as function of the downstream distance. Pref is the production of WTG1 for T1 using
M1.
CHAPTER 4
Actuator line simulations
In this chapter the main results from Paper 6 are presented. The primary aim of
this paper is to validate the near wake characteristics simulated using the ACL
method. For this reason the results from the simulations are compared with
experiments performed on the MEXICO rotor. Three flow cases are evaluated
using λ = 4.2, 6.7 and 10, respectively. The experimental results are based
on phase-averaged PIV data. The tip vortex cores are identified using the λ2
criterion defined by Jeong & Hussain (1995) and Wall & Richard (2006).
The wake expansion is analyzed as depicted in Figure 4.1 and it is observed
that the expansion is, as expected, dependent on the tip speed ratio. The wake
expansion in the different flow cases show a good agreement when comparing
simulations with experiments. Also, it can be seen that both the radial and
axial positions of each tip vortex are fairly well predicted in the simulations
compared to the experiments.
The tip vortex core sizes are shown in Figure 4.2. The simulated cores are
found to be in the order of five times larger than the experimental cores. The
difference in size is believed to be due to the relatively coarse grid used in the
simulations.
The tip vortex and bound circulations are shown in Figures 4.3 and 4.4.
When analyzing the tip vortex circulation, the agreement when λ = 10 is good
while in the other two flow cases the simulations deviate from the experiments.
These differences are explained by limitations in the window sizes of the PIV
measurements, leading to what appears to be an overestimation in the simulations. Furthermore, when λ = 6.7, the last two positions show a large drop in
the circulation values and the reason for this is that parts of the tip vortices
are outside the PIV windows, naturally reducing the circulation values.
The agreement in bound circulation for λ = 6.7 and 10 between simulations
and experiments is found to be satisfactory. For λ = 4.2, an overestimation is
found in the simulations compared to the experiment and this overestimation
is more pronounced towards the root of the blade than in the tip. The reason
for this is that the use of the Kutta-Joukowski theorem, Γ = L/(ρUrel ), in the
experiment is questionable due to an increasingly detached flow towards the
root of the blade. In the simulation a detached flow is impossible and the lower
lift due to stall is taken into account through the airfoil data.
31
32
4. ACTUATOR LINE SIMULATIONS
Furthermore, velocity components at different radial positions as functions
of the downstream distance (not shown here) are found to be in good agreement
in the simulations compared to the measurements.
1.25
sim, λ=10
sim, λ=6.7
sim, λ=4.2
exp, λ=10
exp, λ=6.7
exp, λ=4.2
1.2
x/R
1.15
1.1
1.05
1
0
0.5
1
1.5
2
z/R
Figure 4.1. Wake expansion as function of downstream distance.
0.08
sim λ=10
sim λ=6.7
sim λ=4.2
exp λ=10
exp λ=6.7
exp λ=4.2
0.07
Rcore /R
0.06
0.05
0.04
0.03
0.02
0.01
0
0
0.5
1
1.5
2
z/R
Figure 4.2. Tip vortex core radius as function of downstream distance.
8
sim λ=10
sim λ=6.7
sim λ=4.2
exp λ=10
exp λ=6.7
exp λ=4.2
7
Γtip [m2 s−1 ]
6
5
4
3
2
1
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
z/R
Figure 4.3. Tip vortex circulation as function of downstream distance.
8
sim, λ=10
sim, λ=6.7
sim, λ=4.2
exp, λ=10
exp, λ=6.7
exp, λ=4.2
7
Γbound [m2 s−1 ]
6
5
4
3
2
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
r/R
Figure 4.4. Bound circulation as function of the local radius
on the blade.
1
CHAPTER 5
Conclusions
The wake flow behind wind turbines was analyzed numerically using large-eddy
simulations. The wind turbine rotors were modeled using either the actuator
disc method or the actuator line method in which the blades were represented
by body forces computed by using airfoil data. Using these models, the boundary layers of the turbine blades were not resolved and most of the computational
power could be preserved for the wake flow.
The power production of the Lillgrund wind farm was estimated using
the actuator disc method and compared with measurements from the farm.
Generally, the results were in very good agreement with the measurements.
However, some uncertainties were found in the numerical model, concerning
the rotor configuration and the power determination.
Three different rotor configurations subject to different inflow conditions
(level of ambient turbulence) and turbine spacing were simulated. The aim
with this study was to increase the understanding of the uncertainties of using
a generic rotor such as in the simulations of the Lillgrund wind farm. As a
general conclusion it can be stated that the choice of rotor configuration in
actuator disc simulations is not crucial if the intention of the simulations is
to compute the mean wake characteristics subject to turbulent inflow. The
results from this study could be helpful for other researchers dealing with the
same issue of not being granted real turbine data. Furthermore, using the
knowledge obtained from this study, a generic turbine model is currently under
development which will simplify actuator disc simulations further.
Prior to this project, it was only possible to perform simulations using a
fixed angular velocity of the turbines in the used application of the actuator
disc method. Real turbines operating below rated power are controlled so that
they maintain a constant tip speed ratio. Therefore assuming a fixed angular
velocity increases the uncertainties when comparing with real turbines. For
this purpose, a power control strategy was adapted to and implemented in the
actuator disc method. When the controller was used, the flow both upstream
and downstream of a turbine was affected as turbulence levels in the wake flow
were observed to be reduced and velocity levels were increased. However, the
use of a controller was observed to have a small impact on the power production
which was explained by the behavior of the power coefficient as function of the
tip speed ratio of the used wind turbine. Future work will be focused on power
35
36
5. CONCLUSIONS
and load optimization and for this purpose an accurate power controller is
imperative.
The effects of using the technique of imposing pregenerated turbulence and
a prescribed boundary layer were prior to this project only known for relatively
short domains. In the simulation of the Lillgrund wind farm, it was shown that
the model yielded accurate results. However, for very long domains the model
was yet unproven. A case in the absence of wind turbines was simulated.
In the downstream region close to where the Mann turbulence planes were
introduced, the flow was observed to change drastically. Far downstream of
the planes, however, the turbulence was found to be in near equilibrium with
the prescribed wind shear. Therefore, in a uniform flow, the turbulence would
decay, while when a shear profile was used, the turbulence was found to stabilize
at a level dictated by the wind profile. When using a shear profile and when
placing a row of 10 turbines in the domain, the dependence of the imposed
turbulence on the production trends and levels was found to gradually decrease
with the downstream distance. Consequently, far downstream the production
was observed to be influenced mainly by the conditions dictated by the shear
profile.
The wake flow behind the MEXICO rotor was predicted by the actuator
line method and compared with PIV measurement data. The validation was
performed by comparing wake expansion, vortex core radius, circulation as
well as axial and radial velocity distributions. All parameters were analyzed
for three different tip speed ratios (4.2, 6.7 and 10). The tip vortex radius was
largely overpredicted in the simulations (up to five times depending on the flow
case) compared to the measurements. The wake expansion was, however, well
predicted, the tip vortex circulation was found to be in reasonable agreement
depending on the flow case and the bound circulation gave a satisfactory agreement for two of the flow cases (6.7 and 10). The worse agreement in the third
case was discussed to be due to the method used to determine the circulation.
The outcome of this project has mainly been focused on increasing the
understanding and knowledge on how to efficiently perform accurate wind farm
simulations using numerical methods. It has provided valuable information
on key aspects of the used model in relation to rotor configuration and power
controlling. Furthermore, it has added insight on the applicability of the model
in long domains. This information is vital if the aim is to simulate the flow
through large wind farms and farm-farm interaction. The project has also
validated the near wake features simulated by the actuator line method.
Acknowledgements
I would like to thank my supervisors Dan Henningson and Stefan Ivanell for
giving me this opportunity and for the guidance in the work leading to this
thesis.
I would also like to thank Jens N. Sørensen, Robert Mikkelsen, Wen Z.
Shen, Kurt S. Hansen and Søren J. Andersen at DTU Wind Energy for providing the numerical model, for invaluable input and for pleasant collaborations
and discussions throughout my PhD project.
I would like to thank my PhD student colleagues and Sasan Sarmast at
Uppsala Universitet Campus Gotland for fruitful discussions.
I would also like to thank all my colleages at Uppsala Universitet Campus
Gotland for the pleasant atmosphere. Andrew Barney is especially acknowledged for his endless patience in reading my articles and correcting my English
when needed. Simon-Philippe Breton is greatly acknowledged for the rewarding
collaborations and interesting discussions.
Antonio Segalini at KTH Mechanics is greatly acknowledged for reading
and commenting on the thesis.
I would like to thank all my colleagues in the Nordic Consortium for Optimization and Control of Large Wind Farms.
Thanks to Jan-˚
Ake Dahlberg at Vattenfall AB for providing measurement
data for the Lillgrund wind farm site.
Finally, I would like to acknowledge the Vindforsk III and Vindforsk IV
programs which together with H¨
ogskolan p˚
a Gotland and Uppsala Universitet
funded the present work. Thanks also to S¨
allskapet De Badande V¨
annerna
(DBW) for its huge generosity.
The computations were performed on resources provided by the Swedish
National Infrastructure for Computing (SNIC) at the National Supercomputer
Centre (NSC).
Sist, men inte minst, s˚
a¨
ar jag skyldig det st¨
orsta tacket till Jenny f¨
or hennes
¨
andl¨
osa st¨
od, k¨
arlek och t˚
alamod under alla dessa ˚
ar. Tack ocks˚
a till min familj
och alla andra som st˚
ar mig n¨
ara och berikar mitt liv, ni vet vilka ni ¨
ar!
37
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