An Experimental Investigation of Wind-farm Flows Blas Muro by

An Experimental Investigation of Wind-farm Flows Blas Muro by
An Experimental Investigation of
Wind-farm Flows
by
Blas Muro
July 2014
Technical report from
Royal Institute of Technology
KTH Mechanics
SE-100 44 Stockholm, Sweden
i
Abstract
In this thesis, fundamental properties of the turbulent flow above different
wind-farm models were determined by means of wind-tunnel measurements.
The assessed wind farms consisted in two staggered configurations, and two
inline configurations, where two different streamwise spacings were evaluated.
The experiment was focused on dense wind farms: the spacing in the spanwise
direction was fixed to approximately 3d (where d indicates the rotor diameters)
for every case, and two streamwise spacings were used: 2.5d and 5d. Freelyrotating turbines were used to perform this experiment. The wind-turbine
models had a diameter of 45 mm and a height of 85 mm from the ground to
the top tip. The wind-farm models were placed one at a time inside the test
section of the KTH NT2011 wind tunnel, where the inflow was completely flat,
i.e there was no simulated atmospheric boundary layer. X-wire anemometry
was the measurement technique to measure the streamwise and wall-normal
velocities above the wind farms. From the results, it could be observed that
close to the turbines, the streamwise mean velocity had variations in the spanwise direction for inline and staggered configurations, even deep downstream
on the wind farms. Horizontal averages were applied to the data to account for
the inhomogeneity of the properties above the wind farms. A scaling behaviour
was found on the flow above all the wind farms studied: in order to have the
streamwise mean velocity profile, all that was needed was the boundary layer
thickness, the free-stream velocity, and the streamwise velocity close to the
top tip of the turbines. Other scaling behaviours were found for the Reynolds
stresses. The dispersive stresses above different wind farms did not scale with
the friction velocity, since it was seen that these stresses are highly dependent
on the wind farm layout. Therefore, it was concluded that Reynolds stresses
and dispersive stresses cannot be compared to each other because they come
from different scales: the small scales and the large velocity scales, respectively.
An equation to estimate the friction velocity above wind farms was derived,
requiring measurements of the angular velocity of the turbines, the free-stream
velocity, and the mean velocities close to the tip of the turbines. Finally, it
was found that the angular velocity of the turbines was 25 % higher for the
staggered arrangement, when comparing it with an inline wind farm.
ii
Acknowledgement
First of all, I am thankful to the almighty God and Mary the Virgin for giving
me the necessary willingness and determination for completing this thesis.
A big thank you to my family, who were always there supporting me, and
cheering me up during the bad times. To my cousin Laura Urbani who helped
me in the reviewing process.To my cousins Carolina and Ricardo, who are not
anymore with us, I love you wherever you are.
A really special thank you to my supervisor Antonio Segalini for giving me
the golden opportunity of writing my master thesis in KTH. I would also like
to thank him for his daily and outstanding help in this project, for having the
needed patience when explaining totally new things to me, and for supporting
me in my career decisions. It was really fun to work with you.
I would like to thank my Alma mater Universidad Simón Bolı́var, that
despite the adversities in my country, they gave me the opportunity to go
abroad, and experience a fruitful exchange year.
I would like to acknowledge Jan Åke Dalberg for lending us his wind-turbine
models. To Ramis Örlü, for having the patience of building the X-wire probes
for this experiment. To my fellows Tomas Rosén and Marcus Winroth for those
good times in the office, and for teaching me Swedish. To all the guys in the
lab for the support, afterworks and Monday’s fikas. I will never forget you.
To my friends that made easier the way until the end, and also for making
special those great weekends in Stockholm, and around Europe. It was such a
wonderful time. Thank you all.
iii
Contents
Abstract
ii
Acknowledgement
iii
Chapter 1. Introduction
1.1. Wind-farm layouts
1.2. Motivations & aims
1
3
4
Chapter 2. Theoretical background
2.1. Basic fluid mechanic governing equations
2.2. Turbulence
2.3. Wind-turbine aerodynamics
2.4. Force model
7
7
9
17
21
Chapter 3. Experimental apparatuses & setup
3.1. Wind tunnel description
3.2. Prandtl-tube measurement
3.3. Photodiode
3.4. Hot-Wire anemometry
3.5. Traversing system
3.6. Wind-farm model
23
23
23
24
25
32
33
Chapter 4. Experimental procedure
4.1. Base flow measurements
4.2. Inflow-RPM correlation
4.3. Box measurements
4.4. RPM measurements
4.5. Farm measurements
36
36
38
38
44
44
Chapter 5.
45
Results and discussions
iv
5.1. Box 2 × 32.5d flow
5.2. Box 3 × 32.5d flow
5.3. Scaled Flow
5.4. Comparison between Reynolds stresses and dispersive stresses
5.5. Farm 2 × 32.5d case
5.6. Farm 3 × 32.5d case
5.7. Comparison between farm 2 × 32.5d and 3 × 32.5d
5.8. Force model validation
Chapter 6.
Summary and conclusions
Bibliography
45
48
51
53
57
62
67
69
72
75
v
CHAPTER 1
Introduction
Over the years, the study of wind energy has become more and more important.
The principal motivation of this trend has been the exponential growth of the
wind-energy industry in the last two decades (see Figure 1.1). The reason of
this growth is the fact that considerable amounts of electricity are generated
using the most clean and cheap fuel, as the wind.
Figure 1.1: Global cumulative installed capacity 1996-2013 (from gwec.net)
The global prosperity of this type of energy source can be considered as
a fact, since many wind farms have been installed in different parts around
the world such as Europe, America, and Asia (see Figure 1.2). Furthermore,
the trends have shown an increment of the wind-energy industry in developing
countries (like those of South America) over the last 3 years, observation that
generates an open question: is wind energy an expensive energy source that
only highly-developed countries can afford?
A wind turbine is a machine that converts the wind’s kinetic energy into
electrical energy. This process is obtained by the extraction of momentum
from the air (work done by the blades). The rotor is connected by a shaft to
a generator which produces electricity and is further distributed to the main
electric grid, and then to different places such as houses, companies, etc.
1
Figure 1.2: Annual capacity installed by region 2005-2013 (from gwec.net)
The evolution of wind turbines has a long history. There are evidences
that two hundred years BC a wind wheel was made by the Persians. But, it
was not until the 7th century that a windmill with practical purposes was built
in Iran. At that time, the windmills, as the name says, were used for milling
different kind of grains to make flour and for drawing up water from wells.
Later, in 1887 the first wind turbine was built by James Blyth in Scotland.
Afterwards, that same year an American scientist (Charles Brush) built the
first automatically operated 12 kW wind turbine. After that moment, it was
not until the middle of 1900s that people in the northern countries of Europe
(Denmark, Germany, and Sweden for instance) started to build high-power
turbines, and started gathering them in the now well known wind farms, with
the purpose of producing significant amounts of electrical power from the wind.
There are several types of wind turbines, but the most common are the
horizontal and vertical axis turbines. As the names show, those turbines only
differ in the shape while keeping their functions the same. In this thesis, only
the Horizontal Axis Wind Turbine (HAWT) is going to be studied and a brief
description of this one is presented. Horizontal turbines consist of a tower on
which a set of blades (rotor) and a generator are connected by a shaft, and
are installed perpendicular to the tower axis. It is also relevant to comment
that horizontal turbines are overall to be seen in wind farms, in contrast to the
vertical axis type because the former occupy less surface area.
2
Figure 1.3: An sketch of a Horizontal Axis Wind Turbine (from englishecoenergy.com)
1.1. Wind-farm layouts
A wind farm is a group of turbines placed close to each other in a certain
space with the purpose of producing energy. Wind farms could be organised in
infinite ways, but several applications and studies (Cal et al. (2010), Chamorro
et al. (2011), Chamorro & Porté-Agel (2011), etc.) show that the most common
layouts in wind farms are the staggered and the inline configurations.
It is called inline configuration when the turbines rows are organised one
behind the other. This configuration is important for the academic world due
to its simplicity to be tested and simulated. For this reason, this arrangement
is assessed in this experimental work. A sketch of this layout is shown in Figure
1.4.
Alternatively, it in a staggered configuration the turbines are organised so
that the wakes of one row do not affect the inflow of the very next row, which
makes this arrangement more efficient than its inline counterpart when referring
to the power produced by each turbine individually. In fact, it is important
to comment that the staggered configuration is the most used configuration
3
Figure 1.4: Inline Configuration of Turbines
in installed wind farms around the world, i.e. the wind farms are designed in
a way that the most probable wind direction faces a staggered arrangement.
For this reason, this arrangement is also assessed in this experimental work. A
sketch of this layout is shown in Figure 1.5. Curiously, if the flow direction in
an inline configuration is changed of 45◦ , the new wind farm layout would be
staggered.
Figure 1.5: Staggered Configuration of Turbines
1.2. Motivations & aims
Having now defined the most basic concepts, it is important to discuss what
other researches have shown in the past. First, it is relevant to comment
that wind turbines are designed to operate independently from each other.
However, in wind farms, turbines work in the wakes of other turbines. For this
reason, many studies about wake models have been done in the fields of layout
optimisations and fatigue prediction. One of the most famous wake model
was presented by Frandsen (1992). Also, other models have been presented as
the one in Barthelmie et al. (2007) and Frandsen et al. (2006). Furthermore,
experimental studies of these flows have been performed by different authors,
such as Chamorro & Porté-Agel (2009) where they demonstrate that behind an
isolated turbine, the near wake deficit is axisymmetric, and that high levels of
4
turbulence intensity are reached in the wake of the turbines. Also, in Medici &
Alfredsson (2008) and Medici & Alfredsson (2009), they studied the behaviour
of the 3D wake of an isolated turbine, finding that for two and three bladed
turbines, there is meandering in the wake at high rotational speeds.
The question “does the flow above a wind farm behave as the flow above
rough elements?” has been around in the community for some time, and some
researches have tried to model the wind farms as roughness applied to the
bottom of the Atmospheric Boundary Layer (ABL). The surface is characterised
by using a roughness length scale, which translates the effect of the wind farm
on the ABL. Some of the studies that have tried to characterise the wind-farm
flows using roughness lengths have been performed by Lettau (1969), Frandsen
(1992), Frandsen et al. (2006) and Calaf et al. (2010).
In many wind-farm studies, horizontal averages are usually used to simplify
the analysis of wind-farm flows. For example, Cal et al. (2010) and Calaf
et al. (2010) used the horizontal averaging to observe the behaviour of the flow
on different wind-farm arrays. However, when using horizontal averages, new
terms appear such as the Reynolds stresses; these terms come up when the
statistical data are not spatially homogeneous where the average is being done.
When studying wind-farm flows, different approaches to characterise the
boundary layer are used. For example, Chamorro & Porté-Agel (2011) performed hot-wire velocity measurements within and above a 3 × 3 inline configuration in a wind tunnel. They observed that there are two flow regions: one
from the base of the turbines to the top tip, and the other from the tip to the
free stream. Furthermore, the first region was fully developed after 4 turbine
rows, unlike the other one that never reach developed flow. Additionally, another approach to wind-farm flows was presented by Calaf et al. (2010), where
they observed that a better description of the flow would be given, if the flow
was decomposed into three different regions: the region above the turbines, the
rotor swept region, and the region between the lower tip and the base of the
turbine.
It is important to comment that the majority of the studies on windfarm boundary layers are performed by measuring the flow velocities on the
centreline of the wind farm. For this reason, the first objective of this thesis is
to perform velocity measurements above different wind-farm models accounting
for the spanwise variation of the flow properties.
As done by different authors, the space average of the flow properties is
going to be performed for each assessed configuration, and comparisons between
these quantities are going to be done, as well as comparisons of the dispersive
stresses for different cases. Furthermore, the growth of the space-averaged
boundary layer is going to be shown and compared between each of the cases
studied.
5
When describing the velocity profile on a wind farm, it is usually done by
calculating the roughness length of the model. However, there is a problem with
this approach, and that is the difficulty of replicating an atmospheric boundary
layer in the wind tunnel. Hence, an expression to estimate the velocity above
different wind-farm models without the necessity to have an ABL is going to
be derived.
6
CHAPTER 2
Theoretical background
In this chapter, the necessary concepts of this work are going to be introduced,
such as the equations of motion, basic fluid mechanics equations, an overview
of turbulence, and time and space averaging.
As in classical dynamics, the Newton’s 2nd law is the governing law for
the movement of an object F = m · a. In fluid mechanics the equations of
movement are the Navier-Stokes equations, which are the Newton’s 2nd law
applied to a control volume, and combined with the continuity equation.
2.1. Basic fluid mechanic governing equations
“The movement of fluids is the movement of a great amount of molecules from
one point in space to another as a function of time” Munson et al. (2009), but
studying each one of these molecules is considered really difficult, hence, the
definition of Fluid Particles came up. A fluid particle or parcel is a very small
amount of fluid, identifiable throughout its dynamic history while moving with
the fluid flow. The mass of the fluid particle is constant as it moves. This
definition states that the motion of a flow is described by particles of fluid
interacting with each other in space and time. Keeping this definition in mind,
and introducing a frame of reference (x, y, z), the velocity field of a particle can
be represented as in Equation 2.1 where u, v and w are the x, y and z velocity
components respectively.
u(t, x, y, z) = u(t, x, y, z)ı̂ + v(t, x, y, z)̂ + w(t, x, y, z)k̂,
(2.1)
In order to solve problems regarding fluid physics, questions about the
movement of the fluid are assessed in a bounded volume. These discrete volumes are called Control Volumes. The later concept is of high importance in
physics, since several fundamental laws of physics are applied on these control
volumes, such as Newton’s laws in a fluid, mass conservation, thermodynamic
laws, etc.
The mass conservation for a fixed, non-deforming control volume (Continuity Equation) is:
∂
∂t
Z
Z
ρ u · n̂ dA = 0,
ρ dV +
CV
S
7
(2.2)
which states that the time rate of change of the mass inside the control volume,
plus the net rate of mass flow passing through the surfaces of the volume is
equal to 0, since mass cannot be created nor destroyed.
Another important governing expression in fluid mechanics is the linear
momentum equation or Newton’s second law which states that in an inertial
frame of reference, the total force applied to a body is equal to the time rate
of change of momentum, namely
Fext =
∂Q
.
∂t
When applying the previous equation to a fluid in a fixed, non-deformable
control volume, the following equation is obtained:
Fext
∂
=
∂t
Z
Z
u ρ u · n̂ dA
ρ u dV +
CV
(2.3)
S
which states that the total external force is equal to the time rate of change
of the linear momentum inside the control volume, plus the net rate of flow of
linear momentum through the control surface.
When studying fluids, it is important to talk about energetic quantities.
In order to account for these contributions, the 1st law of thermodynamics is
used, which states that the time rate of the total energy of a given control
volume is equal to the sum of the rate of work done to what is inside the
control volume, plus the rate of heat supplied by the universe to the control
volume. In symbolic form, this statement is:
Q̇received + Ẇdone =
∂
∂t
Z
ρ (e+kuk2 /2) dV +
CV
Z
(e+kuk2 /2) ρ u· n̂ dA, (2.4)
S
where e is the internal energy per unit mass.
It is also important to note that, all the governing equations shown before are in Eulerian notation, i.e. the equations are defined for a fixed, nondeformable control volume.
The equations mentioned above were focused to describe the properties and
behaviour of a fluid within the boundaries of a control volume. Sometimes,
this approach is not enough when analysing fluids, because it does not take
into account variations of fluid properties such as pressure and velocity inside
the control volume. Therefore, if a detailed study of the fluid is required, a
differential analysis of the flow has to be done. The basics of this approach are
the same as the ones used in the control volume analysis, but in this analysis,
smaller (infinitesimal) control volumes are used to describe the fluid properties.
The mass conservation law (Continuity equation) in differential form is
given by the following expression:
8
∂ρ ∂ρu ∂ρv ∂ρw
+
+
+
= 0.
(2.5)
∂t
∂x
∂y
∂z
For incompressible flows the density is constant, and Equation 2.5 reduces
to:
∂w
∂u ∂v
+
+
=0
(2.6)
∂x ∂y
∂z
and the momentum conservation equation in the three coordinates reduce to:
∂u
∂u
∂u
∂u
1 ∂p
+u
+v
+w
=−
+ν
∂t
∂x
∂y
∂z
ρ ∂x
∂v
∂v
∂v
∂v
1 ∂p
+u
+v
+w
=−
+ν
∂t
∂x
∂y
∂z
ρ ∂y
∂w
∂w
∂w
1 ∂p
∂w
+u
+v
+w
=−
+ν
∂t
∂x
∂y
∂z
ρ ∂z
∂2u ∂2u ∂2u
+ 2 + 2
∂x2
∂y
∂z
∂2v
∂2v
∂2v
+
+
∂x2
∂y 2
∂z 2
− fx
(2.7)
− fy
(2.8)
∂2w ∂2w ∂2w
+
+
∂x2
∂y 2
∂z 2
− fz
(2.9)
where the left hand side of the equations represents the acceleration terms, and
the right hand side represents the force terms (pressure, shear, and body force).
Another important equation in fluid mechanics is the so called Bernoulli
Equation which states that for an incompressible inviscid flow, with conservative volume forces applied, the following expression is constant along a streamline:
P
kuk2
+
+ gZ = constant
ρ
2
(2.10)
It is important to remark that this equation could be seen as an energy
2
balance between kinetic energy kuk
2 , gravitational potential energy gZ, and
pressure potency energy Pρ .
2.2. Turbulence
Most flows in life are turbulent. For instance, flows at the boundaries of cars,
airplanes, ships, wind turbines, etc. Also, most internal flows are turbulent,
namely flows inside a pipe, inside internal combustion engines, and so on. It
is relevant to comment that this kind of flows are as complicated as they are
common. “Turbulent flows are chaotic and unsteady flows with high levels of
vorticity distributed along different sizes of eddies, characterised by high difusivity between fluid particles, and by high dissipation of energy into heat”
9
(From Ferro 2012). The study of turbulence began in 1883 when Reynolds
was involved in a work related to transition. At that time, he defined a nondimensional number (the Reynolds number) Re = ud
ν , and he noticed that
when this number reached a certain value in the flow inside a pipe (where u
is the velocity, d the pipe’s diameter and ν the fluid’s kinematic viscosity),
the flow became irregular. He noticed that, under that regime, the instantaneous flow characteristics were not predictable, but they were for the mean flow
quantities. For this reason, he suggested to divide the flow variables into mean
and fluctuating contributions. After some years of development (1895), he introduced the Time Averaged Navier-Stokes equations or Reynolds averaged
Navier-Stokes (RANS) equations.
Before introducing the RANS equations, it is important to introduce some
statistical tools for the analysis of turbulence.
2.2.1. Statistical analysis of turbulence
In order to understand turbulent flows, it is necessary to introduce a statistical
operator, as it is the time average. The time average of {qn (x, y, z, t); n = 1...N }
is the time integral of this property (q) in a certain time period, but for steady
flows, the time average becomes the arithmetic mean of a large number (N ) of
samples, as
q=Q=
1
T
Z
T
q(x, y, z, t)dt =
0
N
1 X
q(n)
N n=1
(2.11)
This is the usual form used for steady flow situations, but it can also be
used for non-stationary flows if the timescale for the mean flow variation is
much larger than the timescale of the turbulence fluctuations.
After introducing the time average, the Reynolds decomposition can be
written
0
q =Q+q ,
(2.12)
where the time series of a signal (q) is equal to the time-averaged value (Q),
0
plus the fluctuations (q ).
After defining the fluctuations, the variance of q is defined as the average
of the square of the fluctuation q 0 2 , and the fluctuation intensity or “rms”
value (standard deviation) is defined as the square root of the variance σq =
(q 0 2 )1/2 . On the other hand, when measuring two time series at the same time,
a correlation of the two different signals can be done, this correlation is called
covariance. Assuming that two time series of velocities are measured (u and v),
the covariance would be the average of the product of their fluctuations (u0 v 0 ).
Furthermore, when studying turbulence a correlation coefficient of these two
signals is usually computed in the following way:
10
u0 v 0
(2.13)
σu σv
It is important to comment that the flow under study in this thesis is considered steady, and consequently the time averages are going to be performed
by applying Equation 2.11 to the velocities time series. The formulas used for
calculating the time averages in this thesis are shown as follows:
corr =
u=U =
N
1 X
u(n)
N n=1
(2.14)
v=V =
N
1 X
v(n)
N n=1
(2.15)
u0 2 =
N
1 X 02
(u )(n)
N − 1 n=1
(2.16)
v0 2 =
N
1 X 02
(v )(n)
N − 1 n=1
(2.17)
σu = (u0 2 )1/2
(2.18)
σv = (v 0 2 )1/2
(2.19)
u0 v 0 =
N
1 X 0 0
(u v )(n)
N n=1
(2.20)
2.2.2. Reynolds averaged Navier-Stokes equations
The complexity and nonlinearity of the Navier-Stokes equations is the reason
why fluids are described in such a detailed way. Therefore, a simplification of
these equations is usually done by time averaging them. This simplification
was done by Reynolds, and these new equations are the so called Reynolds
Averaged Navier-Stokes equations or RANS.
2.2.2a. Cartesian Coordinates and Einstein Notation. In order to facilitate the handling of the equations a different coordinates system is defined:
u = uı̂ + v̂ + wk̂ = u1 ı̂ + u2 ̂ + u3 k̂
(2.21)
r = xı̂ + y̂ + z k̂ = x1 ı̂ + x2 ̂ + x3 k̂
(2.22)
11
With the aim of having a compact way to express the RANS equation,
Einstein notation is used. Next, some properties needed to understand this
notation are described:
ui ui = u1 u1 + u2 u2 + u3 u3
(2.23)
∂u1
∂u2
∂u3
∂ui
=
+
+
∂xi
∂x1
∂x2
∂x3
(2.24)
∂ 2 ui
∂ 2 ui
∂ 2 ui
∂ 2 ui
+
+
=
∂xj ∂xj
∂x21
∂x22
∂x23
(2.25)
Then, the Navier-Stokes equations and the continuity equation for incompressible flows in Einstein Notation become:
∂ui
∂ui
1 ∂p
∂ 2 ui
+ uj
=−
+ν
− fi
∂t
∂xj
ρ ∂xi
∂xj ∂xj
(2.26)
∂ui
=0
(2.27)
∂xi
By using the concept of dividing the flow variables into mean and fluctuating contributions, the so called Reynolds decomposition becomes:
0
0
ui = Ui + ui and p = P + p ,
(2.28)
0
where ui is the velocity time series, Ui is the time-averaged velocity and ui is
the velocity fluctuation of the ith velocity component.
2.2.2b. Reynolds averaged Navier-Stokes equations. The basic step of
the derivation is to express the Navier-Stokes equation as a function of fluctuating and mean properties, and then taking the average of the whole equation:
0
0
0
1 ∂(P + p0 )
∂ 2 (Ui + ui )
∂(Ui + ui )
0 ∂(Ui + ui )
+ (Uj + uj )
=−
+ν
∂t
∂xj
ρ
∂xi
∂xj ∂xj
and by averages properties, the first term gives:
0
0
0
∂(Ui + ui )
∂Ui
∂ui
∂Ui
∂ui
∂Ui
=
+
=
+
=
,
∂t
∂t
∂t
∂t
∂t
∂t
since the average of a fluctuating contribution is equal to 0.
Following the above properties, the second term becomes:
12
(2.29)
0
0
(Uj + uj )
0
∂u
0
0
0
0
∂uj ui
∂uj
∂(Ui + ui )
∂Ui
∂Ui
0 ∂u
0
= Uj
+ uj i = Uj
+
− ui
,
∂xj
∂xj
∂xj
∂xj
∂xj
∂xj
0
0
where ui ∂xjj = 0 because of
where it is stated that
∂uj
∂xj
∂uj
∂xj
= 0, which comes from the continuity equation,
= 0. Since the former relation is linear, it means that
the same condition holds for the mean velocity
∂Uj
∂xj
= 0, and the fluctuating
0
part
∂uj
∂xj
= 0.
The third and fourth term follow the same properties applied in the first
term. Thus, the final equation has the following form:
∂Ui
∂Ui
1 ∂P
∂
+ Uj
=−
+
∂t
∂xj
ρ ∂xi
∂xj
0
∂Ui
0
0
ν
− uj ui
∂xj
(2.30)
0
Being −ρ uj ui the so called Reynolds Stresses or turbulent stresses, which
have the form of a tensor. It is shown as follows:

0
0
u u
 1 1
0
0
−ρuj ui = −ρ  u02 u01
0
0
u3 u1
0
0
u1 u2
0
0
u2 u2
0
0
u3 u2

0
0
u1 u3
0
0 
u2 u3  .
0
0
u3 u3
The additional terms (Reynolds Stress) that came from averaging the nonlinear terms of the equation provide more unknowns to the equations of motion.
Therefore, the Navier-Stokes expression proceeds to get more unknowns than
the number of equations. This situation is known as the turbulent closure
problem.
It is important to comment that this problem is not going to be an issue
in this scientific work, because the flow properties are going to be measured.
Namely, all the statistical values, such as the mean flow, velocity fluctuations,
variances, covariances, etc. are going to be calculated from wind-tunnel data.
However, when a flow is simulated, the turbulence closure problem becomes
an issue, and different turbulence models have to be applied (See Johansson &
Wallin (2013)), but as it was mentioned before it is not a subject to consider
in this work.
13
2.2.3. Spatial and time averaged Navier-Stokes equations
Figure 2.1: Coordinates System
When measuring the velocity profile above a plate, spanwise invariance in the
flow is usually assumed. Namely, it is considered that the flow above the
whole plate is exactly the same as the flow in the middle of it. However,
when measuring the velocity profile above surfaces when there are objects on
it, as the case of canopies and wind farms, it is known that the flow is highly
three-dimensional, hence, this assumption would be false. But, because of its
simplicity, it is still widely used by many authors.
In order to avoid this assumption, but at the same time “keeping the
definition” of having the same flow characteristics in a certain space, a spatial
average of the flow quantities is done. In Figure 2.2, a schematic representation
of the space-average decomposition is shown. From the image, it can be seen
that the space-average decomposition is similar to the approach used for the
Reynolds decomposition, but in this case the fluctuations are in space and not
in time. Namely, it can be seen that the flow has different velocity values at
different positions, but the flow is divided in a constant part, plus a spacefluctuating one.
The measurement location taken into account for doing this spatial average
is the one surrounded by the red box shown in Figure 2.3, i.e. between two
rows of turbines. It is also important to comment that this average is done at
each Y measurement height, as if there was a horizontal slab (see coordinates
in Figure 2.1).
Raupach & Shaw (1982) presented the spatial and time-averaged NavierStokes equations for flows within a canopy, which in this thesis are being extrapolated for flows within wind farms. Before introducing the final expression
for these equations, it is necessary to introduce the averaging operator they
14
Figure 2.2: Spatial-average decomposition: a time averaged signal (U ) is equal
to the sum of the space averaged-signal (hU i), plus the fluctuations in space
00
(U ).
Figure 2.3: Horizontal Average
used for the derivation of the equations. The formal definition of the average
operator is
hΨi =
1
A
Z Z
Ψ(x) dx dz
(2.31)
R
where Ψ is a fluid property that is observed inside a region R (see Figure 2.3).
Also, it is important to comment that it is assumed that all the properties
being spatial averaged are already time averaged.
The spatial decomposition of the averaged properties are:
00
Ui = hUi i + Ui and P = hP i + P
00
(2.32)
where Ui is the time averaged velocity, hUi i is the spatial average of the time
00
averaged velocity, and Ui is the spatial fluctuation of the velocity.
The basic step of the derivation is to express the time averaged NavierStokes equations as a function of fluctuating and mean properties, and then
15
taking the spatial average of the whole equations (Raupach & Shaw 1982). The
time and space averaged Navier-Stokes equation in Einstein notation is given
by the following expression:
00
00
∂hUi i
∂
∂
∂hUi i
0
0
+ hUj i
+
hU U i +
hu u i =
∂t
∂xj
∂xj i j
∂xj j i
* 00 +
00
1 ∂hP i 1 ∂P
+ ν∇2 hUi i + νh∇2 Ui i
=−
−
ρ ∂xi
ρ ∂xi
(2.33)
where:
00
00
• hUi Uj i = hUi Uj i − hUi ihUj i, and it is called the Reynolds dispersive
stress,
00
• − ρ1 h ∂P
∂xi i is the form drag imposed by the wind turbines.
00
• νh∇2 Ui i is the viscous drag imposed by the wind turbines.
For simplicity, Equation 2.33 is going to be modified by defining
1
− hfi i = νh∇ Ui i −
ρ
2
00
*
00
∂P
∂xi
+
,
(2.34)
where −hfi i is an equally distributed force applied by the wind farm to the
flow.
Finally, the spatial and time averaged Navier-Stokes equations are:
00
00
∂hUi i
∂
∂
∂hUi i
0
0
+ hUj i
+
hU U i +
hu u i =
∂t
∂xj
∂xj i j
∂xj j i
1 ∂hP i
=−
+ ν∇2 hUi i − hfi i
ρ ∂xi
Figure 2.4: Wind farm control volume (side view)
16
(2.35)
When applying this equation to the flow inside the red control volume
shown in figure 2.4 (inside the wind farm), the following assumptions are done:
ii
• ∂hU
∂t = 0 since the flow is assumed to be steady.
i
• hU i ∂hU
∂x ≈ 0 because the flow is assumed not to be accelerating downstream in the wind farm.
• ν∇2 hUi i ≈ 0 because the viscous contributions are considered to be
small.
i
∂hU∞ i
by differentiating the Bernoulli equation in the
• − ρ1 ∂hP
∂x = hU∞ i ∂x
free stream, and by using the thin boundary layer approximation, which
states that the pressure does not change in the wall-normal direction.
The force to take into consideration is the force in x1 or x, which is:
00
00
00
00
∂hU∞ i
∂
∂ 0 0
∂
∂ 0 0
hfx i = hU∞ i
−
hU U i −
hu u i −
hU V i −
hu v i (2.36)
∂x
∂x
∂x
∂y
∂y
where the variations of dispersive and Reynolds stresses in the x direction are
assumed to be negligible, as well as, the contributions of the spanwise velocity
W , leading to the following expression:
00
00
∂hU∞ i
∂
∂ 0 0
−
hU V i −
hu v i
(2.37)
∂x
∂y
∂y
Integrating Equation 2.37 from the floor to the top tip of the turbines (from
G to 0 in Figure 2.4), the final expression of the force applied by the turbines
to the fluid is obtained:
hfx i = hU∞ i
00
00
∂hU∞ i hU V i(0) hu0 v 0 i(0)
Fx = hfx i = hU∞ i
−
−
∂x
h
h
(2.38)
where:
00
00
• hU V i(0) is the dispersive stress at the turbine-tip height.
• hu0 v 0 i(0) is the Reynolds stress at the turbine-tip height.
• h is the height of the turbine from the floor to the tip.
It is important to comment that the Reynolds stresses and the dispersive
stresses are equal to zero close to the floor, and for this reason they do not
appear in Equation 2.38.
2.3. Wind-turbine aerodynamics
A wind turbine is a machine that transforms kinetic energy from the air into
electricity. This process is done by the extraction of momentum with a propeller
(turbine) from the incoming air, obtaining rotational energy that is used to
generate electricity. This extraction of momentum implies a reduction of flow
17
velocity i.e. the velocity behind the turbine (wake) has to be slower. Physically,
the incoming flow perceives the presence of the rotor, and tries to avoid it. The
streamlines try to diverge from the rotor but some are forced to go inside the
rotor, and experience the extraction of momentum.
When the flow pass through the rotor it experiences a continued reduction
in velocity, and by Bernoulli’s theorem this monotonic reduction of velocity
can be expressed as a pressure jump in the rotor due to the extraction of
momentum. This difference in pressure creates a thrust force (T) to the rotor,
which could be expressed by the following expression:
T = Ad (Pu − Pd ),
(2.39)
where Ad is the swept area by the rotor, Pu is the pressure upstream the rotor
and Pd is the pressure downstream the rotor which is lower than its upstream
counterpart.
2.3.1. Actuator disk theory
When observing the behaviour of the flow passing through a HAWT, it is
common to assume that the rotor behaves like a frictionless porous disk with
the same rotor diameter, infinitesimally thin, and uniformly loaded i.e. T is
uniformly distributed in the disk area. This disk was defined by Froude and
Rankine as the actuator disk.
In order to show the theory, the first thing to do is to establish a control
volume. This control volume is defined as a stream tube control volume for the
sake of simplicity, since the surface-normal vector is perpendicular to the flow
direction (U · n = 0) at some boundaries (see Figure 2.5), and as a consequence
the surface terms in the conservation Equations 2.2, 2.3 and 2.4 vanish.
Formally speaking, the control volume should not have the actuator disk
inside it, but since all the properties vary continuously along the rotor except
the pressure, this control volume is taken accounting for the contribution of
the pressure discontinuity, in terms of an uniform force placed in the rotor, as
it is shown in Figure 2.5.
It is assumed that the velocity profile in each cross-section is uniform.
Moreover, the flow is assumed steady, incompressible and azimuthally homogeneous.
The next thing to do to derive the actuator disk theory is to apply the conservation Equations 2.2 (Mass conservation) and 2.3 (Momentum conservation
equation). By applying the mass conservation equation to the control volume
shown in Figure 2.5:
∂
∂t
Z
Z
ρ u · n̂ dA = 0,
ρ dV +
CV
S
18
(2.40)
Figure 2.5: Actuator Disk Control Volume (From Segalini & Ivanell (2013))
the first term becomes 0 due to the incompressibility condition, and the second
term becomes zero at the surface (S) of the stream tube; the remaining terms
are associated to the flow passing through the two caps of the control volume,
and are equivalent to the mass flow passing through the control volume:
Z
Z
Z
ρ Uinf dA −
ρ u · n̂ dA =
S
ρ Uw dA = 0,
(2.41)
Aw
A∞
ṁ = ρUinf A∞ = ρUw Aw
(2.42)
It is important to remark that the flow needs to expand in the wake due
to the reduction of velocity, phenomena that can be explained by looking at
Equation 2.42.
By recalling the momentum conservation equation:
Z
Z
∂
ρ u dV +
u ρ u · n̂ dA
(2.43)
∂t CV
S
and using a similar analysis to that used previously in the mass conservation
equation, the moment conservation becomes:
Fext =
Z
Fviscous + Fpressure + Fvolume − T =
Aw
ρUw2 dA −
Z
2
ρUinf
dA,
(2.44)
A∞
by assuming that the velocity profile is constant on Aw and A∞
Fviscous + Fpressure + Fvolume − T = ṁ(Uw − Uinf ),
19
(2.45)
where pressure forces are 0 because the integral of the pressure on the external
surface (S) is 0 (see Segalini & Ivanell 2013). It is stated that pw = p∞ , because
it is assumed that the pressure is recovered in the far wake, since flows in the
wake cannot sustain pressure gradients in steady conditions, and far from solid
boundaries. Additionally, volume forces in the horizontal axis are 0, and viscous
forces are 0, because the disk was assumed to be porous and frictionless. This
leads to:
T = ṁ(Uinf − Uw )
(2.46)
which is highly important, since the value of the total force applied by the
turbine to the fluid is represented on it.
A relation between the velocities Uw , Uinf and Ud , is needed because in
almost no case all the velocities are known. This relation is achieved by means
of Bernoulli’s theorem downstream and upstream the turbine.
Using Bernoulli’s equation (Eq 2.10) upstream the turbine, i.e. between
points 1 and 2 in figure 2.5, one gets:
2
Uinf
pu
U2
p∞
+
=
+ d,
ρ
2
ρ
2
(2.47)
doing the same between points 3 and 4,
pd
U2
p∞
U2
+ d =
+ w
ρ
2
ρ
2
(2.48)
subtracting both Bernoulli balances, the following expression is obtained:
pu − pd = ρ
2
Uinf
U2
− w
2
2
!
.
(2.49)
By introducing Equations 2.39 and 2.49 in Equation 2.46, the following
expression is obtained:
Ud =
Uinf + Uw
2
(2.50)
which indicates that the disk velocity is equal to the arithmetic mean between
the free stream velocity and the wake velocity. Therefore, this relation removes
one of the variables in the system, and the thrust force can be expressed as:
T = Ad ρ Ud (Uinf − Uw ) = ṁ(Uinf − Uw ) = Ad (pu − pd ),
20
(2.51)
2.3.2. Thrust coefficient
A normalisation of the disk force introduced in Equation 2.51 is done by defining
a characteristic dimensionless value called Thrust coefficient (Ct ), which is
given by:
Ct =
1
2
T
,
2
ρ Ad Uinf
(2.52)
in this case the ideal normalising factor is a thrust force equivalent to the
2
dynamic pressure of the free-stream velocity ( 21 ρ Uinf
) applied in the disk area
(Ad ).
Introducing Equation 2.51 in Equation 2.52, the Thrust coefficient gives:
Ct =
Ad ρ Ud (Uinf − Uw )
1
2
2 ρ Ad Uinf
(2.53)
2.4. Force model
A force model is derived in order to get the force distribution along the streamwise direction of the wind farm. The aim of this model is to provide valuable
information about the force applied by the turbines to the fluid, by using a simple analysis based on the actuator disk theory. So, the force could be related
with flow properties by inserting it in Equation 2.38.
Over the years, the development of force models for the Navier-Stokes
equations have shown a characteristic feature that is, many of them have been
defined as a characteristic velocity squared divided by a length scale i.e −hf i ∼
U2
L , where the length scale in canopy studies is usually the height of the canopy.
In this model, the same logic used for the canopies is going to be applied i.e
the length scale would be represented by the height of the wind farm. The first
step for the derivation of this model is to recall Equation 2.52:
1
2
,
T = Ct ρ Ad Uinf
2
where T indicates the uniformly distributed force applied on the disk area Ad .
In order to apply a similar
approach to wind farms, the expression has to
be dimensioned to sm2 , because the integrated Navier-Stokes equations have
these units (see Eq 2.38). When thinking about the variables of this force, the
following expression comes up to be:
hfx i ∼ f (T, Nt , CV ∗, ρ)
hmi
(2.54)
s2
where Nt is the number of turbines, CV* is the mini control volume bounded
by the region R used for the spatial average (figure 2.3). After observing the
21
variables dependence of this force, the following reasonable expression of the
streamwise force in a wind farm is derived:
Fx =
2
Nt T
Ct Nt Ad Uinf
=
Sx Sz h ρ
2 Sx Sz
h
(2.55)
where:
•
•
•
•
•
•
Sx = turbines spacing in the streamwise direction.
Nt is the number of turbines in this particular row.
Ct is the thrust coefficient of the turbines.
Uinf is the inflow velocity of the turbines.
Ad is the area swept by the turbine rotor.
Sz = Sx [(Nt max )Lz ] where Lz is the spacing in the spanwise direction
and Nt max is the maximum number of turbines in a wind farm row, i.e.
In a staggered arrangement 3 X 2 Nt max = 3.
• h = wind farm height.
22
CHAPTER 3
Experimental apparatuses & setup
The aim of this thesis is to analyse different fluid quantities in the flow above
wind farms, and the way to do this is by performing wind-tunnel experiments of
various wind-farm models. The measurements were carried out in the NT2011
wind tunnel located in the Mechanics department at KTH. In the following
sections, a description of the facility and the measurement techniques is going
to be shown.
3.1. Wind tunnel description
The NT2011 is an open circuit wind tunnel, which means that the flow goes
from the entrance to the exit of the wind tunnel, without being directly recirculated. However, the facility is located inside a closed room, so the flow
gets recirculated anyhow. The principal parts of the facility are the inlet, the
contraction, the test section, the diffusor, the fan and the outlet, as they can
be seen in the schematic model of the wind tunnel shown in Figure 3.1. The
wind tunnel is used at a speed around 8 m/s over the whole experiment. The
system is driven by a 15 kW DC fan located after the diffusor. The height of
the test section is 500 mm, the width is 400 mm and the length is 1400 mm. In
order to have a good flow quality in the test section (low levels of fluctuations),
honeycombs and screens are located at the entrance of the wind tunnel. The
honeycombs break turbulent eddies, and orient the flow to be parallel to the
axis of the test section.
3.2. Prandtl-tube measurement
A Prandtl tube is a pressure-based measurement instrument used to measure
the velocity of a fluid. It consists of a tube with two holes inside it, one of
which is located at the tip of the tube, while the other is located at the pipe’s
side. With this layout, both static and total pressure are measured. The total
pressure is measured by pointing the Prandtl-tube normal to the flow direction,
so that the hole located at the tip of the tube is a stagnation point (U = 0).
On the other hand, the static pressure is measured by the pipe’s side holes. A
representation of the used Prandtl tube can be observed in Figure 3.2. This
device measures the mean velocity of the flow by using Bernoulli’s theorem,
which states:
23
Figure 3.1: NT2011 Wind Tunnel
Pstatic
U2
Pstagnation
+
=
ρ
2
ρ
(3.1)
obtaining that the velocity is given by
s
U=
2(Pstagnation − Pstatic )
ρ
(3.2)
The Prandtl-tube is used for measuring a reference velocity, which is the
free-stream velocity at the inlet of the test section.
3.3. Photodiode
Photodiodes are devices that convert light into electric current. This device is
used to measure the angular velocity of the wind-turbine models. It is done
by pointing a red laser to a blade of the turbine with a certain angle, and
placing a photodiode behind the model for having contact between the laser
and the photodiode when the blade changes position as shown in Figure 3.3.
When this happens, a current is generated by the photodiode. Therefore,
the rotational speed of the turbines can be calculated from the photodiode
output by measuring the time between each pulse, and by relating it with the
displacement of the turbine blade.
24
Figure 3.2: Prandtl-tube
Since the wind turbine models are two bladed, each time that a pulse is
read, the turbine blade has done half a revolution. Thus, the angular velocity
ω of the turbine can be calculated from the following equation:
πf
(3.3)
z
where z is the distance between two peaks (number of points difference)
and f is the sampling frequency.
ω=
3.4. Hot-Wire anemometry
Hot-wire anemometry is a widely-used technique for measuring turbulent flows
with high precision. The principle of this technique is that, a body that is
exposed to a flow is mainly cooled by convective heat transfer, which in turn
is related to the flow velocity.
This apparatus is characterised by its microscopic dimensions. Nowadays,
normal hot-wires have diameters between 3-5 µm and length of about 1-2 mm.
These devices are composed by the microscopic wire mentioned before. The
wire is attached to two supports (prongs) by which current passes through,
heating the microscopic wire. A graphic representation of this device can be
observed in Figure 3.4.
25
Figure 3.3: Laser-Turbine-Photodiode arrangement
Figure 3.4: Single Hot-wire anemometer (From aawe.org)
3.4.1. Hot-Wire operating system
Two ways of operating hot-wire probes have prevailed over the years: constant
current anemometry (CCA) and constant temperature anemometry (CTA).
However, the most common, and the one used in this thesis is CTA. Further
information can be found in Perry (1982), Lomas (1986) and Bruun (1995).
26
Constant temperature anemometry (CTA) is when the hot-wire is kept at
a constant temperature i.e. when it has a constant resistance (since R is a
function of T ). In order keep a constant temperature, it is necessary to apply a
variable current to the hot wire. The principle of a CTA circuit is explained in
Figure 3.5. The hot wire is inserted in a Wheatstone bridge with an adjustable
resistance, and a part of the bridge is connected to an operational amplifier.
To cite Bruun (1995) “As the flow conditions vary the error voltage e2 − e1 will
be a measure of the corresponding change in the wire resistance. These two
voltages form the input of the operational amplifier. The selected amplifier has
an output current, i, which is inversely proportional to the resistance change
of the hot-wire sensor. Feeding this current back to the top of the bridge
will restore the sensor’s resistance to its original value”. An offset value eof f
is applied on one side of the amplifier, which gives a constant current to the
bridge, making the temperature of the wire under no flow conditions dependent
on the value of the variable resistance R3 (Figure 3.5).
Figure 3.5: CTA circuit (Electronic testing sub-circuit, Wheatstone bridge and
a feedback amplifier (G) (From Bruun 1995)
It is important to comment about what is the relation between the obtained
signal and the flow velocity, and for this, it is necessary to apply heat-transfer
principles. Specifically, it is fundamental to know the behaviour of a heated
cylinder located in a stream, which can be seen in many heat transfer books as
Incropera & Dewitt (1981), Cengel (2007) or in hot-wire anemometry books as
Bruun (1995). It is also important to say that the heat transferred to the flow
by a hot-wire is mostly convective, and it is proportional to the power generated
by the resistance and the voltage of the hot-wire. This is represented by the
following equation:
27
Qh + Qr + Qc =
2
Ew
Rw
(3.4)
where Ew is the voltage difference in the hot-wire, and Rw is the hot-wire
resistance. If the heat transferred by radiation Qr , and by conduction to the
prongs Qc are considered small in comparison with the heat transferred by
natural and forced convection Qh , the expression 3.4 becomes:
Qh = π d h L (Tw − Ta ) ≈
2
Ew
Rw
(3.5)
where:
• d = hot-wire diameter.
• h = convective heat-transfer coefficient which depends on the stream
velocity (different correlations between h and the flow velocity can be
found in Incropera & Dewitt (1981))
• L = hot-wire length.
• Tw = hot-wire temperature.
• Ta = ambient temperature.
3.4.2. X-Wire probes
To measure the velocity in the streamwise u and in the traverse (wall-normal)
v direction, a single hot-wire is not enough because it can just measure one
velocity component. For this reason, an X-wire is used, which is composed by
two single hot-wires placed on the same probe. This kind of hot wire allows
to measure two velocity components at the same time, making possible the
calculation of the covariance between these (Reynolds stresses).
The voltage response of a generically oriented hot-wire probe is dependent
on the magnitude of velocity kuk and the yaw angle α, so that:
1
E = f (kuk, α) = A + BVen
(3.6)
where Ve is the effective velocity, which represents an equivalent normal flow
velocity that reproduces the same output voltage, and it is defined as:
Ve = kukf (α)
(3.7)
where f (α) is a yaw function (Bruun 1995), which is usually assumed to be:
1
1
Ve = kukf (α) = kuk(cos2 (α) + k 2 sin2 (α)) 2 = (u2n + k 2 vn2 ) 2
with
28
(3.8)
un = u cos(α) − v sin(α)
(3.9)
ut = u sin(α) + v cos(α)
(3.10)
where it is assumed that the binormal velocity component w is small. The velocities and angles showed in Equations 3.9 and 3.10 are sketched in Figure 3.6.
After, introducing Equations 3.9 and 3.10 in Equation 3.8 for both wires,
doing the algebra, and assuming that v 2 /u2 is small, the following expressions
are obtained:
Ve1
Ve2
q
q
tan(α)(1 − k 2 ) cos2 (α) + k 2 sin2 (α)
(3.11)
= u cos2 (α) + k 2 sin2 (α) − v
1 + k 2 tan(α)
q
q
tan(α)(1 − k 2 ) cos2 (α) + k 2 sin2 (α)
2
2
2
(3.12)
= u cos (α) + k sin (α) + v
1 + k 2 tan(α)
by rearranging these equations, it is possible to get:
U1 = q
U2 = q
Ve1
tan(α)(1 − k 2 )
= u − vg(α)
1 + k 2 tan(α)
(3.13)
tan(α)(1 − k 2 )
= u + vg(α).
1 + k 2 tan(α)
(3.14)
=u−v
cos2 (α) + k 2 sin2 (α)
Ve1
=u+v
cos2 (α) + k 2 sin2 (α)
By isolating both streamwise and vertical (wall-normal) velocity, and by
knowing that the wires are inclined with the same angle but in opposite directions, it can be obtained that
u=
U1 + U2
2
(3.15)
and
v = (U1 − U2 )
1 + k 2 tan(α)
2 tan(α)(1 − k 2 )
(3.16)
3.4.3. Calibration
The last step before starting the measurements is the calibration of the X-wire
probe. It is important to remember at this stage that, in spite of having the
Equations 3.15 and 3.16, no information about the velocities could be obtained
before doing a correlation between the voltages read by the hot-wire, and the
velocity of the flow. Two calibrations have to be performed: a conventional
29
Figure 3.6: Sketch of the velocity components over both hot wires
hot-wire calibration and an angular calibration . The first calibration consists in
placing the hot-wire in an uniform stream, and measuring the voltages obtained
by each wire, meanwhile a reference velocity using a Prandtl-tube is stored.
30
This calibration is done at the beginning and at the end of each measurement
day. In Figure 3.7, it can be seen a characteristic curve taken from this first
calibration.
10
10
U 2 [ m/s ]
15
U 1 [ m/s ]
15
5
5
0
0
1.4
1.6
E1 [ V ]
1.8
2
1.4
1.6
E2 [ V ]
1.8
2
Figure 3.7: Calibration between the voltages E1 and E2 and the free-stream
velocity. The solid and dashed lines represent 3rd degree polynomials that
relate E1 and E2 with the free-stream velocity, respectively.
Since the conventional hot-wire correlation is done in the empty test section of
the wind tunnel, it is expected that the wall-normal velocity is equal to zero
(v = 0). For this reason, the following relations can be used:
U1 = u = f (E1 )
(3.17)
U2 = u = f (E2 )
(3.18)
where it is assumed that U1 and U2 are 3rd degree polynomials, which can be
seen as the solid and dashed lines in Figure 3.7. It is interesting to comment
that the values of the voltage E1 and E2 are almost the same.
The second calibration consists on placing the X-wire probe in an angular
calibration facility (Figure 3.8) that allows to vary the velocity angle and the
velocity magnitude, so that v 6= 0. Therefore, the value of the constant k can be
calculated, by comparing the voltage signals with a reference velocity provided
by a Prandtl-tube located inside the facility. Moreover, rewriting Equations
3.15 and 3.16 in a easier form to make the calibrations, they give:
31
Figure 3.8: Angular calibration facility.
u=
U1 + U2
G1 = uest G1
2
(3.19)
and
v = (U1 − U2 )
U1 − U2
1 + k 2 tan(α)
=
G2 = vest G2
2
2 tan(α)(1 − k )
2
(3.20)
When comparing vest with vref , the slope G2 could be calculated. The
calibration is showed in figure 3.9.
Finally, from the slopes of the 1st degree polynomials in Figure 3.9, it was
found that G1 = 1 as expected, and G2 = 0.9708.
3.5. Traversing system
In order to move the X-wire inside the wind tunnel, it is necessary a traversing
system that allows movement in the three directions (streamwise, spanwise and
traverse (wall-normal)). This system is composed by an electronic traversing
in the y direction, and two manual traversing systems in the streamwise and
spanwise directions. It was possible to traverse either in the streamwise and
traverse direction, or in the traverse and spanwise direction. For this reason
a traversing system capable to be moved in the three directions (Figure 3.10)
was built.
32
16
8
6
14
4
12
v r e f [ m/s ]
u r e f [ m/s ]
2
10
0
−2
8
−4
6
−6
4
0
5
10
15
−8
−10
20
u e s t [ m/s ]
−5
0
5
10
v e s t [ m/s ]
2
Figure 3.9: Angular correlation. vest = U1 −U
and uest =
2
between ve st and vr ef , and between ue st and ur ef .
U1 +U2
.
2
Comparison
Figure 3.10: Traversing System
3.6. Wind-farm model
In order to perform the study of flows above wind farms, a model of such
systems have to be built. Therefore, different objects are placed in the test
33
section to simulate a wind farm. These objects are showed in Figure 3.11 and
described as follows:
• A wooden plate of dimensions 1000 × 390 × 15 mm3 with a rounded
leading edge attached to the floor of the test-section.
• A steel plate of dimensions 990 × 390 × 15 mm3 attached to the wooden
plate.
• Wind turbine models of φ = 45 mm with a hub height of H = 65
mm. These turbines were provided by Jan-Åke Dahlberg (see Hägglund
2013).
Figure 3.11: Wind farm model
The used freely-rotating HAWT models were made of steel with a plastic
rotor, and a magnet placed in the base for attaching to metal surfaces. The
properties of the blades are described by Hägglund (2013). These turbines
have a rotor diameter of φ = 45 mm, a total height from base to the top tip of
h = 85 mm, and a thrust coefficient described in Figure 3.12.
34
Thrust Coefficient
0.63
0.62
CT
0.61
0.6
0.59
0.58
0.57
500
1000
1500
ω [ r ad /s ]
2000
Figure 3.12: Thrust coefficient vs Angular velocity from Hägglund (2013)
It is important to comment that the values of the thrust coefficient at
different angular velocities are between 0.55 and 0.62, similar to the values of
real-scale wind-turbine thrust coefficients (between 0.8 and 0.5) at velocities
around 8 meters per second.
35
CHAPTER 4
Experimental procedure
The experiments done in this thesis were divided in two measurement campaigns. The first one was performed in february 2014, and the other one in
april 2014. In the first measurement campaign, several wind farm configurations were assessed assuming spanwise homogeneity, i.e it was assumed that
the flow in all the spanwise positions had the same behaviour, thus just measurements in the centreline of the wind farms were done.
In almost every inline configuration tested in the wind tunnel, a turbine
was always placed underneath the measurement point i.e there was always a
turbine located at Z = 0 in each row of the wind farm, and for that case the
growth of a boundary layer could be clearly seen. However, when there were
no turbines present in the measurement plane, it was seen that there was no
boundary layer growth, demonstrating that the spanwise homogeneity was not
true for all the studied cases.
These results led to the realisation of the second measurement campaign
where several measurements were done, taking into account the spanwise nonhomogeneity i.e measuring the velocity profiles at different spanwise positions,
and at different streamwise positions (as the first measurement campaign). In
order to perform these measurements, a special traversing system capable to
move in the spanwise direction had to be built. This traversing system was
previously shown in Figure 3.10.
In order to achieve a complete description of the flow in the study, five
different experimental layouts were used in the second measurement campaign,
and they are described as follows:
•
•
•
•
•
Base flow measurements.
Inflow-RPM correlation.
Box measurements.
RPM measurements.
Farm measurements.
4.1. Base flow measurements
In this experiment a characterisation of the flow is done when the wind tunnel
has nothing in its test-section but the ground plate. One important factor is
that the voltage at which the wind tunnel fan is set up for this experiment, is
36
Y /h
the same for all the subsequent experiments. Measurements of the statistics of
the flow were taken at x= 500 mm (the middle of the test section) from the free
stream to a region close to the plate in the wall-normal direction. Moreover, it
is relevant to say that the aim of this experiment is to observe the flow quality of
the wind tunnel. These measurements were done during the first measurement
campaign.
After performing the measurements, profiles of different flow statistics were
computed and shown in Figure 4.1.
0.2
0.2
0.2
0
0
0
−0.2
−0.2
−0.2
−0.4
−0.4
−0.4
−0.6
−0.6
−0.6
−0.8
−0.8
−0.8
−1
0.85 0.9 0.95 1
U /U ∞
1.05
−1
−1.5
−1
−0.5
0
−3
2
u ′ v ′/U ∞
x 10
−1
0
0.05
σ u /U ∞
0.1
Figure 4.1: Base flow characteristics. h is the height of the turbine from the
floor to the tip of the blade. The dashed lines represent the area swept by the
rotor, and the solid lines indicate the turbine’s hub height.
What is important to point out in Figure 4.1 is that the flow conditions
are optimal in the area of interest: the mean velocity profile is uniform within
the turbine rotor, and the shear stresses and turbulent fluctuations are 0 in
this region, which means that there is no external perturbation in the flow.
However, a small boundary layer due to the ground plate is observed.
37
4.2. Inflow-RPM correlation
In this experiment, a correlation between angular velocity and inflow velocity,
and between angular velocity and disk velocity is obtained. This was done by
placing a turbine in the test section of the wind tunnel at different flow velocities; the angular velocity of the turbine was obtained by a laser-photodiode
arrangement, and the disk and inflow velocity were measured using hot-wire
anemometry. In Figure 4.2 a relation between the velocities and the angular
velocity can be seen.
10
Uinf
u [ m/s ]
8
U
disk
6
4
2
0
0
500
1000
ω [ r ad /s ]
1500
2000
Figure 4.2: Correlation between ω and Udisk and Uinf . The solid line and the
dashed lines are third degree polynomials used to correlate the inflow velocity
Uinf and the disk velocity Udisk with the angular velocity ω respectively.
The angular velocity ω is used to calculate the force Fx in the streamwise
direction using the model proposed in this thesis (Equation 2.55). Also, the
value of the angular velocity in each row of turbines is highly important to
determine the behaviour of the flow (accelerations, decelerations, peaks, etc).
4.3. Box measurements
In this experiment the statistics of the flow are measured between two rows
of turbines using a big amount of measurement points in the three directions
(spanwise, streamwise and traverse). The goal of this experiment is to do an
analysis of the statistics in this domain, so the distributions of these quantities
can be known, providing some information about the optimal location of the
measurement points for the farm measurements.
38
It is important to comment that the region between two rows of turbines
is located deep downstream, therefore, valuable data of the distribution of flow
quantities in that region are going to be analysed. This type of experiment was
done in the second measurement campaign.
Four different box measurements were performed on four different wind
farm arrangements. These configurations are as follows:
• 2 × 32.5D : this wind farm arrangement consisted on a series of 9 rows by
2-3 columns of wind-turbine models with a spacing of 2.5 diameters in
the streamwise direction and a spacing of 2.88 diameters in the spanwise
direction.
• 3 × 32.5D : this wind farm arrangement consisted on a series of 9 rows by
3-3 columns of wind-turbine models with a spacing of 2.5 diameters in
the streamwise direction and a spacing of 2.88 diameters in the spanwise
direction.
• 2 × 35D : this wind farm arrangement consisted on a series of 5 rows by
2-3 columns of wind-turbine models with a spacing of 2.5 diameters in
the streamwise direction and a spacing of 2.88 diameters in the spanwise
direction.
• 3 × 35D : this wind farm arrangement consisted on a series of 5 rows by
3-3 columns of wind-turbine models with a spacing of 2.5 diameters in
the streamwise direction and a spacing of 2.88 diameters in the spanwise
direction.
A graphic representation of these arrangements is shown in Figure 4.3.
Figure 4.3: Graphic representation of all the box measurements done in this
thesis.
In order to manage the data in terms of spatial-averages, the minimum
number of measurement points were needed to be known along with their position in a certain region for doing a good spatial average in terms of time and
39
flow description. That was one of the main reasons why the box measurements
were done.
The first measured case was the 2 × 32.5d case, where the normal assumption of symmetry in the spanwise direction was assessed i.e. measurements at
both positive and negative values of Z were done. The measurement matrix
is shown in Figure 4.4, where h = 85 mm is the wind turbine height from the
bottom to the top tip, and d = 45 mm is the rotor diameter. An analysis of
the flow in this region was done and it was found that the peaks in velocity for
this box measurement were located at Z ≈ 0, and at Z ≈ −0.7d in the spanwise direction, and at X ≈ 12.5d and X ≈ 13.7d in the streamwise direction,
respectively.
Z /d
0
−1
−2
−3
12
12.5
13
13.5
12.5
13
13.5
14
14.5
15
14
14.5
15
1
Y /h
0.5
0
−0.5
−1
12
X/d
Figure 4.4: 2 × 32.5d case measurement matrix
Since the definition of the spatial average operator (in Equation 2.31) seems
to be practically complex, a simplification can be done by defining the spatialaverage operator as the arithmetic average of certain measurement points located inside the region of study, as it was done by Cheng & Castro (2002).
In order to confirm that the arithmetic average can be used as the spatialaverage operator, both methods were used and compared. A quadrature scheme
was used to numerically solve the integration shown in Equation 2.31. Furthermore, the arithmetic mean was calculated for the points where the peaks in
40
velocity were located. The comparison between each average is presented in
Figure 4.5.
80
Y [ mm ]
60
Integration
Many points
4 points
40
20
0
6
6.5
7
7.5
U [ m/s ]
Figure 4.5: Averaging method comparison for U (staggered case)
From this comparison, it could be concluded that the simplification of using
an arithmetic average (Many points) instead of the formal definition of spatialaverage can be used, because the variations of averaged velocity between the
two methods are small. Moreover, it was observed that by averaging four points
where peaks in velocity were found, a good spatial-average can be obtained. For
this reason, just 4 points per row were going to be used for the spatial-average
in the Farm measurements.
The same analysis was done for the variance of the mean velocity (σu2 )
using the same locations for the 4 points average. The comparison between the
methods is shown in figure 4.6.
Since it was proved that the arithmetic average of certain points over a
region is a good approximation for the spatial-average integral operator, the
same method was used to calculate the spatial average of an inline 3×32.5d configuration. But, as it is known, flows over inline arrangements have a different
behaviour in comparison to the flows over staggered configurations. Therefore,
the locations for the 4 points average has to be modified.
By assuming spanwise symmetry, measurements are going to be performed
on just one side of the plate. The measurement matrix for these measurements
is shown in Figure 4.7.
An analysis of the flow in this region was performed, and it could be seen
that the peaks in velocity were now located at Z ≈ 0, Z ≈ −1.5d in the
spanwise direction, X ≈ 12.5d and X ≈ 13.7d in the streamwise direction. The
41
80
Y [ mm ]
60
Integration
Many points
4 points
40
20
0
0
0.1
0.2
0.3
σ u2 [ m 2 /s 2 ]
0.4
0.5
Figure 4.6: Averaging method comparison for σu2
1
Z /d
0
−1
−2
−3
−4
12
12.5
13
13.5
12.5
13
13.5
14
14.5
15
15.5
14
14.5
15
15.5
1
Y /h
0.5
0
−0.5
−1
12
X/d
Figure 4.7: 3 × 32.5d case measurement matrix
comparison between the arithmetic average of all the points, and the arithmetic
average of the points where the peaks were located is shown in Figure 4.8. A
difference on the velocity values between the two methods can be seen, but it is
42
considered small. Therefore the 4 points average was used for space averaging
inline configurations in the Farm measurements.
80
Many points
4 points
Y [ mm ]
60
40
20
0
6
6.5
7
7.5
U [ m/s ]
Figure 4.8: Averaging method comparison for U (Inline case)
The same analysis was done for the variance of the mean velocity (σu2 )
using the same locations for the 4 points average. The comparison between the
methods is shown in figure 4.9.
80
Many points
4 points
Y [ mm ]
60
40
20
0
0
0.2
0.4
σ u2 [ m 2 /s 2 ]
0.6
0.8
Figure 4.9: Averaging method comparison for σu2 (inline case)
43
4.4. RPM measurements
In these measurements the angular velocity of the turbines is calculated by
using the system explained in section 3.3. The arrangement showed in Figure
3.3 is used in every row of turbines in order to measure the voltage pulses,
which later are transformed in angular velocities using expression 3.3. The
angular velocity was measured in just one turbine per row, and it was further
assumed that the angular velocity was constant throughout the row. The
angular velocity was then related to the inflow velocity of each turbine by
the use of the correlation in Figure 4.2, so that the force model showed in
equation 2.55 could be used. The cases assessed in this experiments were the
2 × 32.5D , and the 3 × 32.5D cases previously described in Box Measurements.
4.5. Farm measurements
One of the aims of this thesis is to observe the velocity profiles above different
wind farm arrays, taking into account the spanwise non-homogeneity of the
flow, by measuring the velocity time series in specific spanwise, and streamwise
positions at different heights.
The optimal places for measuring the velocities above the different wind
farms were selected from the Box Measurements by observing where the velocity maxima and minima were located. The motivation for measuring in the
points where the peaks in velocity were located was to obtain good spatial
averages using the minimum amount possible of measurement points, so less
time-consuming measurements could be performed.
The growth of the boundary layer above the different assessed wind farms
is to be measured, together with some turbulent properties of the flow. Furthermore, calculations of the force applied by the turbines to the fluid are to
be performed and, as it can be seen in Equation 2.38, space-averaged values of
the statistics are required. After, this force is to be compared with the force
proposed in Equation 2.55.
The assessed cases in this experiment were the 2 × 32.5D , and 3 × 32.5D
wind farm configurations.
44
CHAPTER 5
Results and discussions
5.1. Box 2 × 32.5d flow
A contour plot of the velocity distribution at Y = 0 (close to the turbine tip)
was done (Figure 5.1) with the aim of observing the velocity distribution at
the turbines’ tip.
1
0.5
0.95
Z /d
0
0.9
0.85
−0.5
0.8
−1
0.75
−1.5
12.5
0.7
13
13.5
X/d
14
Figure 5.1: U/U∞ at Y = 0 for the 2 × 32.5d case
There are 3 important observations regarding Figure 5.1. The first one is
that the assumption of symmetry in the spanwise direction is not completely
true. Variations smaller than 5 % in velocity could be seen between two opposite spanwise locations (e.g Z = 0.6d and Z = −0.6d) but, the spanwise
symmetry assumption can still be used, since these variations are not big.
The second important observation is the presence of a Venturi effect between Z = −0.6d and Z = −0.8d, which corresponds to half the spanwise
distance that separates two turbines of two different rows. It is called Venturi
effect when an acceleration in the flow is observed; in this region this effect is
caused by the turbine’s wake interaction
45
The third relevant observation is that the velocity exactly in between two
turbines, located on the same row (Z = −1.5d), is approximately the velocity
on top of the turbine in the centre, which means that the wake recovery between
two rows is quite small. This fact could also be seen when observing that the
velocity change in the streamwise direction was almost zero at a fixed spanwise
position.
Another information from the Box measurement 2 × 32.5d which is interesting to show is the velocity distribution over the 6th (12.5d) row of turbines
in the Z − Y plane, as shown in Figure 5.2. From Figures 5.1 and 5.2 large
velocity variations in the spanwise direction can be observed (see velocity at
Z/d ≈ 0 and at Z/d ≈ −0.7 at Y ≈ 0). It is important to remark that
these measurements are done in a place were developed flow is assumed to be
reached. The observation of variations on the streamwise mean velocity under
developed regime is important, because the assumption of spanwise invariance
of the streamwise mean velocity cannot be used for staggered configurations,
contradicting what was stated by Chamorro et al. (2011). Also, large differences in the Reynolds stresses at different spanwise positions can be seen in
Figure 5.3.
0.5
1
0.4
0.95
0.9
Y/h
0.3
0.85
0.2
0.8
0.1
0.75
0
0.7
0.5
0
−0.5
Z /d
−1
−1.5
Figure 5.2: U/U∞ at X = 12.5d for the 2 × 32.5d case
It is interesting to show the contributions of the dispersive, and the spaceaveraged Reynolds stresses above the turbines. These are shown in Figure 5.4,
where it can be seen that the contribution of the Reynolds stresses is higher
than the contribution of the dispersive stresses at low heights, and they both
tend to zero after a certain height. In Figure 5.4, the friction velocity is defined
0
0
00
00
as u2∗ = −hu0 v 0 i(0). Also, the term U ∗∗ is equal to U ∗∗ = huuv2 i + hU uV2 i , and
∗
∗
shows the contribution of the dispersive and Reynolds stresses, where in this
case the Reynolds stresses dominates.
46
−3
0.5
x 10
−0.5
0.4
−1
−1.5
Y/h
0.3
−2
0.2
−2.5
0.1
−3
0
−3.5
0.5
0
−0.5
Z /d
−1
−1.5
2
at X = 12.5d for the 2 × 32.5d case
Figure 5.3: u0 v 0 /U∞
0.4
Reynolds st
Dispersive st
U**
Y /h
0.3
0.2
0.1
0
−1
−0.5
0
0.5
Figure 5.4: Comparison between the Reynolds stresses (hu0 v 0 i/u∗ ) and the
00
00
dispersive stresses (hU V i/u∗ ) for Box 2 × 32.5d measurements. Also, the
sum of both contributions is represented by U ∗∗ .
47
5.2. Box 3 × 32.5d flow
A contour of the velocity profile at Y = 0 for the 3 × 32.5d case was done (figure
5.5). It is inferred that the velocity exactly in between two turbines at the
same row is approximately the free-stream velocity i.e. the flow is unperturbed
at that spanwise location, which means that the wake interaction between the
turbines located in the same row is small, despite the fact that this experiment
was set up with small spacings in both streamwise and spanwise directions,
in comparison with other experiments performed in the past (Chamorro &
Porté-Agel 2011).
0
1
0.95
−0.5
Z /d
0.9
0.85
−1
0.8
0.75
−1.5
12.5
0.7
13
13.5
X/d
14
Figure 5.5: U/U∞ at Y = 0 for the 3 × 32.5d case
The Venturi effect due to wake interaction observed in Section 5.1 is visible
exactly in between two turbines at the same row. However, what can be seen
here is that the flow velocity exactly in between two turbines is the free-stream
velocity, i.e. the flow is not perturbed by the turbines at this location. Additionally, inside the wake of the turbine a small decrease on the streamwise
velocity in the streamwise direction could also be seen.
Another information from the Box Measurement 3 × 32.5d which is interesting to show is the velocity distribution over the 6th (12.5d) row of turbines
in the Z − Y plane, as shown in Figure 5.6. In this picture, large velocity variations in the spanwise direction can be seen. The velocity discrepancies reach
values up to 30 % at different spanwise locations. It is important to remark
that these measurements are done in a place were developed flow is assumed to
be reached. Furthermore, when comparing the velocity variations in this configuration with the ones in the staggered case, it could be seen that the velocity
variations are higher, and the wake interaction is reduced for the inline case.
Therefore, it can be stated that a bigger wake interaction is observed in the
48
staggered case, despite the fact that the turbines spacings between the inline
and staggered configurations were exactly the same.
0.5
1
0.4
0.95
0.9
Y /h
0.3
0.85
0.2
0.8
0.1
0
−1.5
0.75
0.7
−1
−0.5
0
Z /d
Figure 5.6: U/U∞ at X = 12.5d for the 3 × 32.5d case
Large differences in the Reynolds stresses in the spanwise direction could
be determined in Figure 5.7. When observing this figure and comparing it with
Figure 5.3, it is found that higher values of Reynolds stresses are present in
the inline case at the position [Y, Z] = [0, 0], but it could also be seen that
the Reynolds stresses are more evenly distributed in the spanwise direction for
staggered configurations.
−3
x 10
0.5
−1
0.4
−2
Y /h
0.3
−3
0.2
−4
0.1
0
−1.5
−5
−1
−0.5
0
Z /d
2
Figure 5.7: u0 v 0 /U∞
at X = 12.5d for the 3 × 32.5d case
49
The contributions of the dispersive and the space-averaged Reynolds stresses
above the turbines are shown in Figure 5.8, where it can be seen that the contribution of the dispersive stresses are much higher in comparison with the
staggered case shown in Figure 5.4. Therefore, when summing the Reynolds
and dispersive stresses contributions up, they practically cancel each other.
This can also be observed when comparing the plotted U ∗∗ term between Figures 5.4 and 5.8 .
0.4
Reynolds st
Dispersive st
U**
Y /h
0.3
0.2
0.1
0
−3
−2
−1
0
1
0
0
Figure 5.8: Comparison between the Reynolds stresses ( huuv∗ i ) and the disper00
00
sive stresses ( hU uV∗ i ) for Box 3 × 32.5d measurements. Also, the sum of both
contributions is represented by U ∗∗ .
50
5.3. Scaled Flow
5.3.1. Scaled velocity profile
After performing all the box measurements mentioned in Chapter 4, and plotting the spatial-averaged streamwise mean velocity (hU i) at different heights,
as it is shown in Figure 5.9, it was decided to plot these profiles using a characteristic scaling as it is usually done for boundary layers over flat plates.
80
2 × 32.5d
2×3
Y [ m ]
60
5d
3 × 32.5d
3 × 35d
40
20
0
6
6.5
7
7.5
h U i [ m/s ]
8
Figure 5.9: Space-averaged velocity profiles of the different assessed box measurements.
By trying to normalise the velocity profile with a velocity scale, and the
wall-normal position with the the boundary layer thickness, the following expression was suggested for flows above wind farms:
hU i = hU i(0) + [hU∞ i − hU i(0)]f
Y
δ∗
(5.1)
where δ ∗ is similar to the displacement thickness used in the integral boundary
layer equation (Kundu et al. (2012)). This definition is shown as follows:
δ∗ =
Z
∞
1−
0
hU i − hU i(0)
hU∞ i − hU i(0)
dy.
(5.2)
An important observation is obtained when analysing Figure 5.10. All the
assessed wind-farm configurations follow the same profile shape, thus the spaceaveraged velocity profile above a wind farm (at least for developed sections) can
51
be completely described when having the thickness of the boundary layer (δ ∗ ),
the space-averaged free-stream velocities, and the space-averaged streamwise
velocities close to the turbines (hU∞ i and hU i(0)), as it is shown in Equation
5.1.
12
2 × 32.5d
10
2×3
5d
Y /δ ∗
8
6
3 × 32.5d
3 × 35d
4
2
0
0
0.5
1
1.5
h U i − h U i ( 0)
h U ∞ i − h U i ( 0)
Figure 5.10: Universality of the normalised velocity profile over wind farms.
5.3.2. Scaled Reynolds stresses
After observing that the scaled velocity profile above all the assessed wind farms
followed the same trend, a similar approach was attempted with the second
order flow statistics, such as the space-averaged Reynolds stresses and velocity
variances (hu0 v 0 i, hσu2 i and hσv2 i). For this reason the following expressions were
assumed:
,
hu0 v 0 i
Y
=
f
2
u∗
δ∗
hσu2 i
Y
=f
u2∗
δ∗
(5.3)
(5.4)
and
hσv2 i
=f
u2∗
where u2∗ is the friction velocity.
52
Y
δ∗
(5.5)
When observing Figure 5.11, it can be noted that, if the friction velocity
and the boundary layer thickness are known, a complete characterisation of the
Reynolds stress above a wind farm can be achieved.
12
Y /δ ∗
2 × 32.5d
10
2 × 35d
8
3 × 32.5d
6
3 × 35d
4
2
0
−1
−0.8
−0.6 −0.4
h u ′ v ′i /u 2∗
−0.2
0
Figure 5.11: Universality of the normalised Reynolds stress over wind farms.
The same analysis is done for the variance of the streamwise and vertical
velocities, and it is shown in Figures 5.12 and 5.13 respectively. What is relevant
to observe from these figures is that they follow the same profile for every windfarm configuration assessed. Moreover, it is also important to comment that
other scaling factors were assessed. In particular, the following one
00
00
u2∗ = −hu0 v 0 i(0) − hU V i(0)
was tried several times due to the its logical appearance in the space-averaged
equations, but no flow generalisation was achieved which such a scaling. It is
believed that, the contributions of the Reynolds and dispersive stresses cannot
be compared because they come from different scales: the Reynolds stress
contribution comes from the small scales (turbulence) , and the dispersive stress
comes from the large scales (turbines layout).
5.4. Comparison between Reynolds stresses and dispersive
stresses
Since horizontal averages are being used in this experiment, additional terms
in the equations of motion come up, as they are the dispersive stresses. So, it
is important to observe the contributions of such stresses, and compare them
with the Reynolds stresses.
53
12
Y /δ ∗
2 × 32.5d
10
2 × 35d
8
3 × 32.5d
6
3 × 35d
4
2
0
0
1
2
h σ u2 i /u 2∗
3
4
Figure 5.12: Normalised streamwise-velocity variance over wind farms.
12
2×3
10
Y /δ ∗
8
6
2.5d
2 × 35d
3×3
2.5d
3 × 35d
4
2
0
0
1
2
h σ u2 i /u 2∗
3
4
Figure 5.13: Normalised vertical-velocity variance over wind farms.
00
00
Figure 5.14 shows a comparison between hu0 v 0 i and hU V i. It can be
seen that the Reynolds stresses collapse on the same curve for the two configurations, but the dispersive stresses do not. The reason of this is that the
dispersive stresses come from the inhomogeneity of the turbine spatial distribution. Therefore, using the friction velocity as a scaling for these stresses is
54
not correct because these two contributions do not come from the same scales:
The Reynolds stresses come from the small velocity scales, and the dispersive
ones come from the mean velocity variation scale.
It can also be seen that the magnitude of the dispersive stresses for the
staggered configurations is smaller than the contributions for the inline ones.
Therefore, it can be concluded that, when the turbines are placed in an inline
layout, the flow above the turbines is changing at different positions of the
averaged space, i.e. the flow is less homogeneous than the one in staggered
arrangements. However, it cannot be said that the flow is perfectly homogeneous above staggered configurations as well, since dispersive stresses were also
present for such arrangement.
When comparing the contributions of both stresses at different heights, it
is seen that the dispersive stresses close to the turbines can be as high as the
Reynolds stresses for inline arrangements. The reason why it is interesting to
know the contributions of this terms is that, when integrating the Navier-Stokes
equations in the streamwise direction (see Equation 2.36), derivatives of these
terms in the wall-normal direction come up, and therefore, it is necessary to
understand the influence of each contribution.
10
Y /δ ∗
8
6
4
2
0
−1
−0.5
0
0.5
′′
′′
h u ′ v ′i /u 2∗
h U V i /u 2∗
1
Figure 5.14: Comparison between the contributions of the Reynolds stresses
00
00
(hu0 v 0 i) and dispersive stresses (hU V i). Two wind-farm configurations are
shown: an inline () and a staggered (4) with the same streamwise and spanwise spacings. The color blue represents the dispersive stresses, and the red
one the Reynolds stresses.
The same comparison was done for other dispersive stresses. For exam00
00
ple, Figure 5.15 shows the comparison between hσu2 i and hU U i. It can
be seen that close to the turbines, the dispersive stresses were quite higher
55
than the Reynolds stresses for both layouts. However, after integrating the
space-averaged Navier-Stokes equations in the streamwise direction (Eq 2.36),
derivatives of these terms in the streamwise direction come up and, since it is
known that the flow is not changing too much in the streamwise direction, these
contributions of the Reynolds and dispersive stresses become less important.
10
Y /δ ∗
8
6
4
2
0
0
2
4
h σ u2 i /u 2∗
6
8
′′
′′
h U U i /u 2∗
10
Figure 5.15: Comparison between the contributions of the Reynolds stresses
00
00
(hσu2 i) and dispersive (hU U i) stresses. Two wind-farm configurations are
shown: an inline () and a staggered (4) with the same streamwise and spanwise spacings. The color blue represents the dispersive stresses, and the red
one the Reynolds stresses.
56
5.5. Farm 2 × 32.5d case
From the data reported in Section 5.1, an efficient way of measuring staggered
arrangements was adopted, and measurements above a whole wind farm model
were performed. This was done for a 2 × 32.5d configuration which had the
same wind-farm layout assessed in the box measurements.
The measurement matrix shown in Figure 5.16 was applied to this wind
farm arrangement. It can be seen that the measurement points are located in
the middle of the wind farm, and where the acceleration of the flow produced
by the interaction between the wakes of the turbines is expected (as it could be
seen in Figure 5.2). Namely, an analysis to determine the most representative
4 points was discussed in the previous chapter.
Z /d
0
−1
−2
−3
−4
0
2
4
6
8
10
12
14
16
18
20
0
2
4
6
8
10
X/d
12
14
16
18
20
1
Y /h
0.5
0
−0.5
−1
Figure 5.16: Farm 2 × 32.5d case measurement matrix
The distributions of the streamwise velocity along the farm and close to
the tip of the turbines are shown in Figure 5.17. From the plot, it can be seen
that the velocity is approximately the free-stream one between the 1st and 2nd
row of turbines in both spanwise locations. This effect clearly shows that the
turbines are not interacting at the very beginning of the wind farm. Also, when
observing the spanwise variability of velocity at different streamwise positions,
it can be seen that this difference gets smaller as the number of rows increases.
The achieved value for this difference was of approximately 6 % at the last
assessed streamwise location.
57
1.05
1
U /U ∞
0.95
0.9
0.85
0.8
0.75
0.7
0
5
10
X/d
15
20
Figure 5.17: U/U∞ along the farm at Y = 0 for the 2×32.5d case. The black line
represents the average of the velocity in each streamwise position, the bottom
of the vertical bars represent the velocity values at Z = 0, and the top of them
represent the velocities at Z ≈ −0.6d.
The same data analysis is done for the Reynolds stresses, and it is shown in
Figure 5.18. In the figure, it can be seen a similar effect at the beginning of the
wind farm, but here the Reynolds stress is almost zero between the 1st and 2nd
row of turbines at the centre of the wind farm, and it is almost zero between
the 1st and 4th row of turbines at Z ≈ −0.65d. Another important feature is
that the spanwise variability of Reynolds stresses is much higher in comparison
with the variability of the streamwise mean velocity. This means that, when
performing absorption of momentum calculations (force calculations) for wind
farms just by taking into account the flow in the middle of the wind farm, a
high overestimation is done due to the fact that this force (See Equation 2.38)
is highly dependent on the Reynolds stresses on top of the turbines.
After the application of the spatial-average operator to the measurement
points at different heights, a distribution of the normalised averaged velocity
above the farm was plotted in Figure 5.19. It can be seen that the boundary
layer starts to grow between the 2nd and 3rd row. Thus, it is important to
comment that, if measurements would have been performed just in the centre
of the wind farm, the boundary layer would have started to grow at the same
position, but the thickness of it would have been higher at every streamwise
position. Therefore, what the space-averaged boundary layer shows is actually
a boundary layer which never reaches a parallel condition.
58
−3
x 10
1
0
2
u ′ v ′/U ∞
−1
−2
−3
−4
−5
−6
0
5
10
X/d
15
20
2
along the farm at Y = 0 for the 2 × 32.5d case
Figure 5.18: u0 v 0 /U∞
0.5
1
0.4
0.95
Y /h
0.3
0.9
0.2
0.85
0.1
0
0.8
5
10
X/d
15
Figure 5.19: hU i/hU∞ i. In this figure, the boundary layer growth for the
2 × 32.5d space-averaged case is shown.
The same analysis performed for the space-averaged streamwise velocity is
done for its standard deviation. This distribution is shown in Figure 5.20, where
it can be clearly seen that the turbulence is developing with the downstream
distance. By observing Figures 5.19 and 5.20, it is noted that when the velocity
reached its lowest value, the turbulent intensity reached its maximum.
59
0.5
0.12
0.4
0.1
0.08
Y /h
0.3
0.06
0.2
0.04
0.1
0.02
0
0
5
10
X/d
15
Figure 5.20: hσu2 i/hU∞ i2 . In this figure, the turbulent boundary layer growth
for the 2 × 32.5d space-averaged case is shown.
Figure 5.21 shows the turbulence correlation coefficient along the wind
farm. It can be seen from this figure that, close to the turbines after the
4th row, the value of this correlation coefficient is equal to −0.4, which is a
characteristic value for turbulent flows near walls. This coefficient has been
shown in the past by several authors as Priyadarshana & Klewicki (2004) and
Monin & Yaglom (1973), and they have concluded that this value can be found
above different types of surfaces (smooth and rough). What it is interesting is
the fact that above a wind farm the non-slip condition is not true (i.e. there
is no wall on top of the turbines), and still this value (−0.4) of the correlation
coefficient is found.
5.5.1. Angular velocity of the turbines
Another important aspect from a wind farm is of course the rotational speed of
the turbines, because this quantity is related to many other ones, like its power
generation, inflow velocity, thrust coefficient, torque, etc. Since the turbines
in this thesis work are freely-rotating, a relation between rotational speed and
power generation cannot be developed, however useful information can be taken
from the trend of the rotational speed along the farm. A curve showing the
location of the turbines and a normalised angular velocity is shown in Figure
5.22, where it can be clearly seen that the first and second row have more or
less the same angular velocity, consistently with the fact that both rows are
submerged by the free stream. Another interesting feature of this curve is the
sudden decrease of angular velocity after the second row, due to the interaction
of these turbines with the wake of the upstream turbines. It can also be seen
that, after this decrease in velocity, the velocity increases of a small amount,
and then it remains almost constant.
60
0.5
0
0.4
−0.1
Y /h
0.3
−0.2
0.2
−0.3
0.1
0
−0.4
5
10
X/d
15
Figure 5.21: The turbulence correlation coefficient (hu0 v 0 i/(hσu ihσv i)) for the
2 × 32.5d space-averaged case is shown.
Since the angular velocity is directly related with the inflow velocity, it can
be concluded that after 3 rows, the flow between the top tip of the turbines
and the base reaches a developed condition for this staggered arrangement.
An gu l ar Ve l o c i ty
Y /h , ω /ω ( 0)
1
0.5
0
−0.5
−1
0
5
10
X/d
15
20
Figure 5.22: Angular velocity distribution for the 2 × 32.5d case
61
5.6. Farm 3 × 32.5d case
From the data reported in Section 5.1, an efficient way of measuring staggered
arrangements was adopted, and measurements above a whole wind farm model
were performed. This was done for a 3 × 32.5d configuration which had the
same wind-farm layout assessed in the box measurements.
The measurement matrix shown in Figure 5.23 was applied to this wind
farm arrangement. It is worth to notice that, the measurement points are
located in the middle of the wind farm and exactly in between the turbines (as
seen in Figure 5.6) at two different streamwise locations where the maximum
velocity variations were found.
Z /d
0
−1
−2
−3
−4
0
2
4
6
8
10
12
14
16
18
20
0
2
4
6
8
10
X/d
12
14
16
18
20
1
Y /h
0.5
0
−0.5
−1
Figure 5.23: Farm 3 × 32.5d case measurement matrix
A distributions of the streamwise velocity along the farm and close to the
tip of the turbines are shown in Figure 5.24. It can be seen that, the velocity is
equal to the free-stream velocity in the first row of turbines at Z = 0, and it can
also be seen that the velocity at Z = −1.5d is approximately the free-stream
velocity along the whole wind farm, which means that the wake interaction in
the spanwise direction between the turbines is really weak, despite the high
wind farm density. Furthermore, when observing the spanwise variability of
velocity at different streamwise positions, it can be seen that it practically
remains the same. The velocity difference between Z = 0 and Z = −1.5d
is approximately 25% along the whole wind farm. Leading to the conclusion
62
that the streamwise invariance assumption is not valid for inline configurations
either.
1
U /U ∞
0.9
0.8
0.7
0
5
10
X/d
15
20
Figure 5.24: U/U∞ along the farm at Y = 0 for the 3×32.5d case. The black line
represents the average of the velocity in each streamwise position, the bottom
of the vertical bars represent the velocity values at Z = 0 and the top of them
represent the velocities at Z ≈ −1.5d
The same analysis is done for the Reynolds stresses as shown in Figure
5.25. In this figure, it can be seen a big variation between the measurements
at Z = 0 and Z ≈ −1.5d as well, but it can be clearly seen that, at Z = 0
the magnitude of the Reynolds stress is increasing as the streamwise position
increases. Also, it can be observed that the Reynolds stresses are almost zero
at Z ≈ −1.5d at all streamwise positions i.e. the variation between Reynolds
stresses is almost 100 % between the two spanwise positions.
After the application of the spatial-average operator to the measurement
points at different heights, a distribution of the averaged streamwise velocity
above the farm was plotted (Figure 5.26). It can be seen that the boundary
layer starts to grow just after the 1st row of turbines, and it never stops growing.
Therefore, what the space-averaged boundary layer shows is actually a boundary layer which never reaches a parallel condition, result that is in agreement
with the statement of Chamorro & Porté-Agel (2011) for inline configurations.
The same analysis performed for the space-averaged streamwise velocity is
done for its standard deviation. This distribution is shown in Figure 5.27, where
it can be clearly seen that the turbulence is increasing with the downstream
distance. When observing Figures 5.26 and 5.27, it is noted that, when the
velocity reached its lowest value, the turbulent intensity reached its maximum.
63
−3
x 10
1
0
2
u ′ v ′/U ∞
−1
−2
−3
−4
−5
−6
−7
0
5
10
X/d
15
20
2
along the farm at Y = 0 for the 3 × 32.5d case
Figure 5.25: u0 v 0 /U∞
0.5
1
0.4
0.95
Y /h
0.3
0.9
0.2
0.85
0.1
0
0.8
5
10
X/d
15
Figure 5.26: hU i/hU∞ i. In this figure, the boundary layer growth for the
3 × 32.5d space-averaged case is shown.
Figure 5.28 shows the turbulence correlation coefficient along the wind
farm. It can be seen from this figure that close to the turbines, from the 2nd
row in the wind farm, the value of this correlation coefficient is again equal to
−0.4, which is a characteristic value for turbulent flows near walls, as it was
observed for the staggered case.
64
0.5
0.12
0.4
0.1
0.08
Y /h
0.3
0.06
0.2
0.04
0.1
0.02
0
0
5
10
X/d
15
Figure 5.27: hσu2 i/hU∞ i2 . In this figure, the turbulent boundary layer growth
for the 3 × 32.5d space-averaged case is shown.
0.5
0
0.4
−0.1
Y /h
0.3
−0.2
0.2
−0.3
0.1
0
−0.4
5
10
X/d
15
Figure 5.28: The turbulence correlation coefficient (hu0 v 0 i/(hσu ihσv i)) for the
3 × 32.5d space-averaged case is shown.
5.6.1. Angular velocity of the turbines
A curve showing the location of the turbines and a normalised angular velocity
is shown in Figure 5.29, where it can be clearly seen that the first row of
turbines has a much higher velocity in comparison with the rest, due to the
fact that these turbines are submerged by the free stream directly. Another
interesting feature of this curve is the sudden decrease of angular velocity after
the first row, due to the interaction of these turbines with the wake of the
65
upstream turbines. It can also be seen that after this velocity decrease, the
velocity increases a small amount, and then it remains almost constant.
Since the angular velocity is directly related with the inflow velocity, it can
be concluded that after 4 rows for this inline arrangement, the flow between the
top tip of the turbines and the base reach a developed condition, in agreement
with the result of Chamorro & Porté-Agel (2011).
An gu l ar Ve l o c i ty
Y /h , ω /ω ( 0)
1
0.5
0
−0.5
−1
0
5
10
X/d
15
20
Figure 5.29: Angular velocity distribution for the 3 × 32.5d case
66
5.7. Comparison between farm 2 × 32.5d and 3 × 32.5d
In the last sections, values and characteristics of different flow quantities were
shown for two different wind-farm configurations. These two layouts only differ
in the organisation of the turbines (staggered and inline arrangement), because
the streamwise and spanwise spacing were exactly the same for both of them.
Therefore, the objective of this section is to observe the differences between
flow properties attributed to the wind-farm layouts.
The streamwise mean velocity distribution just above the turbines is shown
in Figure 5.17 for the staggered case, and in Figure 5.24 for the inline case. The
velocity difference between the two spanwise positions tends to converge with
the streamwise distance for the staggered case. On the other hand, this difference does not converge at all for the inline case, demonstrating that the wake
interaction in the staggered case is much stronger than in the inline case. But,
as it was said before, the variations in velocity do not reach yet negligible values, therefore the assumption of spanwise velocity invariance cannot be done
for either configuration. Moreover, It can also be seen that for the inline configuration, the streamwise mean velocity becomes almost constant after four
rows, unlike the staggered case which does not seem to reach a plateau in its
velocity at any location, in fact, it decreases monotonically with the streamwise
distance.
When comparing the Reynolds stresses (Figures 5.18 and 5.25) just above
the turbines, higher differences can be seen. In the staggered case, these differences in Reynolds stress have high values (approximately 50 %), and the
values at the centre and at Z ≈ −0.6 do not appear to converge at any location. However, when observing the inline case a divergence in the Reynolds
stresses at Z = 0 and Z ≈ −1.5d can be detected i.e. the difference between
this covariance between the two spanwise locations increases as the streamwise
distance increases.
Another difference between the flow above the two layouts is the spaceaveraged boundary layer (Figures 5.19 and 5.26), where it can be seen that in
the inline case, the boundary layer starts before than in the staggered case. It
can also be seen that lower values in velocity are reached for the boundary layer
above the staggered arrangement. Moreover, the streamwise mean velocity
just above the turbines for the inline case is almost constant, unlike the one
above the staggered configuration which tends to decrease as the streamwise
distance increases. However, when comparing the space-averaged boundarylayer thickness at the last row between the two arrangements, it appears to
have similar values with a magnitude of approximately 0.35h, where h is the
wind-turbine height from the floor to the top tip, leading to the conclusion that
for staggered cases the growth of the space-averaged boundary layer is faster,
because it starts after, and ends up with the same boundary layer thickness
than the inline case. But, in both of them the boundary layer never stops
67
growing, showing that the flow never reaches a developed condition above the
wind farms.
After the comparison of the growth of the space-averaged boundary layer,
it was convenient to study the turbulence above these two wind farms configurations. This was done by comparing Figures 5.20 and 5.27, where two things
were important to remark: first, higher values of turbulence were found at the
last row for the staggered configuration, and second, the turbulent boundary
layer started before in the inline case, which means that the turbulence starts
to grow earlier for inline configurations.
A similar finding was seen when the turbulence correlation coefficient was
plotted for both configurations in Figures 5.21 and 5.28, where it was found
that for the inline case the characteristic correlation = −0.4 was reached in the
early stages of the wind farm, unlike the staggered case where this correlation
coefficient was found after the fourth row of turbines, which means that the
turbulence starts to grow first for inline configurations. Furthermore, the fact
that this turbulence correlation coefficient reached a value of −0.4 where the
boundary layer is, represents an interesting finding, because the flow above
these wind farms behaves as the flow above walls. Therefore, this result leads
to think that this coefficient is more related to the turbulence in general than
to boundary layer flows.
A comparison between the angular velocity for both configurations was
done. In order to do that, the values shown in Figures 5.22 and 5.29 were
replotted in the same figure (Figure 5.30) for the sake of clarity. In this figure,
it can be clearly seen that the angular velocity for staggered configurations is
higher in overall than the angular velocity for inline configurations: in fact, in
the staggered configuration the angular velocity reaches a plateau at a value
approximately 25 % lower than the velocity in the 1st row, and in the inline
configuration the angular velocity reaches a plateau at a value approximately
50 % lower than the one in the first turbine. In spite of the fact that these are
freely rotating turbines, it can be said that there is a direct relation between
power generation and angular velocity, which means that in terms of efficiency
per turbine, staggered layouts perform better than inline layouts, as it was
discussed by Chamorro et al. (2011). However, when there are space restrictions
to place the turbines, a specific answer for which configuration is more efficient
cannot be determined with the obtained data in this thesis work. Further
information on layout efficiency and optimisation can be found in Alfredsson
& Dahlberg (1981), Mosetti et al. (1994) and Frandsen & Jørgensen (2009).
Another important observation is that developed flow inside the wind farm
is first reached for staggered arrangements. This conclusion can be achieved by
observing when the angular velocity reaches a plateau for both configurations.
68
An gu l ar Ve l o c i ty
Y /h , ω /ω ( 0)
1
0.5
Staggered
Inline
0
−0.5
−1
0
5
10
X/d
15
20
Figure 5.30: Angular velocity comparison between the 2 × 32.5d and 3 × 32.5d
case
5.8. Force model validation
One of the aims of this thesis is to estimate in a simple way the momentum
absorption in wind farms with different layouts (staggered and inline), in order
to relate it with second-order flow properties, such as the friction velocity. For
this reason, a simple force model (Equation 2.55) was developed in Chapter
2. In order to validate this model, the momentum conservation equation was
integrated (Equation 2.38) to obtain an expression for the force applied by
the turbines to the flow. Before going further, it is important to recall both
expressions. First, the force model:
2
Ct Nt Ad Uinf
Fx =
2 Sx Sz
h
where:
•
•
•
•
•
•
Sx is the turbines spacing in the streamwise direction.
Nt is the number of turbines in this particular row.
Ct is the thrust coefficient of the turbines.
u∞ is the inflow velocity of the turbines.
Ad is the area swept by the turbine rotor.
Sz = Sx [(Nt max )Lz ] where Lz is the spacing in the spanwise direction
and Nt max is the maximum number of turbines in a wind farm row, i.e.
In a staggered arrangement 3 X 2 Nt max = 3.
69
• h is the wind farm height.
And second, the integrated space-averaged Reynolds Navier-Stokes equation:
00
Fx = hU∞ i
00
∂hU∞ i hU V i(0) hu0 v 0 i(0)
−
−
∂x
h
h
where:
00
00
• hU V i(0) is the dispersive stress at the tip turbine height.
• hu0 v 0 i(0) is the Reynolds stress at the tip turbine height.
The measurements over the farms 2 × 32.5d and 3 × 32.5d were used for
doing the validation, and this one is shown in Figure 5.31. In this figure, it
can be seen that this force, as its expressions shows, is constant between rows
of turbines. The motivation of defining the force model as a constant force
between two rows is to be consistent with the domain of the spatial-average.
When observing Figure 5.31, it can be noted that the force model (solid
line for both cases) almost matches the value of the integrated force after 5
turbine rows, fact that is interesting since the Equation 2.38 is valid when the
flow is completely developed i.e. when there is no acceleration of the flow.
As a conclusion, this model can be utilised for estimating the force applied
by the turbines to the fluid for staggered or inline developed wind farms with
a certain confidence. But, the most relevant result from this model is that,
by having the free-stream velocity, the angular velocity of the turbines, and
the dispersive stresses close to the turbines, an estimation of the friction velocity can be obtained. This is helpful, since just high-scale measurements are
required to obtain micro-scale values such as the friction velocity.
70
F or c e [ N / k g ]
14
Force Model (Stag)
Navier−st (Stag)
12
10
8
6
4
2
0
2
4
6
8
10
12
14
16
18
20
F or c e [ N / k g ]
20
Force Model (Inli)
Navier−st (Inli)
15
10
5
0
0
2
4
6
8
10
X/d
12
14
16
18
Figure 5.31: Force model validation for 2 × 32.5d and 3 × 32.5d cases
71
20
CHAPTER 6
Summary and conclusions
The flow above different wind-farm arrangements was analysed experimentally
by means of wind-tunnel measurements. As wind-farm models, small freelyrotating turbines were placed above a metallic plate inside the test section of
the KTH NT-2011 wind tunnel. The assessed wind-farm arrangements were
different in terms of turbine spacings and turbine layout (inline and staggered).
Three types of measurements were performed. First, the time series of
streamwise and wall-normal velocities were measured between two rows of turbines deep downstream on the wind farm (Box Measurements). Second, the
time series of streamwise and wall-normal velocities were measured above the
whole wind farm, in this experiment two different wind farm layouts with the
same streamwise and spanwise spacing (Farm Measurements) were assessed.
Third, the angular velocity of the turbines was measured in each row of turbines (RPM Measurements). To measure the velocities time series, X-wire
anemometry was used, while to measure the rotational speed of the turbines,
a laser-photodiode device was used.
In the Box Measurements 4 different wind farm arrangements were assessed, two inline and staggered configurations with two different streamwise
spacings but the same spanwise spacing (2 × 32.5D , 2 × 35D , 3 × 32.5D and
3×35D respectively). However, just two wind farm configurations were assessed
in the Farm Measurements and RPM Measurements: an inline and a staggered
configuration with the same streamwise and spanwise spacing (2 × 32.5D and
3 × 32.5D ).
From the Box Measurements, it was found that close to the turbines, there
is a high spanwise variation of streamwise velocity for both staggered and inline
configurations, in spite of the fact that the measurements were performed after
6 rows of turbines, and that the wind farms had quite high densities in comparison with the cases studies before in Cal et al. (2010), Chamorro & Porté-Agel
(2011) and Chamorro et al. (2011). Moreover, it was found that a good spatial
average could be achieved by only measuring in four different points in each
plane above the wind farm.
It was found that even for these high densities and developed flow conditions, the velocity between two turbines in the same row is the free-stream
velocity for inline configurations. Also, It was found that an acceleration of the
72
flow (Venturi effect) takes place at half the spanwise distance that separates
two turbines of two different rows for staggered arrangements.
A scaling behaviour of the streamwise mean velocity profile and the Reynolds stresses was found. First, it was found that, to know the streamwise velocity profile above a wind farm, it was only necessary to know the free-stream
velocity, the streamwise velocity close to the turbines, and the boundary layer
thickness (δ ∗ ). Second, it was also found that to know the Reynolds stresses
profiles above a wind farm, it was only necessary to know the friction velocity
close to the turbines and the thickness of the boundary layer. On the other
hand, it was found that the dispersive stresses did not scale with the friction
velocity, because they do not come from the same contributions: the friction
velocity comes from the small scales (turbulence) while the dispersive stresses
come from the large scales (mean flow). Also, it was found that the value of the
00
00
dispersive stresses hU V i are higher for inline configurations, since the turbines in this layout are not distributed as homogeneously as for the staggered
counterpart.
From the Farm Measurements, it was observed that the boundary layer
above the assessed wind farms did not stop growing, which means that developed conditions were not reached for any of the wind farms. Also, it was
observed that turbulence starts to grow faster for inline cases, and that in
the zones where the streamwise velocity was the smallest, the turbulence was
the highest. Moreover, it was found that after 5 rows of turbines, the spaceaveraged velocities close to the wind farm top were lower for staggered configurations. Furthermore, it was found that the turbulence correlation coefficient
above the two assessed wind farms was equal to −0.4, which is a characteristic
value for flows above walls, leading to think that this value is more related to
the turbulence per se, rather than to boundary layer flows.
From the RPM Measurements, it was found that the angular velocity of
the turbines was around 25 % higher for staggered configurations along the
whole wind farm, and as it is intuitive, by having more angular velocity, more
power can be produced. Therefore, having a given number of turbines, the
best configuration to place them is the staggered one. Plateaus of the angular
velocities could be seen for both configurations. However, this plateau was
reached faster for the staggered configuration. What is important to conclude
is that, since the angular velocity is directly related to the inflow velocity, it can
be stated that the condition of developed flow inside the wind farm is reached
for both configurations.
A model for calculating the absorption of momentum by the wind farm
was derived, and then it was compared with the values of the estimated forces,
obtained by introducing the measurements on the time and space-averaged
integrated Navier-Stokes equations. From the results of this comparison, it
could be concluded that the force model was good enough for developed wind
73
farms, which means that by just knowing the angular velocity of the turbines,
the thrust coefficient of them and the geometry of the wind farm, the force
applied by the turbines to the flow can be estimated. But, the most relevant
result is that, by having the free-stream velocity, the properties of the turbines,
the angular velocity of the turbines, and the dispersive stresses close to the
turbines, an estimation of the friction velocity can be obtained. This is helpful,
since just large-scale measurements are required to obtain small-scale values
such as the friction velocity.
74
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