Introduction
The study of wake properties is important for assessing the optimal layout of
modern wind farms. Wind turbine wake development may be studied using field
experiments, small-scale wind tunnel measurements or numerical simulations
with computational fluid dynamics (CFD). There are several advantages of CFD
over field experiments and small-scale wind tunnel measurements, e.g. no
violation of similarity requirements, control over inflow conditions and
information about the relevant parameters, e.g. wind velocity, over the entire flow.
However, as CFD results are sensitive to the experience and knowledge of the
user of the CFD code and to the numerous computational parameters and
assumptions involved, it is imperative to perform validation studies.
Previous work on validating CFD wake models using a wind turbine tested in
wind tunnels has been presented by and by
. These studies demonstrated that there was a
significant deviation between the various prediction tools and the wind
tunnel measurements. Similar results for a small-scale model wind turbine are
reported by and by ,
indicating the importance of validating existing wind turbine modelling tools
and methodologies.
Advanced methods of wake modelling with CFD may be implemented by using
large-eddy simulation (LES) techniques in which the wind turbine forces may
either be prescribed with an actuator line (ACL) method or an actuator disc
(ACD) method. Work along these lines has been performed by numerous
researchers such as , ,
, ,
and . Although LES
provides high-fidelity results comparable to field measurements, the
computational requirements of the method
(; )
are still too expensive and therefore not yet suitable for engineering
practices of wake computations for whole wind farms. A less computationally
expensive alternative to LES are Reynolds-averaged Navier–Stokes (RANS)
simulations. RANS simulations have been used with the ACD method to simulate
wind turbine wakes by numerous researchers, e.g. ,
, and .
The aim of the present study is the validation of the implementation of an
ACD model in the CFD code PHOENICS . In
order to do so, computational and experimental results are compared for three
cases. Case A consists of a single wind turbine in a low-turbulence-intensity
environment with a uniform wind inflow. Case B is composed of two wind
turbines positioned in-line in the same low-turbulence-intensity environment
with a uniform wind inflow. Case C again uses two wind turbines positioned
in-line, but in this case multiple inflow conditions are studied for
different spacings of the wind turbines. This is done to investigate the
influence of the inlet conditions on the wind turbines' thrust.
As this method is intended to be used for industrial purposes, it therefore
needs to provide accurate and reliable results with low computational effort.
The simulations are performed according to the “Blind test 1”, “Blind
test 2” and “Blind test 4” invitation workshops organized by NOWITECH and
NORCOWE (). The goal of these three workshops is to serve as an
ideal test case for CFD tools by providing detailed measurements of the
thrust coefficient and the wake properties behind the rotor both in terms of
mean flow and turbulence kinetic energy within a controlled wind tunnel
environment. Note that this work is an extension of the
preceding work of .
Illustration of wind tunnel layout for (a) the set-up with one model wind
turbine (case A) and (b) the set-up with two in-line model wind
turbines
(case B). The three downstream positions x/R=2, 6, 10 and x/R=2, 5, 8 are
where the measurements are extracted; radius R=0.447 m.
The paper unfolds as follows: Sect. 2 presents the experimental set-up of
the workshops, in which the three test cases are outlined. This is followed by
a description of the numerical method and of the computational settings used
to perform the simulations. The results from the numerical simulations are
introduced and discussed in Sect. 3. Lastly, in Sect. 4 the main
conclusions of this study are presented.
Methods
Experimental set-up
The experiments are performed in the large closed-return wind tunnel
facility at the Norwegian University of Science and Technology (NTNU). The
test section for all three cases has the width (W) and length (L)
dimensions of W×L = 2.710 m × 11.150 m; Fig. . To
maintain a zero pressure gradient and maintain a constant velocity along the
stream-wise direction, the height H of the wind tunnel increases from 1.801 m at the inlet to 1.851 m at the outlet. Velocity measurements are performed
using both hot-wire anemometry (HWA) and laser Doppler anemometry for
verification purposes. The tip Reynolds number for these three cases is
approximately Rec,tip≈105 for the upstream wind turbine. This
tip Reynolds number is based on the velocity of the tip and the chord length
at tip. For full-scale experiments a typical tip Reynolds number is on the
order of 106. The air density ρ is equal to 1.2 kg m-3. In all
cases the results are only considered where the wind turbines operate
at their design condition, i.e. tip speed ratio of 6 (TSR = 6).
In case A, the three-bladed wind turbine is positioned at a distance of
3.660 m from the inlet. The model wind turbine has a tower that consists of
four cylinders of different radii. The hub height is Hhub=0.817 m and
the rotor radius is R=0.447 m. The rotor blades are designed to produce a
constant pressure drop across the rotor, which resembles a uniformly
distributed thrust, when operating at their design condition
(). The airfoil used is the NREL S826 and to increase
the Reynolds number of the blades, a chord length of approximately 3
times longer than normal was used. The blades have a circular shape close to
the nacelle primarily to allow them to be attached to the hub. The transition
from the airfoil section of the blade to the circular section is abrupt. An
asynchronous generator of 0.37 kW is located under the tunnel floor and is
connected to the wind turbine rotor by a belt located behind the tower. The
total blockage effect, defined herein as the fraction of the total tower- and
rotor-swept area to the wind tunnel cross section, is approximately 13 %.
As a result the flow will be impacted by the walls and this interference will
lead to artificial speed-up effects. The stream-wise inlet velocity is
Uref,A=10 m s-1 and the stream-wise turbulence intensity at the
turbine position is Iu,A=0.3 %. A thin boundary exists near the wall of
the wind tunnel. This boundary layer has been measured in an empty tunnel
using pitot tubes for four distances downstream of the wind tunnel inlet, i.e.
1.80, 4.50, 6.30 and 8.10 m. Further information on the details of the
experimental investigations are reported by
and .
Overview of cases.
Case
Separation distance
Wind profile
Turbulence
Hub height
Measurement position
x/R
intensity
(m)
x/R
A
–
Uniform 10 m s-1
0.3 %
0.817
2, 6, 10
B
6
Uniform 10 m s-1
0.3 %
0.817
2, 5, 8
C1
18.00
Uniform 11.5 m s-1
10 %
0.827
5.54, 10.36, 17.00
C2
10.36
Sheared 11.5 m s-1 at hub
10.1 %
0.827
5.54
For case B the stream-wise inlet velocity and turbulence intensity are
similar to case A. Here, two in-line wind turbines horizontally centred in
the wind tunnel are investigated, where the downwind wind turbine is the same
one as used in case A. The upstream wind turbine hereafter is always referred
to as T1 while the downstream wind turbine is referred to as T2. Both
wind turbines rotate in a anticlockwise direction as seen from the inlet and
are three bladed with the same blade geometry and airfoil, i.e. the NREL S826
airfoil. As the nacelle diameter of T1 is somewhat larger than T2, the
turbines have slightly different rotor diameters. The rotor radius of the
upstream wind turbine, T1, is RT1=0.472 m and the radius of the
downstream wind turbine, T2, is RT2=R=0.447 m. For T1 as
opposed to T2, the belt connected to the 0.37 kW asynchronous generator
is located within the tower. To calculate the thrust force of the turbines,
they are mounted on a six-component force balance. Further information on the
experimental investigations is reported by
.
Case C is divided into two sub-cases, C1 and C2. For both sub-cases the same
wind turbines as in case B are used with a hub height of 0.827 m instead of
0.817 m. The distinction between sub-cases is made because the wind turbines
are exposed to different inflow conditions in terms of the wind velocity
profile and turbulence intensity as seen in Table . For both
sub-cases the upstream wind turbine is positioned at 4R from the inlet.
Looking at each case individually, case C1 has a uniform inflow velocity of
Uref,C=11.5 m s-1 measured at the inlet and a turbulence intensity
of 10 % measured at the first wind turbine position. The turbulence in the
wind tunnel is created by a bi-planar grid built from wooden bars installed
at the inlet. To estimate the effect of unintended stream-wise velocity
gradients, horizontal stream-wise velocity values at four positions
downstream of the inlet are measured in an empty domain. For case C1 the
thrust values along with the velocity and turbulence kinetic energy at three
distances downstream of T1 are measured. Regarding case C2, a sheared
inflow is considered with a turbulence intensity of 10.1 % at the hub height
position of T1. The inlet velocity as a function of height z is
expressed by the power law used for atmospheric flows, which is given as
U(z)=U(zref)zzrefa,
where a is the shear exponent equal to 0.11, the reference height is
zref=0.827 m and the reference velocity is U(zref)=Uref,C=11.5 m s-1 . For C2, similar to case C1, empty-domain measurements are
conducted at the same four positions from the inlet for the horizontal
stream-wise velocities and turbulence intensity. The position of the second
wind turbine is fixed to 10.36R and wake measurements for stream-wise
velocity and turbulence kinetic energy are taken at a downstream distance of
T1 equal to 5.54R.
Numerical method
The simulations are performed with the commercial CFD code PHOENICS in which
the RANS equations are solved using four different turbulence models. The
turbulence models are (1) the standard k-ε
, (2) the re-normalization group (RNG)
k-ε , (3) the KL
k-ε and (4) Wilcox's k-ω
turbulence model . The flow variables are
stored in a uniform fully structured staggered grid and the Cartesian
coordinate system is used. The SIMPLEST algorithm
is used to solve the RANS equations, and the
hybrid differencing scheme is used to discretize
the convective terms. The diffusion terms are discretized using the central
differencing scheme. In the computations, the wind tunnel conditions are
replicated accordingly for each case and a zero static pressure is applied at
the outlet plane. The lateral, top and bottom faces of the domain are set to
be impermeable and a wall function method according to
is employed to introduce the effects of the wind
tunnel walls into the numerical simulation. This particular method is
preferred for its advantages in terms of low computational requirements and
storage needs. The wall function method for a flow in local equilibrium obeys
the relations
Up=U∗κlnEY+,k=U∗2Cμ,ε=Cμ34k32κY,
where Up is the absolute value of the velocity parallel to the wall at the
first grid node, U∗ is the friction velocity calculated as
U∗=τw/ρ, κ is the von Kármán constant equal to
0.41, E is a roughness parameter dependent on the wall roughness taken
equal to 8.6 for smooth walls, Y+ is a dimensionless near-wall quantity
for length determined as Y+=U∗Yν, ν is the turbulence
viscosity, Y is the distance of the first grid node to the wall and
Cμ is a dimensionless constant equal to 0.09 in the standard
k-ε turbulence model. If the wall is considered to be rough (not
smooth) then the roughness parameter E is a function of the Reynolds
roughness number defined as Rer=U∗hrν, where hr is the
sand grain roughness height. The relation between the roughness parameter E
and Reynolds roughness number Rer follows the empirical laws proposed by
:
E=8.6when Rer<3.7,E=1aRerb2+1-a8.62when 3.7<Rer<100,E=bRerwhen Rer>100,
where b=29.7, a=(1+2x3-3x2) and
x=0.02248⋅(100-Rer)/Rer0.564.
Thrust distributions over the disc, where r is the distance from
the centre of the disc and R is the radius of the disc. Note that f on
the left-hand side of the equations has dimensions of force per unit
area.
Distribution
Equation
b
Range of application
Uniform
funi,i=bFtot1∑Ai
b=1
0≤r≤R
Polynomial
fpol,i=bFtotrR21-rR21∑Ai
b=6
0≤r≤R
Trapezoidal∗
ftra,i=bFtot4rR+11∑Ai
b=27
0.2R≤r≤R
Triangular
ftri,i=bFtotrR1∑Ai
b=32
0≤r≤R
∗ For the trapezoidal distribution there is no force
applied in the region of 0≤r<0.2R.
For the simulations no tower or hub effects are considered. The presence of
the rotor is modelled using an ACD method based on the 1-D momentum
theory. The thrust force Fi of each individual cell of the disc is
calculated according to
Fi=CTU1,i12ρU1,i1-αi2Ai,
where U1,i is the velocity of the flow at the individual cell numbered
i of the disc, αi is the axial induction factor calculated for each
individual cell of the disc, Ai the surface area of the cell facing the
undisturbed wind flow direction and CT(U1,i) is a modified
thrust coefficient curve dependent on the velocity at the disc. The modified
thrust coefficient curve is created in a preprocessing step by replacing
the undisturbed wind velocity values of the thrust coefficient curve with the
wind velocity values at the disc U1. To do this Eq. () is used,
where CT is the thrust coefficient for the respective undisturbed wind
velocity U∞.
U1=U∞1-121-1-CT
The total thrust force applied to the flow is calculated by summing the
individual thrust forces according to Ftot=∑iFi over the disc
area. This total thrust force may then be distributed in different ways over
the disc. In this work, apart from using Eq. () as it is to
prescribe the forces in each individual cell, referred to as the
undistributed thrust, four different thrust distributions are tested: a
uniform, a polynomial, a triangular and a trapezoidal distribution. Their
equations are presented in Table . Figure presents a
normalized plot of all four distributions along a diameter of the disc. The
uniform distribution is chosen to match the thrust distribution of the actual
rotor of this case. Full-scale wind turbines however have a zero thrust value
at the hub and at the tip of the blades. The polynomial distribution, which is
a fourth order polynomial, is intended to respect this by having a zero thrust
at the hub and at the tip of the disc. The triangular distribution is
designed to have a zero thrust at the hub and to linearly increase the thrust
force along the radius, up to the tip of the disc. Lastly, the trapezoidal
distribution is set up to resemble the thrust distribution produced using the
ACL method presented in . While it is possible
to determine a thrust distribution given the rotor geometry and airfoil data
through a blade element momentum theory, it is somewhat impractical for
industrial applications. Airfoil data of commercial wind turbines are
generally not available to the typical industrial user. The purpose of
testing different thrust distributions with this ACD method is that these
will probably produce different wake properties, e.g. with respect to the
velocity deficit and turbulence kinetic energy of the wake. Two questions thus
arise, i.e. which thrust distribution within this ACD method better captures
the wake produced by a wind turbine and up to which distance does the thrust
distribution have an effect on the wake? Here it should be noted that the
primary goal is not to isolate the influence of the thrust distribution on
the wake flow, as the total thrust over the disc will intrinsically vary
depending on the thrust distribution used within the method. Here the goal is
to investigate the effect the ACD method with different thrust distributions
has on the wake flow.
Normalized plot of all four distributions along a diameter of the
disc.
For the first part of the simulations (case A) the numerical domain was
defined according to the wind tunnel geometry as reported in
. Initially, empty-domain simulations were conducted
to assess the extent of unintended stream-wise gradients for the mean
velocity and turbulence parameters. For this purpose horizontal profiles of
U, k and ε are extracted at the inlet, turbine location and
x/R=10 downstream of the turbine position. As the roughness height of the
wind tunnel walls is not known a priori, a comparison between the
experimental boundary layer profile and the simulated boundary layer profile
for different roughness height values is conducted in a trial and error
fashion until the appropriate value for the roughness height is found. When
considering the wind turbine in the simulation, the computed results are
compared against the HWA measurements for the normalized axial velocity
U/Uref,A and normalized turbulence kinetic energy k/Uref,A2 at
the three downstream positions mentioned in Table along the
horizontal line through the centre of the wake in the crosswise direction.
For case B the domain geometry and the positioning of the wind turbines are
in accordance with the invitation sent out by .
The equilibrium wall function method for smooth walls, that is E=8.6, is
used to introduce the effects of the wind tunnel walls into the numerical
simulation; this applies as well for case C. The computed results when the
ACDs are considered are compared to HWA measurements for the normalized
axial velocity U/Uref,B and normalized variance of the axial velocity
component u′2‾/Uref,B2 at the three downstream positions
shown in Table along the horizontal line through the centre of the
wake in the crosswise direction. Further, the thrust coefficients
CT=2FtotρUref,B2A of the two wind turbines are
compared with the experimental results, where A is the rotor cross section
of each individual wind turbine. Here it should be noted that even though the
thrust coefficient curve is an input to the simulation, the thrust
coefficient value applied against the flow depends upon the velocity at the
disc, which changes as the simulation progresses.
Lastly, for case C the domain geometry and the positioning of the wind
turbines are in accordance with the invitation sent out by
. Prior to the simulations with the ACD, empty-domain
simulations are performed for the sub-cases (C1, C2). This is to match the
inlet wind profile and turbulence intensity with the experimental
measurements at four downstream positions from the inlet, that is x/R=4.00,
9.54, 14.36 and 22.00. When considering the wind turbines in the numerical
simulation via the ACD method, the computed results are compared with the HWA
measurements for the normalized axial velocity U/Uref,C and normalized
turbulence kinetic energy k′=k/Uref,C2, where Uref,C=11.5 m s-1. The thrust coefficients of the two turbines are calculated as
CT=2FtotρUref,C2A.
Grid convergence analysis
A grid independence study is carried out according to the recommended
procedure of for mixed-order schemes. For these
simulations a uniform grid is used based on the cells per rotor diameter.
Table presents information regarding the different grid levels used
in the grid independence study. Even though the grid independence study is
performed solely for case A it is considered to apply for the other cases as
well.
According to the series that represents the discrete
solution for each grid is given by
fk=fexact+g1hk+g2hk2+Ohk3,
where fk is the discrete value solution of grid k, gi is the ith
order error term coefficient and hk is a measure of the grid spacing. The
three unknowns (fexact, g1 and g2) may be found by expanding Eq. () for three consecutive grids and by solving the resulting three-set equation.
f1=fexact+g1h1+g2h12+Oh13,f2=fexact+g1h2+g2h22+Oh23,f3=fexact+g1h3+g2h32+Oh33.
The spatial discretization error is calculated according to the following
formula:
Grid levels and size.
Grid
Cells per rotor
Cells in the
level
diameter
domain
1
40
48×105
2
20
6×105
3
10
3×105
Overview of the cases for which results will be presented.
Case
Grid and wall
Thrust distribution used for
Turbulence model used for
Thrust coefficient
Empty-domain
function study
different turbulence models
different thrust distributions
comparison
study
A
yes
Uniform
k-ε
–
–
B
–
Undistributed
k-ε
yes
–
C1
–
Undistributed
k-ε
yes
yes
C2
–
Undistributed
k-ε
yes
yes
spatial error(%)=fk-f̃exactfexact×100,
where fk is the discrete value solution of grid k and
f̃exact is an approximation to the exact solution
fexact,
which is found by disregarding the higher-order terms of Eqs. () to
(). The normalized magnitudes of the first- and second-order
error terms and the magnitude of their sum (mixed order) is given by
g̃1hf̃exact×100,g̃2h2f̃exact×100,g̃1h+g̃2h2f̃exact×100,
where g̃1 and g̃2 are approximations to g1 and
g2, which are found as mentioned previously by solving and disregarding
the higher-order terms of Eqs. () to ().
Normalized axial velocity and spatial discretization error computed
behind a single model wind turbine (case A) for three grids at
(a, d) x/R=2, (b, e) x/R=6 and
(c, f) x/R=10 for the k-ε turbulence model and the
undistributed distribution.
Normalized axial velocity for different wall functions plotted
against the experimental data for four downstream distances from the inlet.
Results and discussion
A summary of the cases for which results will be presented is shown in Table . For cases A and B, empty-domain results for the axial velocity
extracted at cross-sectional horizontal profiles at the inlet, turbine
location and at a position x/R=10 downstream of the turbine location show
an approximately 2.7 % increase in the axial velocity. The turbulence
parameters k and ε, however, decrease steadily from
the inlet to x/R=10, which is due to the lack of a turbulence generating
mechanism along the domain, e.g. shear. The decrease in k and ε
along the empty domain is 5 orders of magnitude lower than their average
value when an ACD model is present in the computations.
Normalized axial velocity contours for the k-ε
turbulence model using the polynomial thrust distribution. (a) One
model wind turbine, case A, and (b) two in-line model wind turbines,
case B.
Figure presents the spatial discretization error results obtained
for the normalized axial velocity profiles at three distances downstream of
the wind turbine position for case A. The error is estimated to be less than
2.4 % for finest grid (grid 1). Therefore, for the purpose of this
investigation, a uniform grid resolution of 40 cells per rotor diameter is
found suitable for all cases. Also shown in Fig. are the normalized
magnitudes of the first- and second-order error terms and of their sum, which
is given by Eq. ().
(a) Normalized axial velocity and (b) normalized
turbulence kinetic energy computed behind a single model wind turbine (case A)
at x/R=2, 6 and 10 for different turbulence models using the uniform
distribution.
Figure illustrates the computed stream-wise velocity results for
two different wall function values, against velocity measurements conducted
in the wind tunnel with pitot tubes. A quite good match between experimental
measurements and the computed results exists when considering a smooth wall,
i.e. E=8.6. Therefore, the equilibrium wall function for smooth walls is
used in the simulations, as it is found to have the best agreement with the
measurements. By setting a sand grain roughness height other than that for a
smooth wall causes the discrepancy between the simulated boundary layer
development and the measurements to increase for the axial velocity profile.
(a) Normalized axial velocity and (b) normalized
turbulence kinetic energy computed behind a single model wind turbine (case A)
at x/R=2, 6 and 10 for different thrust distributions and the
k-ε turbulence model.
Figure illustrates the normalized axial velocity contours for cases A and B. The polynomial thrust distribution is used along with the
k-ε turbulence closure model. It is clearly seen that the method
reproduces what is expected, that is, by positioning a second turbine in the
wake of the first, the axial velocity of the flow is further reduced. This
reduction is due to the further energy extraction of the second wind turbine
from the mean flow. The dashed lines in Fig. indicate the positions
at which flow values are extracted and compared with the HWA measurements.
For case A, the computed results are validated against HWA measurements for
the normalized axial velocity and normalized turbulence kinetic energy; see
Fig. . These results are computed using different turbulence models
and the uniform thrust distribution. To investigate the influence of the
thrust distribution on the wake development, simulations using the
k-ε turbulence model with different thrust distributions were
conducted; results are shown in Fig. .
In Fig. a it is observed that the k-ε and the KL
k-ε turbulence models produce results similar to the
measurements, with the KL k-ε model being less diffusive than
the k-ε model in the crosswise direction. Apart from the
undistributed thrust, all thrust distributions used in this study assume
axisymmetry; therefore, the simulated profiles are symmetrical to the rotor
centre. However, this is not the case with the measurements, which exhibit
asymmetric profiles as seen in Fig. a. According to
this asymmetry may be produced by the slowly
rotating tower wake as seen at the downstream position of x/R=10, for example. As
the ACD is non-rotating in the simulations, it is expected that the
predictions will not capture this asymmetry in the wake. Further, as the
effects of the nacelle and tower are not considered, it is also anticipated to
find small deviations of the predictions from the measurements in the near
wake. On average the blockage effect is captured by the simulations as seen
in Fig. a. This effect is apparent outside the wake region
(|y/R|>1.5) where the simulated and measured normalized axial velocity
values are higher than one. Considering the normalized turbulence kinetic
profiles in Fig. b, the shape of the profiles is not successfully
predicted by any of the turbulence models. The k-ω turbulence model
tends to over-predict the turbulence kinetic energy production in this environment with low background turbulence. As a result, the simulated wake recovery in
comparison to the velocity measurements is too high. The discrepancy between
the measured profile shape of the turbulence kinetic energy from that
predicted at the downstream position of x/R=2 for the wake region of
|y/R|<0.5 is mainly due to the presence of the nacelle and abrupt change of
the blade shape from the airfoil profile to a cylinder near the nacelle.
(a) Normalized axial velocity and (b) normalized
stream-wise variance of the velocity computed for the two in-line model wind
turbines (case B) at x/R=2, 5 and 8 downstream of the second wind turbine
for different turbulence models using the undistributed thrust.
(a) Normalized axial velocity and (b) normalized
stream-wise variance of the velocity computed for the two in-line model wind
turbines (case B) at x/R=2, 5 and 8 downstream of the second wind turbine
for different thrust distributions and the k-ε turbulence
model.
When keeping the turbulence model constant and changing the thrust
distribution, it is observed in Fig. that the effect of the thrust
distribution is pronounced in the near-wake region and diminishes further
downstream. also observed that the effect of
representing the forces of a wind turbine differently, such as by a rotating
or non-rotating ACD or ACL was more pronounced in the near-wake region,
rather than in the far-wake region. Regarding the turbulence kinetic energy,
all thrust distributions seem to capture the position of the tip vortex apart
from the polynomial. The increased turbulence production due to the breakdown
of the tip vortex at the x/R=2 position is not captured by any combination
of thrust distribution and turbulence model. This is also observed by other
researchers such as , who concluded
that the ACD method lacks the ability to simulate the turbulence structures
present in the near-wake region. results also
show that in an environment with low background turbulence intensity there
seemed to be a perceptible dependency of the wake development on the
turbulence closure used, in terms of velocity deficit and turbulence kinetic
energy.
For case B when using the undistributed thrust, the normalized axial velocity
and stream-wise variance of the velocity at three positions downstream of
the second turbine are shown in Fig. for different turbulence
models. In Fig. , the effect of the thrust distribution on the wake
downstream of the second wind turbine is investigated by varying the thrust
distribution while keeping the same turbulence model. Results of the thrust
coefficient values for the upstream and downstream wind turbines are
summarized in Table for when the k-ε turbulence model
is used.
Empty-domain stream-wise velocity and turbulence intensity results
for case C1 (a, b) and case C2 (c, d) for four vertical
profiles downstream of the inlet when using the k-ε turbulence
model.
(a) Normalized axial velocity and (b) normalized
turbulence kinetic energy computed for the two in-line model wind turbines
(case C1) at x/R=5.54, 10.36 and 17.00 downstream of the first wind turbine
for a separation distance of 18.00R. Different turbulence models are used
along with the undistributed thrust.
For case B in which the wakes of both wind turbines interact, similar to the
results of case A, the k-ε and the KL k-ε
turbulence models produce axial velocity results in agreement with the
measurements. The KL k-ε seems however to underestimate the
normalized stream-wise variance of the velocity. The k-ω turbulence
model here again over-predicts the wake recovery and overestimates the
normalized stream-wise variance of the velocity. Conversely, the RNG
k-ε under-predicts the wake recovery and underestimates the
normalized stream-wise variance of the velocity. The blockage effect is on
average captured for all turbulence models, apart from when the k-ω
turbulence model is used, and thrust distributions are as seen in Figs.
and . The wake expansion is accurately predicted when using the
k-ε turbulence model. Further, when keeping the turbulence model
constant and changing the thrust distribution (Fig. ), it is observed
that the effect of the thrust distribution is less pronounced in the case with two
in-line wind turbines than for the case with a single wind turbine due to the
higher turbulence diffusion. The wake predicted when using the undistributed
or uniform thrust distribution seems to be in closer agreement with the
measurements. This is possibly due to the fact that these distributions
produce a fairly constant pressure drop over the disc, as do the blades when
operating at their design condition. The thrust coefficient values summarized
in Table for the different thrust distributions and the
k-ε turbulence model agree quite well with the measured data.
There is, on average, a 5 % difference between the measured thrust and the
results for the upstream wind turbine and less than a 10 % difference for
all cases concerning the downstream wind turbine, with the exception of the
results when using the undistributed or uniform thrust. These differences
increase when considering different turbulence models; this is due to the
different associated wake development corresponding to the different
turbulence models. When considering the k-ω turbulence model, the
simulations greatly overestimate the values of the thrust coefficient for the
second wind turbine and vice versa for the RNG k-ε.
Thrust coefficients for case B and sub-cases C1 and C2 when using
the k-ε turbulence model for the first and second wind
turbines.
Case B
Sub-case C1
Sub-case C2
CT,T1
CT,T2
CT,T1
CT,T2
CT,T1
CT,T2
Experimental
0.883
0.363
0.833
0.569
0.785
0.486
Polynomial
0.797
0.379
0.825
0.608
0.809
0.454
Trapezoidal
0.825
0.367
0.856
0.621
0.841
0.478
Triangular
0.822
0.390
0.837
0.600
0.824
0.457
Uniform
0.830
0.269
0.854
0.571
0.840
0.396
Undistributed
0.829
0.268
0.853
0.580
0.840
0.400
(a) Normalized axial velocity and (b) normalized
turbulence kinetic energy computed for the two in-line model wind turbines
(case C1) at x/R=5.54, 10.36 and 17.00 downstream of the first wind
turbine. Different thrust distributions and the k-ε
turbulence model are used.
(a) Normalized axial velocity and (b) normalized
turbulence kinetic energy computed for the two in-line model wind turbines
(case C2) at x/R=5.54 downstream of the first wind turbine. Different
turbulence models are used along with the undistributed thrust.
(a) Normalized axial velocity and (b) normalized
turbulence kinetic energy computed for the two in-line model wind turbines
(case C2) at x/R=5.54 downstream of the first wind turbine. Different
thrust distributions and the k-ε turbulence model are used.
Concerning sub-cases C1 and C2, empty-domain vertical profile results for
the stream-wise velocity and the turbulence intensity at four axial
positions downstream of the inlet are shown in Fig. . Illustrated in
Fig. are the measured and simulated axial velocity and turbulence
kinetic energy at three distances downstream of turbine T1 when using the
undistributed thrust and different turbulence models for case C1. Figure presents the same quantities when using the k-ε
turbulence model but varying the thrust distribution. Lastly, for case C2
results for the stream-wise velocity and the turbulence intensity profiles
are shown in Fig. for different turbulence models when using the
undistributed thrust. The effect of the thrust distribution on the wake is
investigated by varying the thrust distribution while keeping the same
turbulence model; see Fig. . Results of the thrust coefficient
values upstream and downstream from the wind turbine are summarized in Table for cases C1 and C2; the k-ε turbulence model is used
here.
The empty-domain simulations presented in Fig. give a reasonably
good agreement between simulations and measurements for both the uniform and
the sheared inflow condition. Concerning the sub-case C1, the simulated
stream-wise velocity is in quite good agreement with the measurements for all
turbulence models and thrust distributions; see Figs. and
. In this case of an environment with high background
turbulence, the k-ω turbulence model shows remarkably better
agreement with the measurements when compared to the previous cases. The
shape and level of the normalized turbulence kinetic energy is now captured
by all turbulence models, though small differences are observed between the
simulated wakes when different turbulence models are used. This finding is in
agreement with results from the studies performed by
and , in
which small differences between the simulated wakes are also found when
different RANS turbulence models are used in an environment with high
background turbulence intensity. It should be recalled that empty-domain
simulations were performed to match the background turbulence intensity with
the experimental measurements when using different turbulence models for all
cases prior to the simulations with the ACD. It appears that for the cases
with high turbulence intensity this procedure has a significant effect on the
computational results compared to the cases with low turbulence intensity. As
the purpose of using different turbulence models in this study is to
investigate the effect of the turbulence model with its defined constants on
the wake development, it is crucial to set the background turbulence
intensity throughout the domain in accordance with the experimental set-up
when using different turbulence models. This is achieved here by varying the
inlet turbulence parameters (k, ε or ω). In this way the
background turbulence intensity is similar when using different turbulence
models and the effect of the turbulence model on the wake development may be
clearly accounted for. When the cases with higher background turbulence
intensity are considered, it is observed that the effect of the thrust
distribution is less apparent further downstream compared to the lower
turbulence intensity of cases A and B. Similarly, for sub-case C2 (see
Figs. and ) the axial velocity in the wake is predicted
quite well for all turbulence models and thrust distributions. Because the
measurement position is in the near-wake region, the effect of the thrust
distribution is apparent on the turbulence kinetic energy and axial velocity
profile. From Table it is found that thrust coefficient values are
estimated on average to have less than a 10 % difference from the
measured values for both sub-cases when using the k-ε
turbulence model.
Lastly, in terms of computational time or CPU hours, herein defined as the
number of CPUs × wall clock time needed to perform the simulation,
results are shown in Table . These results present the CPU hours
needed to perform the simulations using this method and a LES with the
ACL method described in for case A.
It is found that the RANS–ACD method is significantly faster in simulating
this one wind turbine case compared to the LES–ACL method. Although the
LES–ACL method provides high-fidelity results comparable to the measurements,
the computational requirements of this method, as for now, are still too
demanding to make it usable for wake modelling in industrial applications.
Averaged computational effort in CPU hours to perform the
simulations of case A.
Description
Cells per rotor
Cells in the
CPU
diameter
domain
hours
RANS ACD
40
4.8×106
20
LES ACL
86
24.5×106
1280