Introduction
The performance of a wind farm does not only depend on the ability of its
wind turbines to convert available kinetic energy into electric energy but is
also largely influenced by the fluctuation of the atmospheric winds and the
wakes created by the turbines. Wind turbine wakes are areas of lower wind
speed and enhanced turbulence that result from the extraction of kinetic
energy from the flow by the turbine and can have a significant impact on the
wind conditions up to 10–15 rotor diameters downstream. To minimize the
losses due to wind turbine wakes, the wind rose measured at a location is
usually taken into account during the design process of the wind farm layout.
However, in most locations, in particular in mid-latitudes with alternating
low- and high-pressure systems, the unsteady wind direction creates a high
occurrence of situations for which wake losses remain large.
Multiple studies, e.g., , have shown that the
wake losses in wind farms depend on the turbulence intensity of the ambient
wind, with decreasing efficiency of the wind farm for low turbulence. Sources
of turbulence in the atmospheric boundary layer are mechanical shear and
buoyancy. The latter depends mainly on the thermal stratification and can
also be a sink of turbulence. In a stably stratified atmospheric boundary
layer (SBL) turbulence is suppressed by the stable thermal stratification
that decelerates the vertical movement of air masses while in a convective
atmospheric boundary layer (CBL) the source of energy at the bottom of the
atmosphere enhances the turbulent motion. Studies of atmospheric stability
have shown that convective and stable conditions occur at least as often as
neutral conditions (NBL) at onshore and
offshore wind farms and
that wind farms are least efficient in stable conditions
.
The observation of a change of wind farm performance with different
atmospheric stability has been supported by wind tunnel experiments and
numerical studies. It has been either related to a generally different level
of turbulence or to the presence of large-scale
fluctuations that enhance the so-called meandering of the wakes in less
stable situations .
and argue that the thermal stratification
above the wind farm becomes important for large wind farms as the vertical
momentum transport becomes the only kinetic energy source to refill the wake
deficit. Apart from the energy yield, the structural loads on turbines in the
wake also differ with atmospheric stability as they are influenced by up- and
downdrafts and large coherent structures in a CBL
and by sharp velocity gradients in an SBL .
With increasing capacity of wind turbines the value of every additional
percentage of energy that can be harvested from the wind becomes larger. As a
consequence the interest to increase the power output for unfavorable wake
situations is growing. Recent studies focus on the control of upwind turbines
to minimize wake losses of downwind turbines by either reducing the induction
or by an intentional yaw angle of the turbine to the wind
direction . The first approach
aims on less extraction of energy from the wind by the upwind turbine and
therefore more remaining energy that can be extracted by downwind turbines.
The second approach relies on an induction of a cross stream momentum by the
upwind turbine to change the trajectory of the wake with the goal to deflect
it away from the downwind turbine. While in both approaches the upwind
turbine experiences a loss in power and possibly an increase in structural
loads, the additional gain at the downwind turbine is assumed to exceed this
loss, thus leading to a surplus of total power output of the wind farm. Based
on this assumption, simple models for a joint control of wind turbines to
increase power output during operation for a fixed layout have been proposed
. even suggest including
power yield optimization by wind farm control in the design process of new
wind farm layouts.
Crucial for wind farm control models is a proper description of the wake
trajectory as a wrong description would almost certainly lead to a reduction
of energy yield of the wind farm due to the lower energy yield of the upwind
turbines. However, magnitudes of the wake deflection differ already in the
parameterizations of and . Possible
reasons for the differences include the use of different turbine models, the
method to extract the wake trajectory from the measured wind field and the
ambient wind conditions. Apart from the differences in the description of the
mean wake trajectory, an aspect that is not considered yet in current wind
farm control models is the stochastic nature of the wake trajectory.
show not only that the movement of the wake becomes more and
more stochastic for small averaging intervals, but also that these motions
are linked to atmospheric stability. Considering that the potential to
improve wind farm efficiency through wind farm control appears to be
dependent on atmospheric stability, little knowledge exists on how the
control would need to adapt to changes of the wind conditions as influenced
by atmospheric stability.
In this study we analyze multiple sources that contribute to the uncertainty
of the estimation of the wake deflection downstream of yawed wind turbines in
different ambient wind conditions. The ambient wind conditions are created by
Large Eddy Simulations (LES) of atmospheric boundary layers of neutral,
stable and unstable stability. The simulations are run with the same mean
wind speed and wind direction but changing the stability produces differences
in the shear and turbulence of the wind. The wind turbine wakes are created
by enhanced actuator disc models with rotation . We
use the data from these simulations not only to analyze if the stability
changes the magnitude of the wake deflection but also to compare different
fitting routines to extract the wake center. In addition to these aspects,
that we already consider as contributors to the uncertainty of the wake
deflection estimation, we also look at the influence of different temporal
averaging intervals on our results.
Methods
Estimating the wake deflection
We assume that the wake position μy at a certain distance downstream of
a wind turbine can be predicted when the hub height wind direction
αh and the wake deflection Δyγ are known.
μy=y0(αh)+Δyγ,
where y0 is the displacement of the wake in a fixed coordinate system by
the change of wind direction (Fig. ).
Conceptual image of the method to calculate the wake deflection Δyγ(x2)
by using the inflow wind direction αh(x1) of the wind speed uh(x1) at hub
height and the position of the wake center μy(x2). Here, the x axis is the mean
wind direction. The yaw angle γ is defined relative to αh, with γ>0
for a clockwise turning of the rotor.
Inflow wind speed and direction are averaged along Δy.
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The advantage of LES is that the wake position and the wind direction can be
assessed directly from the flow field to estimate the unknown deflection of
the wake by the yawed turbine. For a fixed thrust coefficient, turbine site,
wind speed and wind direction, the wake deflection is assumed to be a
function of the yaw angle γ and the atmospheric stability, e.g., given
by the Monin-Obhukov length L.
Δyγ=Δyγ(γ,L)
The relationship of Δyγ on the yaw angle and the atmospheric
stability is estimated from multiple LES with different γ and L.
<Δyγ>|γ,L=<μy(fi)>-<y0(αh)>
Here we consider that the estimate of μy depends on the algorithm fi
used to estimate the wake center position from the simulated flow field. To
calculate the temporal variation of the wake deflection we divide the time
series into shorter intervals Δt and calculate the variance of this
individual estimates about the mean.
Estimating the wake displacement by the change of wind direction
We consider the wind conditions at x1=2.5 rotor diameters (D) upstream
as reference inflow conditions to a wind turbine. This distance is chosen as
the wind field closer to the turbine might be modified by the induction of
the rotor . More precisely our inflow information is hub
height wind speed uh and wind direction αh
averaged at x1 on a line extending Δy=2 D perpendicular
to the expected mean wind direction (Fig. ). We choose cross
stream averaged variables instead of a point measurement as we consider them
more representative for the wind conditions for the wind turbine rotor.
To estimate the wake displacement y0 we assume an advection of the wake
with the ambient wind. If the wind direction coincides with the x axis
(αh=0), the wind flows along the x axis and interacts
with the wind turbine to form a wake structure that is advected downstream,
supposedly centered around y0=0. For wind directions αh≠0 the x axis and wind direction differ and the center μγ
of the wake is expected to be shifted by y0=Δx2tanαh along the y axis (Fig. ). As
we only consider deviations of the wind direction from the x axis of less
than 10∘, the change of x2 with αh is neglected.
This simple consideration already allows for a first estimation of how the
uncertainty from the calculation of the wind direction can propagate into the
error of the wake deflection estimation. For an error of the wind direction
estimation of σαh=±5∘(10∘) the
wake center displacement y0 at x2=6D downstream would have
an uncertainty of σy0≈±0.5D(1.0D).
Estimation of the wake center
Three different methods to estimate the wake center position are compared in
this study to assess the bias introduced to μy by the choice of the
method fi. As a first approach the position of the wake is calculated by
fitting the mean wake deficit at hub height to a Gaussian-like function.
fh(y)=uaexp-(y-μy)22σy2
The center μy of the Gaussian is considered as the horizontal wake
center, the amplitude ua as the wake deficit and σy as a
measure of the width of the wake.
As we also have information about the vertical structure of the wake, a two dimentional Gaussian-like fit as proposed
by is used as alternative fitting routine.
f2 D(y)=uaexp-12(1-r2)(y-μy)2σy2-2ρ(y-μy)(z-μz)σy2σz2+(z-μz)2σz2
with μz the equivalent to μy on the vertical axis and r2<1 a correlation factor.
For a perfect circular shape of the wake r=0, whereas for an elliptic wake shape r≠0.
Both functions are fitted to the data through a least-squares approach.
We introduce a third method to determine the wake position based on the
available mean specific power in the wind (AP). As the main interest
of wind farm control is the increase of the power output of downstream
turbines, we consider the position along the y axis of a hypothetical
turbine placed at x2 that feels the lowest AP as the center point
of the wake. For this purpose the cube of the mean flow in wind direction is
averaged on circular planes of diameter D centered around hub height
zh. The AP is normalized by the air density, as density
variations are not considered.
fAP(y)=1/2∫y1y2∫z1z2u3(y′,z′)dz′dy′,(y′-y)2+(z′-zh)2≤(D/2)2
The wake center μy is the value of y that minimizes
Eq. ().
Temporal averaging interval
To study the uncertainty of the wake deflection by the used temporal averaging interval,
we divide time series of inflow at x1 and wake flow at x2 in multiple time intervals Δt.
We chose time intervals of respectively Δt=10, 3 and 1min as we consider them realistic for wind farm control.
For small Δt the wind conditions at x1 and x2 become more and
more uncorrelated, thus the advection time of the turbulent structures
between these points is considered for each averaging interval. Turbulent
structures in the wind field are expected to be transported by the mean wind
following Taylor's hypothesis of frozen turbulence. To describe the time
τ it takes for a structure to be advected from the position x1 to the
position x2 we use the following approximation:
τ=(Δx1+Δx2)/uh
with Δx1 and Δx2 being the distances from x1 and x2
to the wind turbine, respectively. In presence of a turbulent structure of
lower velocity like a wind turbine wake, the advection velocity downstream of
the turbine along Δx2 is not well studied. Following
we assume that the wake is moved like a passive tracer by
the ambient wind field. Thus the advection velocity downstream of the turbine
remains the same as upstream.
Combining the methods presented in previous subsections we find multiple
estimates of the wake deflection Δyγ by calculating the wind
direction αh and the wake center μy for different
averaging intervals Δt, with the time series at x2 shifted by
τ, and for different methods fi to identify the wake center from the
wake flow.
LES model
The simulations presented in here are conducted with the LES model PALM .
PALM is an open source LES code that was developed for atmospheric and oceanic flows and is optimized for massively parallel computer architectures.
It uses central differences to discretize the non-hydrostatic incompressible Boussinesq approximation of the Navier-Stokes equations on a uniformly spaced Cartesian grid.
PALM allows for a variety of schemes to solve the discretized equations.
The following schemes are used in this study:
advection terms are solved by a fifth-order Wicker-Skamarock scheme,
for the time integration a third-order Runge-Kutta scheme is applied.
For cyclic horizontal boundary conditions a FFT solver of the Poisson equation is used to ensure incompressibility,
while for non-cyclic horizontal boundary conditions an iterative multi-grid scheme is utilized.
A modified Smagorinsky sub-grid scale parametrization by is used to model the impact of turbulence of scales smaller than the model grid length on the resolved turbulence.
Roughness lengths for momentum and heat are prescribed to calculate momentum and heat fluxes at the lowest grid level following Monin-Obukhov similarity theory.
The simulations in PALM are initialized with a laminar flow field. Random
perturbations of the flow during the start of the simulation initiate the
development of turbulence. The statistics of the steady turbulence that
develops after some spin-up time depend on the initial conditions provided
for the fluid, e.g., the temperature profile, and the boundary conditions
during the simulation, e.g., surface heat fluxes. For more information about
the general capabilities of the model the reader is referred to
.
Wind turbine model
The effect of the wind turbine on the flow is parameterized by means of an
enhanced actuator disk model with rotation (ADM-R) as in
. The rotor disk is divided into rotor
annulus segments with changing blade properties along the radial axis. The
blade segments positions are fixed in time but each owns an azimuthal
velocity due to the clockwise rotation of the rotor. Local velocities at the
segment positions are used in combination with the local lift and drag
coefficients of the blade to calculate lift and drag forces. The forces are
scaled for a three bladed turbine and are afterwards projected onto the grid
of the LES by a smearing function with a Gaussian kernel as described in
. In internal sensitivity studies we found that a
value of twice the grid size is a good choice for the regularization
parameter as also concluded by . The rotor can be rotated
around the y axis and the z axis enabling a free choice of yaw and tilt
configuration. The influence of tower and nacelle on the flow is represented
by constant drag coefficients.
The blade properties as well as the hub height of zh=90m and the rotor diameter of D=126m originate
from the NREL 5MW research turbine . A variable-speed
generator-torque controller is implemented in the same way as described in
. Note that no vertical tilt is applied to the rotor to
exclude the wake displacement that might result from a mean vertical momentum
of the wake.
Domain of the main simulations. Lxp is the length of the prerun domain.
The turbulence at the recycling surface S is used as input at the inflow again. LI is the distance from the recycling surface to the wind turbine.
![]()
Setup of the three simulations and results by the end of the prerun.
Domain dimensions (see Fig. ) are given in multiples of rotor diameter D.
The number of turbines in the main simulation is nT.
Results consist of wind speed uh and turbulence intensity TIh at hub height,
wind shear coefficient αs and veer δα, both evaluated
between lower and upper rotor tip, Monin-Obukhov-Length L, and boundary layer height zi.
Setup
Results
Lx
Lxp
Ly
LI
Lz
nT
uh
TIh
αs
δα
L
zi
[D]
[D]
[D]
[D]
[D]
[ms-1]
[%]
[]
[∘]
[m]
[m]
SBL
30.5
11.4
7.6
3.0
4.5
1
8.4
4.0
0.30
8.2
170
300
NBL
61.0
23.7
20.3
6.0
13.6
1
8.3
8.3
0.17
2.2
∞
550
CBL
132.0
81.3
50.8
8.0/20.0
11.6
8
7.8
13.3
0.08
0.6
-180
650
Precursor simulations
Precursor simulations of the atmospheric boundary layer for the
representation of three different atmospheric stabilities, stable, neutral
and convective, are conducted with the goal of creating different shear and
turbulence characteristics but with the same mean wind speed and direction at
hub height. All domains have a horizontal and vertical grid resolution of
Δ=5m up until the initial height of the boundary layer in
each simulation. Above this height the vertical grid size increases by 6 %
per vertical grid cell. The roughness length is kept constant in all
simulations at z0=0.1m, representing a low onshore roughness
representative for low crops and few larger objects. The Coriolis parameter
corresponds to 54∘N. Cyclic lateral boundary conditions
are used and the simulations are initialized with a vertically constant
geostrophic wind. Due to Coriolis forces, bottom friction and stratification,
height-dependent wind speed and wind direction profiles evolve after several
hours of spin-up time.
For the generation of a SBL, a constant cooling of the lowest grid cells is
prescribed. The initial temperature profile of the potential temperature
Θ and the rate of bottom cooling (dΘ/dt=1K/4h) are set as in . A CBL is
established by prescribing a constant kinematic sensible heat flux of
60Wm-2 at the bottom boundary. The bottom heat flux is
fixed to zero for the NBL. The initial potential temperature profiles of the
NBL and CBL are constant up to 500 m height with a strong inversion
of dΘ/dz=8K/100m between 500 and
600 m and a stable stratification of dΘ/dz=1K/100m up to the upper model boundary.
The results of the precursor simulations are shown in
Figs. , and Table . The
simulations differ in their horizontal and vertical extent (see
Table ), a consequence of the different heights of the mixing layers
and the different sizes of the largest eddies that need to be explicitly
resolved. These simulations are afterwards used as initial wind fields for
the main simulations described in Sect. 2.8 that include the impact of the
wind turbine on the flow by the ADM-R parametrization. As intended, the
domain averaged profiles have similar mean wind speed and direction at hub
height but differ in vertical shear of the wind speed, wind veer and
turbulence intensity (Fig. ). The SBL is characterized by a
strong vertical shear of wind speed and wind direction over the height of the
rotor. Shear coefficient αs=0.30 and Monin-Obhukov length
L=170m correspond to a stable to highly stable stability class
following . The wind direction changes by 8∘ from
the lower rotor tip to the upper rotor tip. Below the top of the SBL at
around zi = 300 m, the wind speed has a super-geostrophic maximum,
an event called low level jet, that has been documented in measurements
onshore as well as offshore
.
Flow statistics during the last hour of the
precursor simulations. (a) Horizontally averaged vertical profiles
of wind speed, flow direction, potential temperature and turbulence
intensity. Horizontal lines denote the height of the blade tips and the hub.
(b) Distribution of the 1 Hz wind direction from point measurements
at hub height.
![]()
Energy spectral density of the three different
wind components at hub height during the last hour of the precursor
simulations. The gray line denotes the slope of the Kolmogorov cascade.
Vertical lines are at T=10 min, 3 min and 1 min.
![]()
The NBL and the CBL exhibit only low vertical dependency of the wind vector
above the lower rotor tip. Responsible for the low vertical wind speed
gradient is the increased amount of turbulent kinetic energy that leads to a
stronger mixing. The spectra of the three velocity components at hub height
shown in Fig. reveal that not only the total amount of
turbulent kinetic energy is larger in the neutral and convective case, but
the most energetic motion also occurs on larger scales.
The CBL represents a rather moderate convective boundary layer with L=-180m and a ratio between the boundary layer height zi and L
of zi/L=-3.6. Characteristic for such moderate convective boundary
layers in flat terrain are large roll-vortices, whose axes of rotation are
approximately aligned with the mean wind direction and that have a vertical
extension up to the top of the boundary layer
. The presence of these vortices can be seen
in the highly energetic low frequently motion of the v- and w-components
and the large variance of the wind direction.
The meteorological conditions of the CBL and SBL simulation cases are
regularly occurring at wind farm sites
. Numerical simulations
comparable to the CBL and NBL case are studied in ,
while simulate even stronger stable and convective
conditions, which are motivated by measured events.
Setup of the wind turbine wake simulations
For the main simulations a turbulence recycling method is
used at the upstream domain boundary instead of a cyclic boundary
(Fig. ). This allows for studying a single turbine along the
x axis instead of an infinitively long row of turbines. Undisturbed outflow
at the right boundary is ensured by a radiation boundary condition. For the
use of the turbulent recycling method the model domain from the precursor
simulations is extended along the x axis and the recycling surface is
positioned at the domain length Lxp of the precursor run. Test
simulations showed a minimum of Lymin≈8D to prevent
blockage of the flow by the turbine and a minimum distance between recycling
surface and turbine of LImin≈3D to prevent an
influence of the induction zone on the turbulence at the recycling surface.
(a–c) Mean wake deficit 6 D
downstream of a wind turbine in the NBL. The turbine is yawed by
(a) 30∘, (b) 0∘ and
(c) -30∘. Straight contours denote the position of the
upstream turbine. Dashed contours are the isolines of constant
f2 D. (d) Cross sections of normalized udef
at hub height (thin) and results of fAP (bold) and fh
(dashed).
![]()
Wake deflection trajectories in the NBL from different fits to the data.
Numbers on the right denote the turbine yaw angle for the different trajectories.
![]()
The main simulations of the NBL and SBL are conducted for single turbines
with a different yaw angle to the x axis. For each change in yaw angle a
separate simulation of 25 min length is conducted from which the
first 5 min, during which the wake still develops, are discarded from
the analysis. Yaw angles ranging from -30 to 30∘ in steps of
10∘ are chosen. Positive yaw angles are defined as a clockwise
turning of the rotor when seen from above and the wind coming from the left-hand side.
In the CBL the domain width Ly is more than 6 times larger than the
minimum size of Lymin. We use this to include all different turbine
yaw angle configurations in one simulation consisting of two staggered rows
of four turbines each, separated by more than Lymin in y and
12D in x direction. The distances are chosen large enough that a
mutual interaction of the turbines can be excluded. Each of the turbines had
a different yaw angle to the x axis and the simulation was run for 65 min
from which the first 5 min were discarded. The longer simulation time
of the CBL is motivated by the larger turbulence length scales of the flow
that cause longer necessary averaging intervals to get information about mean
properties. Note that due to the cyclic lateral boundary conditions, the
turbines in all simulations are part of an infinite row along y separated
by more than Lymin.
(a–c) Residual cross stream component of the flow at
x2=2D downstream of the wind turbine for the same simulations as in Fig. .
Positive (negative) values stand for a flow to the right (left). Dashed contours denote the position of the wake deficit.
(d) Vertical profile of the total v component at y0 and x2, and the average inflow profile.
![]()
Scatter plot of the horizontal wake deflection in the NBL from the f2 D-fit over yaw angle γ at
different downstream positions x2 and for different averaging intervals.
![]()
Results
In this section we compare the results of the main simulations with presence
of wind turbines. The vertical planes of the LES flow that are shown on the
following pages represent the view of an upstream observer looking
downstream. If not explicitly noted otherwise, the zero coordinate of the
x axis coincides with the x position of the rotor center and the zero
coordinate of the y axis with y0, i.e., the zero coordinate of y
corrected by the measured inflow wind direction αh. The
y axis is positive to the left-hand side of the upstream observer.
Neutral atmospheric boundary layer
We start the analysis with the NBL, as this case is the most studied case in
wind energy applications. Figures a–c show vertical planes of
the wake deficit udef, averaged over the whole simulation time,
for three different yaw angles γ at x2=6D. The velocity
udef is defined as the difference between the inflow velocity
profile of u(y,z) measured as inflow at x1 and averaged along Δy=2D and the velocity field u(y,z) at x2 downstream of the
wind turbines (Fig. ). The isolines of the 2-D fitting
method f2 D are denoted by dashed contours. The wake deflection
Δyγ that results from this routine is visible as the
innermost ring. Cross sections of Fig. a–c at hub height are
shown together with the results of fh and fAP in
Fig. d. The wake centers are the positions along y for which
the functions take the smallest values.
As apparent in Fig. the wake deficit is lower for the two
cases of turbines with a large yaw angle, a consequence of the loss of energy
yield and induction, if a wind turbine is yawed out of the wind direction.
For a positive (negative) yaw angle the wake deficit is deflected to the left
(right) when looking from upstream. Figure shows the mean
deflection Δyγ of the wake center for multiple distances
downstream of the rotor using the three different approaches fi. The
Gaussian-like fit at hub height fh returns the largest deflection
of the wake. The smallest deflection is found when the wake is approximated
by the 2 D normal fit f2 D while the wake position of
minimal fAP lies mostly between the two curves.
The reason for the different output of the three methods is the deviation of
the wake from a perfect symmetric shape as evident in Fig. .
The crescent shapes of the wakes indicate that the lateral displacement is
largest at the height around the rotor center while it is lower around the
upper and the lower rotor tip, which explains the largest magnitude of wake
deflection for fh.
A look at the cross stream component of the flow reveals the origin of the
crescent shape of the wakes of a yawed turbine. Figure shows
the residual cross stream component of the flow in the near wake. The
residual component is the difference between the inflow profile and the
downstream wind field. For γ=0∘, the dominant feature of the
cross stream flow is the counterclockwise rotation of the wake that is
induced by the clockwise rotation of the rotor. For γ≠0∘,
the rotation is superimposed by the induction of cross stream momentum caused
by the yawed turbine. Figure a, c show that this cross stream
momentum is either opposing the rotor rotation below or above the hub, which,
together with the influence of wind veer, leads to the asymmetries further
downstream as evident in Fig. a, c.
As apparent in Fig. the induced cross stream momentum also
triggers a counter momentum above and below the rotor area. The opposing
cross stream velocities appear to be responsible for the varying magnitude of
lateral displacement at different heights and the crescent shape of the wake
further downstream. The counter momentum is stronger below the rotor area,
which is likely to be related to the presence of the bottom just 27 m
below the blade tip.
To assess the influence of the temporal averaging interval on the standard
deviation of the wake deflection, Δyγ is calculated for
different time intervals. Advection of frozen ambient turbulence between
x1 and x2 is considered by shifting the second time interval by τ
(Eq. ). To have more than two estimates for the 10 min
interval, the intervals are overlapping to a large degree resulting in seven
individual estimates per yaw configuration. Figure shows the
spread of the estimates of f2 D at two different positions x2.
We find that the standard deviation of the wake deflection appears to be
independent of the yaw angle but depends on the temporal averaging interval.
The used fitting method has little influence on the standard deviation of the
mean wake deflection in the NBL (Table ).
Stable atmospheric boundary layer
As shown earlier in Fig. , the simulated SBL is
characterized by lower TI and a stronger vertical shear of wind speed and
direction than the NBL. For the simulated wind turbine wake in the SBL, the
strong wind veer leads to a strong slanted shape of the wake deficit, even if
the rotor plane is perpendicular to the wind direction at hub height
(Fig. b). Below the rotor center, the wake is shifted towards
the left-hand side and above towards the right-hand side. Thus, the extend of
the wake cross section at hub height (Fig. d) is less
representative for the whole wake extension than in the NBL simulation
(Fig. ). The amplitude at x2=6D of the wake
deficit udef is larger than in the NBL. The larger amplitude can
be related to the lower ambient turbulent kinetic energy and to the lower
fluctuation of the inflow wind direction.
Standard deviation of the wake deflection at x2=6D for different Δt[min]. Values are averages over
all seven yaw configurations. Note that the 10 min standard deviation might
be biased as the intervals are not strictly independent.
std(fh)
std(f2 D)
std(fAP)
[10-1D]
[10-1D]
[10-1D]
Δt
10
3
1
10
3
1
10
3
1
SBL
0.1
0.3
0.5
0.1
0.3
0.5
0.1
0.3
0.5
NBL
0.4
1.2
2.2
0.4
1.3
2.2
0.3
0.7
1.6
CBL
1.4
2.4
2.8
1.3
2.4
3.0
2.0
2.2
2.3
Same as in Fig. but for the SBL simulation.
![]()
The wakes for γ≠0∘ show a similar crescent shape to the
wakes in the NBL. The differences between the deficit position at hub height
and around the upper and lower rotor tips are even larger, a consequence of
the addition of induced momentum by the yawed turbine and ambient wind veer.
In the case of a yaw angle of γ≈-30∘ the lower part of
the wake detaches from the rest of the structure. In contrast to the fit
f2 D of the wake at γ≈30∘ this detached
part is neglected by the optimal fit.
The trajectories of the wake deflection shown in Fig. have a
distinct bias to the right of the rotor. This appears in all trajectories but
is strongest in the f2 D trajectory where basically no deflection
to the left is found. The wake deflection to the right may be related to two
different mechanisms. Firstly, it can be related to advection of lower
momentum from below the rotor to one side and advection of high momentum from
above the rotor to the other side of the wake by its rotation. The second
effect that could be responsible for the deflection of the wake to the right
is the stronger veer of the wind in the upper rotor half, where the mean flow
is towards the right, compared to the lower rotor half, where the mean flow
is slightly towards the left. Trajectories of simulations with a reversed
rotation of the rotor show that the sense of rotation is not exclusively
responsible for the bias to the right as this would lead to a mirroring of
the trajectories about the wind direction for opposite rotor rotations
(Fig. ). As apparent in Fig. , the wake center
is located a little higher than hub height, therefore the ambient wind
direction at wake center height could also lead to a slight advection towards
the right. Thus both effects seem to be responsible for the difference
between the wake deflection in the SBL and the NBL.
Same as in Fig. but for the SBL simulation.
Crosses mark the wake trajectories for simulations with opposite sense of rotation of the rotor.
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The uncertainty of the estimate of the wake deflection is much smaller in the
SBL than in the NBL for all time intervals (Fig. ). Compared
to the NBL, the variance of the wind direction (Fig. b) is
lower and the energy of the cross stream motion (Fig. ) is
already low on the minute scale. Thus, a 1min averaging window
filters most of the cross stream fluctuation that might be responsible for
the uncertainty of the prediction of the flow field between x1 and x2
and therefore the uncertainty of the wake deflection.
Same as in Fig. but for the SBL simulation.
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Convective atmospheric boundary layer
The deflected wakes in the CBL show a completely different behavior than in the previous presented boundary layer simulations.
Figure shows the yz transects as in Figs. and but for the CBL.
The results are averaged over 1 hour of simulation time instead over 20 min like in the other simulations.
The large deficit width in Fig. is mainly a consequence of the large variance of wind direction (Fig. b) during the averaging time interval,
that leads to a strong fluctuation of the wake position .
A consequence is a much weaker mean deficit than in the NBL and SBL simulations.
Same as in Fig. but for the CBL
simulation and for a time series of 60 min.
t]
As Fig. shows, the wake deflection to the left (right) for a positive (negative) yaw angle is not found in the results of the CBL simulation.
This does not only hold for the long time average but also for shorter time intervals Δt as apparent in Fig. .
The uncertainty of the estimated wake deflection is less dependent on the averaging interval than in the other simulation (Table ).
Same as in Fig. but for the CBL simulation and for a time series of 60 min.
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Following the considerations made in Sect. 2.3 about the uncertainty of the wake deflection due to the uncertainty of the wind direction,
an approximate error of ±2.5∘ of the 3 min wind direction αh can be derived from the spread of the 3 min results (Table ).
A large spread of yaw angles of the turbines to the wind is encountered during the simulation (Fig. ).
The reason for the spread are the wide streaks of the convection rolls that create strong cross stream components (Fig. ),
a feature that distinguishes the CBL from the other simulated cases.
Due to this feature, the local inflow wind direction usually differs from the domain-averaged wind direction,
shown in Fig. , to which the turbines are originally yawed.
These streaks explain the spread of identified wind directions but can not explain the high variance of the wake deflection for the same yaw and inflow angle.
Moreover, the averaged wind speed and direction measured in front of the turbine appears to be insufficient to characterize the flow further downstream.
Example of the instantaneous v component at hub
height in the CBL. Turbine wakes are denoted by black contours. Black lines
denote the rotor positions, gray lines denote the position of the inflow
measurement for each turbine.
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To test the similarity of the free stream flow at different streamwise locations we calculate the root mean square error (RMSE) of two time series in undisturbed flow
with and without considering the time shift τ (Fig. ).
Wind speed and wind direction are averaged at hub height along a cross stream distance as described in Sect. 2.3.
A shift of the downstream time series by τ has the largest effect on the similarity of the wind conditions
in the CBL, where especially the variance of the wind direction is large.
On the other hand that means that a bad estimation of τ introduces the largest error to the estimation of y0 in the CBL.
RMSE of the time series of 3 min averaged
ah and uh at two different positions in the model
domain separated by Δx=8D, with a advection time shift of
the downstream time series of τ (Eq. ) and without time
shift (τ=0).
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Discussion of the wake deflection estimation
Three different sources of uncertainty of the wake deflection estimation are evaluated in this study.
First we show that the incoming wind shear and veer has to be well known by comparing the results
from the neutral and stable thermal stability situation.
The influence of shear and veer is not considered yet by studies of potential improvement of the
wind farm efficiency with wind farm control like and .
Table shows the coefficients derived from the two simulations for the analytical
description proposed in and compared to their results.
show that the energy yield of a small wind farm can be well predicted by a simplified parametric model,
which is fitted to simulated atmospheric conditions of neutral stability, and that the energy
yield of a small wind farm can be improved by more than 10 % for certain scenarios.
Assuming the same parameters for the stable wind field from our study would lead to a miscalculation
of the wake position which corresponds to a yaw induced deflection by a yaw angle of about 10∘.
Thus, the described parametrization of the model would likely propose an unfavorable control for stable situations.
A proper description of the wake trajectory in stable situations is important as the interest to apply
wind farm control in stable atmospheric stability should be higher than in more turbulent conditions due to the increased wake losses.
With the high occurrence of stable situations onshore as well as offshore
the difference in the wake trajectory might be even worth considering in the design process of a wind farm.
As a second source of uncertainty we consider the choice of the method to derive the wake position.
These methods are most often dependent on the measurement device thus we do not expect that it will
be possible to establish a universally applicable method in the near future.
For future studies aiming to study the deflection of the wake we emphasize that the choice of wake
fitting routine for the measured wind field has significant influence on the results
in particular when the turbine yaw angle is large.
Best fit parameters to the wake deflection output of
the different methods using , Eq. (12). Comparison with the
results of the aforementioned study and with . The
parameter kd defines the recovery of the wake trajectory to the
mean wind direction, and ad and bd the displacement
due to the interaction of wind shear and rotation of the wake.
fh
f2 D
fAP
kd
ad, bd
kd
ad, bd
kd
ad, bd
SBL
0.14
-7.7, -1.4
0.23
-8.1, -2.1
0.19
-6.0, -2.4
NBL
0.16
-3.1, 0.4
0.25
-2.8, 0.9
0.18
-2.4, 0.3
Jim.
0.06
–
–
–
–
–
Geb.
0.15
-4.5, -1.3
–
–
–
–
The third source of uncertainty that is considered in this study is the influence of
the time averaging interval to find the wake deflection.
The underlying question behind this analysis is: at what timescales makes wind farm control
sense and what needs to be taken into account at the different timescales.
In the NBL and SBL cases the estimation of the wake deflection on a 10 min scale shows only little variance.
However, here we benefit from the steady wind field in the LES where we do not expect a change of wind direction over this time interval.
In practice, meso-scale wind fluctuations might cause a change of the wind direction on this time scale.
For smaller time intervals than 10 min the variance of the wake deflection increases,
thus a prediction of the wake position by measuring the inflow becomes more uncertain.
The CBL analysis differs from the two other cases as we find no correlation between yaw angle of
the turbine and wake deflection on any of the tested time averaging intervals.
This makes a prediction of the wake position more uncertain and makes an interference by yaw control unreasonable.
Apparently, the stochastic fluctuation of the wake caused by the large fluctuations of the cross stream
component are superimposing the trajectory change of the wake caused by the induction of the turbine
to a degree that the latter signal is not detectable any more.
The larger fluctuation of the wake trajectory in convective conditions has been shown before in measurements
and simulations but has not been related to the applicability of wind farm control, yet.
Investigating the hypothesis of frozen turbulence in flow undisturbed by the wind turbine shows that the consideration
of the time delay between the time series at two streamwise positioned measurements is especially important in the CBL.
However, in flow with a wake structure of lower mean velocity than the ambient wind field, the advection velocity
relevant for the lateral movement of the structure is not well-defined.
Thus, the time delay between inflow measurement and wake measurement can not be estimated accurately.
A better understanding of the relevant advection velocity of the wake might improve a prediction of the wake position in highly turbulent environments.
Attempts to improve the description of the advection velocity are made for example in .
A source that we do not address in this study is the uncertainty of the wind direction estimate by the error of the measurement device that is used.
The cross stream average of hub height flow upstream of the turbine, that we use here, is just one possibility to measure the inflow.
The only way to apply this method in the field would be by using nacelle based lidar systems like proposed in .
The shown simulations represent only examples of thermal stability conditions for stationary and barotropic flow.
In addition to atmospheric stability other factor like baroclinicity and topography influence the wind profile.
Thus, from the shown simulations we can conclude little about the influence of atmospheric stability at a specific location.
For the fine-tuning of wake models it would be beneficial to study the exact effect of shear and veer on the wake position and shape in more detail.