Introduction
Allowing insufficient space between wind turbines in an array leads to
decreased performance through wake interaction, decreased bulk flow velocity,
and an increase in the accumulated fatigue loads and intermittency events on
downstream turbines . Wind
turbine wakes lead to an average loss of 10–20 % of the total potential power
output of wind turbine array . Extensive
experimental and numerical studies focus on wake properties in terms of the
mean flow characteristics used to obtain estimates of power production
. Wake growth depends on the shape
and magnitude of the velocity deficit, surface roughness, flow above the
canopy, and spacing between the turbines.
Although there are many studies dealing with the effect of the density of
turbines on the wake recovery, it is still a debated question. The actual
spacing of wind turbines can vary greatly from one array to another and
depending on the direction of the bulk flow. For example, in the Nysted farm,
spacing is 10.5 diameters (D) downstream by 5.8D spanwise at the exact
row (ER). The wind direction at the ER is 278∘ and mean wind direction
can deviate from ER by ±15∘ .
Variation in the wind direction is evident through wake statistics, including
wake width, center line, and orientation with respect to the array.
showed that the spacing in the Nysted farm
is responsible for 68–76 % of the farm efficiency variation. In the Horns Rev
farm, spacing between devices is 7D, although aligned with the bulk flow
direction spacing is as much as 10.4D. pointed out
that variations in the power deficit are almost negligible when spacing is
approximately 10D at the Horns Rev farm, in contrast to limited spacings
that present a considerable power deficit. found
that when the streamwise and spanwise spacing increased, the wake
coefficient, which represents the ratio of total power output with and
without wake effects, is increased. performed large-eddy simulations (LESs) of the Lillgrund wind farm, where pre-generated
turbulence and wind shear are imposed in the computational domain to simulate
realistic atmospheric conditions. In the Lillgrund wind farm, the actual
spacing is 3.3 and 4.6D in the streamwise and spanwise directions. A
turbine is missing near to the center of the wind farm, demonstrating the
effects of a farm with limited spacing and one with sufficient spacing in
otherwise identical operating conditions. The results of
are highly applicable in the current study, although
their foci are on turbulence intensity effects and yaw angle.
Further, the effect of the incoming flow direction on the wake coefficient
increased when the spacing of the array was reduced.
studied the optimal spacing in a fully developed wind farm under neutral
stratification and flat terrain. The results highlighted that, depending on
the ratio of land and turbine costs, the optimal spacing might be 15D
instead of 7D. pronounced that the optimal
spacing depends on the length of the wind farm in addition to the factors
suggested in . Orography and wind direction are
relevant when deciding distance between turbines as well as layout, as shown
by .
Further investigations in array optimization have been undertaken by changing
the alignment of the wind farm, often referred to as staggered wind farms.
compared aligned versus staggered wind farms, the
latter yielding a 5 % increase in extracted power.
used LES to study the influence of the
streamwise and spanwise spacing on the power output in aligned wind farms
under a fully developed regime. Their work confirmed that power produced by the
turbines scales with streamwise spacing more than with the spanwise spacing.
investigated turbulent flow within and above aligned
and staggered wind farms under neutral conditions. Cumulative wakes are shown
to be subject to strong lateral interaction in the staggered case. In
contrast, lateral interaction is negligible in the aligned wind farm.
quantified the influence of wind farm layout on
the power production, verifying that increasing the turbine spacing in the
predominant wind direction maximized the power production, regardless of
device arrangement in the wind farm. investigated
the power output and wake effects in aligned and staggered wind farms with
different streamwise and spanwise turbine spacings. In the staggered
configuration, power output in a fully developed flow depends mainly on the
spanwise and streamwise spacings, whereas in the aligned configuration, power
strongly depends on the streamwise spacing.
As wind farms become larger, the land costs and availability represent
critical factors in the overall value of the wind farm. Spacing between the
turbines is an important design factor in terms of overall wind farm
performance and economic constraints. Investigation of wind farms with
limited spacing is important in order to quantify the effects of wind turbine
wake interaction on the power production. The current work compares the
turbulent flow in various configurations of the array, where the streamwise
and spanwise spacings are varied. The tunnel-scaled wind farm is restricted
to a flat surface and topographic influences are not considered, although the
inflow to the wind farm includes modifications to resemble an atmospheric
boundary layer. The performance of the arrays is characterized by analyzing
the mean velocity, Reynolds shear stress, and power production. Proper
orthogonal decomposition (POD) is employed to identify coherent structures of
the turbulent wake associated with variations in spacing. The Reynolds
stresses are reconstructed from a POD basis, demonstrating variation in rates
of convergence according to wind turbine spacing. Finally, the Reynolds stress
anisotropy tensor is employed to differentiate the balance of energy in the
turbulence field for the test cases.
Theory
Snapshot proper orthogonal decomposition
POD is a mathematical tool that derives optimal basis functions from a set of
measurements, decomposing the flow into modes that express the most dominant
features. The technique, which was presented in the frame of turbulence by
, categorizes structures within the turbulent flow
depending on their energy content. presented the snapshot POD,
which relaxes the computational difficulties of the classical orthogonal
decomposition. POD has been used to describe coherent structures for
different flows, such as axisymmetric mixing layer
, channel flow ,
atmospheric boundary layer , wake behind disk
, and a wind turbine wake flow
.
The flow field, taken as the fluctuating velocity after subtracting time-average mean velocity from instantaneous velocity, can be represented as
u=u(x,tn), where x and tn refer to the spatial
coordinates and time at sample n, respectively. A set of the orthonormal
basis functions, ϕ, can be presented as
ϕ=∑n=1NA(tn)u(x,tn),
where N is the number of snapshots. The largest projection can be
determined using the two-point correlation tensor and Fredholm integral
equation
∫Ω1N∑n=1Nu(x,tn)uT(x′,tn)ϕ(x′)dx=λϕ(x),
where the left-hand side of the equation presents a spatial correlation
between two points x and x′, T signifies the
transpose of a matrix, Ω is the physical domain, and λ
represents
the eigenvalues. To acquire the optimal basis functions, the problem is
reduced to an eigenvalue decomposition denoted as [C][G]=λ[G],
where C, G, and λ are the correlation tensor,
basis of eigenvectors, and eigenvalues, respectively. The matrix [G] is
related to the time coefficient as [G]=[A(t1),A(t2),⋯,A(tN)]T. The POD eigenvectors illustrate the spatial structure of the
turbulent flow and the eigenvalues measure the energy associated with
corresponding eigenvectors. The summation of the eigenvalues presents the
total turbulent kinetic energy (E) in the flow domain. The cumulative
kinetic energy fraction η and the normalized energy content of each
mode ξ can be represented as ηn=∑j=1nλn/∑j=1Nλn
and ξn=λn/∑j=1Nλn. POD is particularly useful in rebuilding the Reynolds shear
stress using a limited set (Nlm) of eigenfunctions as
〈uiuj〉=∑n=1Nlmλnϕinϕjn.
Reynolds stress anisotropy
Following the development presented by , the
Reynolds stress anisotropy tensor is written
aij=uiuj‾-23kδij, where δij is the Kronecker delta and
k represents the turbulence kinetic energy and is defined by
k=0.5∑i=13〈uiui〉. The deviatoric tensor is then
bij=uiuj‾/ukuk‾-13δij, of which the second and third scalar invariants are determined
as 6η2=bijbji and 6ξ3=bijbjkbki, respectively
. The second invariant, η, measures the
degree of the anisotropy and the third invariant, ξ, specifies the state
of turbulence. Alternatively, the eigenvalue decomposition of the normalized
Reynolds stress anisotropy tensor bij can be used to derive the
second and third invariants as
η2=13(λ12+λ1λ2+λ22)
and ξ3=-12λ1λ2(λ1+λ2). In
an attempt to further facilitate the study of turbulence anisotropy,
presented a linearized anisotropy tensor
invariant, termed barycentric map (BM), as
b^ij=C1c2/3000-1/3000-1/3+C2c1/60001/6000-1/3+C3c000000000,
where C1c, C2c, and C3c are the coefficients that
represent the boundaries of the barycentric map. The BM coefficients are
determined as C1c=λ1-λ2, C2c=2(λ2-λ3), and C3c=3λ3+1. The basis matrices in Eq. ()
represent the vertices of an equilateral triangle with
coordinates (x1c,y1c), (x2c,y2c), and (x3c,y3c).
Table presents the states of turbulence that correspond to each
vertex of the BM, describing either isotropic (three-component), one-component, or
two-component turbulence. As a result, any realizable turbulence state can be
represented as follows:
xnew=C1cx1c+C2cx2c+C3cx3c,ynew=C1cy1c+C2cy2c+C3cy3c.
Summary of the special turbulence cases described by the barycentric map.
Cases
Eigenvalues
Three components
λ1=λ2=λ3=0
Two components
λ1=λ2=16,λ3=-13
One component
λ1=23,λ2=λ3=-13
also introduced a color-map-based visualization
technique that aids in interpreting the spatial distribution of the normalized
anisotropy tensor. In this case, they attributed to each vertex of the
barycentric map an RGB (red–green–blue) color scale; see Fig. for
more details. This color map technique combines the coefficients C1c,
C2c, and C3c to generate an RGB map such that
RGB=C1c*100+C2c*010+C3c*001,
where Cic* are the modified coefficients that can be
determined as Cic*=(Cic+0.65)5. The coefficient with a value of
0.65 and 5 is applied as it provides the optimal visualization; other
coefficients are tested with less success in terms of marking differences. As
a result, one-component turbulence is associated with the red color,
two-component turbulence with green, and three-component (isotropic turbulence)
with blue; see Fig. . The anisotropy has been examined in
different types of flow, including pipe and duct flows
, the atmospheric boundary layer
, and the wake of a wind turbine
.
Here the anisotropy stress tensor is employed to quantify the
effect of the spacing on the turbulence states.
Schematic representation of the barycentric map (BM) with color.
Experimental design
A 4×3 array of wind turbines was placed in the closed-circuit wind
tunnel at Portland State University to study the effects due to variation in
streamwise and spanwise spacing in a wind turbine array. The dimensions of
the wind tunnel test section were 5 m (long), 1.2 m (wide), and 0.8 m (high).
The blockage ratio comparing the frontal area of the model wind turbines to
the cross-sectional area of the test section was less than 5 %. The entrance
of the test section was conditioned by the passive grid, which consists of
seven
horizontal and six vertical rods, to introduce large-scale turbulence. Nine
vertical acrylic strakes, located 0.25 m downstream of the passive grid
and 2.15 m upstream of the first row of the wind turbine, were used to modify
the inflow. The thickness of the strakes was 0.0125 m and they are spaced every
0.136 m across the test section. Surface roughness was introduced to the wall
as a series of chains with a diameter of 0.0075 m, spaced 0.11 m apart.
Figure shows the schematic of the experimental setup.
Streamwise and spanwise spacing of the experimental tests.
Cases
Sx
Sz
Occupied area
C6×3
6D
3D
18D2
C3×3
3D
3D
9D2
C3×1.5
3D
1.5D
4.5D2
C6×1.5
6D
1.5D
9D2
Experimental
setup. Dashed gray lines indicate the placement of the laser sheet relative
to the model wind turbine array. Filled gray boxes indicate measurement
locations discussed below.
Sheet steel 0.0005 m thick was used to construct the three-bladed wind turbine
rotors. The diameter of the rotor was D=0.12 m, equal to the height of the
turbine tower. The scaled turbine models were manufactured in-house. Based on
full-scale turbines with a 100 m rotor diameter and a 100 m hub height, the models were built on a 1:830 scale. In this study, the Reynolds number in the
entrance row turbines was approximately the same order of magnitude of the
independent range detailed in . The rotor blades
were steel sheets laser cut to shape and were 0.0005 m thick. The blades were
shaped using a die press. The die press was designed in-house to produce a
15∘ pitch from the plane of the rotor and a 10∘ twist at the tip.
Figure presents the schematic of the wind turbine model. The wind
turbine model design used is that presented in ,
, and . Operating conditions
for the wind turbines were also scaled, namely the power coefficient,
Cp,
and tip-speed ratio, λ, which were detailed in
. The streamwise integral length scale is
approximately 0.13 m, which was the same order of magnitude as the turbine
rotor and representative of conditions seen by full-scale turbines in
atmospheric flows. A DC electrical motor of 0.0013 m diameter and 0.0312 m long
formed the nacelle of the turbine and was aligned with the flow
direction. A torque-sensing system was connected to the DC motor shaft
following the design outlined in . The torque sensor
consists of a strain gauge, Wheatstone bridge, and data acquisition with
measuring software to collect the data.
Schematic representation of the wind turbine model.
The flow field was sampled in four configurations of a model-scale wind
turbine array, classified as CSx×Sz, shown in Table .
Permutations of the streamwise spacing (Sx) of 6 and 3D
and spanwise spacing (Sz) of 3 and 1.5D are examined. Stereoscopic
particle image velocimetry (SPIV) was used to measure streamwise,
wall-normal,
and spanwise instantaneous velocity upstream and downstream of the
wind turbine at the center line of the fourth row as shown in Fig. .
At each measurement location, 2000 images were taken to ensure convergence
of second-order statistics. The nominal sampling rate of the SPIV system is
fixed at 5 Hz. The SPIV system consists of a Nd: YAG (532 nm, 1200 mJ, 4 ns
duration) double-pulsed laser and four
4 Mpx Imager ProX CCD cameras arranged
in pairs upstream and downstream of the wind turbine. Neutrally buoyant fluid
particles of diethyl hexyl sebacate were introduced to the flow and allowed
to mix. Consistent seeding density was maintained in order to mitigate
measurement errors. The laser sheet was approximately 0.001 m thick with a divergence angle of less
than 5 mrad. Each measurement window was 0.2 m × 0.2 m
aligned with the center of each turbine, parallel to the bulk flow. A
multi-pass fast Fourier transformation was used to process the raw data into
vector fields. Erroneous measurement of the vector fields was replaced using
Gaussian interpolation of neighboring vectors. Based on the variability
estimator , the error of the SPIV measurements was
on the order of 3 % with the greatest uncertainty pertaining to the
out-of-plane (spanwise) component.
Top view of the 4 by 3 wind turbine array. The dashed lines at the last row centerline turbine
represent the measurement locations.
Normalized streamwise velocity, U/U∞, upstream and downstream of cases
C6×3, C3×3, C3×1.5, and C6×1.5.
Normalized
Reynolds shear stress, -uv‾/U∞2, upstream and
downstream of each measurement case.
Results
Statistical analysis
Characterization of the wind turbine wake flow is presented by the streamwise
mean velocity and Reynolds shear stress, with the aim to understand the
influence of turbine-to-turbine spacing. Figure presents the
streamwise normalized mean velocity, U/U∞, upstream and downstream
of each wind turbine for the cases C6×3, C3×3,
C3×1.5, and C6×1.5. The inflow mean velocity at the hub
height U∞=5.5 m s-1 is used in the normalization. For each
turbine, the flow upstream and downstream is shown by the contour plots on
the left and right, respectively. In the upstream region, case C6×3
exhibits the largest streamwise mean velocities due to greater recovery of
the flow upstream of the turbine. Although the streamwise spacing of case
C6×1.5 is the same as case C6×3, the former shows reduced
hub height velocity. The normalized mean velocity is about 0.567 compared
with 0.66 in case C6×3, showing the influence of the spanwise
spacing on wake evolution and flow recovery. Variations between case
C3×3 and C3×1.5 are small. Downstream of the turbine, the
four cases show differences outside of the rotor area, where case
C6×3 shows the greatest velocities by approximately 20 %. Case
C3×3 also shows higher velocities below the bottom tip compared
with cases C3×1.5 and C6×1.5. The normalized mean
streamwise velocity and the turbulence intensity in
showed similar compound wakes from the upstream and downstream turbines and
confirmed the current result of cases C3×3 and C3×1.5. In
that study, there was one location with an absent turbine and the flow was
given extra space for recovery. The recovered wake flow in
is similar to the present cases C6×3 and
C6×1.5.
Figure compares the in-plane normalized Reynolds shear stress
-uv‾/U∞2 for all test cases. The fluctuating
velocities in the streamwise and wall-normal directions are denoted as u and
v, respectively. In the upstream window, cases C3×3 and
C3×1.5 display higher values of the stress compared with
C6×3 and C6×1.5 cases. Although the spanwise spacing of case
C3×1.5 is half of case C3×3, no relevant differences are
apparent. In the downstream window, comparison indicates that reducing
streamwise spacing increases the Reynolds shear stress. The average value of
the shear stress in the wake is 16 % greater for C3×3 than for
C6×3. A similar effect is observed in case C3×1.5, in
which the
average value of the stress is 2 % greater than that of C6×1.5. The
effect of spanwise spacing is more pronounced when the streamwise spacing is
3D; the average shear stress is approximately 20 % greater in
C3×1.5 than in C3×3.
Averaged profiles
Spatial averaging of the flow statistics is undertaken by moving the upstream
domain of each case beyond its corresponding downstream domain and performing
streamwise averaging, following the procedure in . Through
spatial averaging, it is possible to compare key data from different cases
taking into account the different streamwise spacings. Streamwise averaging
is denoted by 〈⋅〉x. Figure a shows profiles of
streamwise-averaged mean velocity for all four cases. Cases C6×3
and C3×1.5 show the largest and smallest velocity deficits,
respectively. At hub height, the velocity of case C6×3 is
approximately 2.25 m s-1 whereas case C3×1.5 shows a velocity
of approximately 1.6 m s-1. Comparing to C6×3, the change seen
in the spatially averaged velocity is greater in C3×3 than in
C6×1.5, confirming that the impact of reducing streamwise spacing
is greater than changing the spanwise spacing. Interestingly, when the
spanwise spacing is fixed to Sz=1.5D, changing the streamwise spacing
has an effect smaller than expected. Constraining the wake suppresses
development of the mean velocity in the streamwise and spanwise directions.
Figure b contains the streamwise-averaged Reynolds shear stress
〈-uv‾/U∞2〉x for cases C6×3
through C6×1.5. Slightly decreased values of
〈-uv‾/U∞2〉x are seen in case
C6×1.5, where the spanwise spacing is reduced, especially below the
turbine hub height y/D=1. Reducing spanwise spacing shows a more pronounced
effect when the streamwise spacing is Sx=3D. The streamwise spacing
plays a larger role than the spanwise spacing, i.e., the maximum differences
between the Reynolds shear stress profiles are detected between cases
C6×3 and C3×3. Interestingly, the largest difference
between the spatially averaged Reynolds shear stress is found between cases
C6×3 and C3×3, located at y/D≈0.7 and y/D≈1.4. Furthermore, the four cases have approximately zero Reynolds
shear stress at the inflection point located at hub height. In addition, case
C3×3 displays the maximum Reynolds stress and case C6×1.5
presents the minimum stress.
Streamwise-averaged profiles of streamwise velocity and Reynolds
shear stress for four different cases: C6×3 (blue □),
C3×3 (red ○), C3×1.5
(black ◇), and C6×1.5 (pink △).
Proper orthogonal decomposition
Eigenvalues produced in the POD express the integrated turbulence kinetic
energy associated with basis function describing the flow. The normalized
cumulative energy fractions ηn
for upstream and downstream measurement
windows are presented in Fig. a and b, respectively. Inset
figures exhibit the normalized energy content per mode, ξn. Upstream
of the turbine, cases C6×3 and C6×1.5 converge more toward
ηn than cases C3×3 and C3×1.5, respectively.
These results are attributed to the reduction in the streamwise spacing. The
convergence of case C3×3 is approximately coincident with that of
case C3×1.5. In the downstream measurement window, case
C6×1.5 converges faster than the other cases, followed by
C6×3, C3×3, and C3×1.5. The comparison between
the upstream and downstream windows reveals that energy accumulates in fewer
modes upstream in every test case, e.g., case C6×3 requires 14
modes to obtain 50 % of the total kinetic energy in the upstream window,
whereas 26 modes are required to obtain the same percentage of energy
downstream of the turbine. Cases C6×1.5 and C3×1.5 show
the respective maximum and minimum variations in λ1 between upstream
and downstream measurements. This observation can be attributed to the
structure of the upstream flow of case C6×1.5, which is more
recovered compared to the downstream flow, where the turbulence is high in
energy content and more complex. However, the upstream and downstream windows
of case C3×1.5 are more similar in terms of turbulence and
organization. From mode 2 through 10, the starkest difference between the
upstream and downstream is found in case C6×3. Increasing the
characteristic area per turbine provides room for the flow to become more
homogeneous in the upstream window and exhibit the most significant momentum
deficit in the wake, accounting for the differences seen in ηn
upstream and downstream.
Energy content of the POD modes for four different cases:
C6×3 (blue), C3×3
(red), C3×1.5 (black), and
C6×1.5 (pink).
The first
mode upstream and downstream of the each case.
The fifth mode upstream and downstream of the each case.
The 20th mode upstream and downstream of the each case.
The streamwise component of the selected POD modes is shown for all cases in
Figs. through . These modes are selected because they
provide a range of large and intermediate scales and highlight the
discrepancies among the cases. Figure presents the first POD mode upstream and downstream of the considered cases. The four cases show
small gradients in the streamwise direction compared to a large gradient in
the wall-normal direction. Although the four cases show a divergence between
the eigenvalues of the first mode, the eigenfunctions demonstrate very
similar structure. For case C6×3, the energy of the first POD mode
decreases by 1.25 %, comparing the upstream eigenvalue to the downstream
one;
see Fig. . Smaller variations of 0.68 and 0.32 % are observed in cases C3×3 and C3×1.5, respectively. Consequently,
the structures upstream and downstream of these cases are approximately
equivalent. The upstream measurement domain of cases C6×3 and
C6×1.5 is representative of the recovering part of the flow, in
contrast to the downstream part that presents the wake region. This difference in
the physical space has an impact on the low-number POD modes that show a
discrepancy in the coherent structures between the upstream and downstream
windows. In the C3×3 arrangement, upstream and downstream regions
exhibit similar behavior, thus pointing to the resemblance in the structure.
Alike observations can be extracted from case C3×1.5. Of note, a
difference in sign of the eigenvectors is present, which is one of the POD
properties.
Figure presents the fifth POD mode of the four cases that show a
combination of POD and Fourier (homogenous) modes in the streamwise
direction. Although the fifth mode of the four cases contains ≈ 74 %
less energy than the first mode, large scales are still pronounced.
Smaller features also appear in the upstream and the downstream windows. The
upstream window of cases C6×3, C3×3, and C3×1.5
is shifted horizontally in the downstream window. The upstream and downstream
windows of case C3×1.5 look like the first mode, reduced in size, as
is observed in the downstream window of the case C6×1.5.
Figure presents the 20th POD mode, in which small structures
become noticeable in both upstream and downstream windows. The upstream
measurement window of cases C6×3 and C6×1.5 shows larger-scale structures compared to the other two cases. Although, after mode 10,
there is no significant difference in the energy content from case to case,
the structure of the modes shows a significant discrepancy between the cases,
confirming that the intermediate modes are associated with the inflow
characterizations. Thus, the intermediate modes are responsible for carrying
the significant part of flow dynamic and cooperative behavior in the energy
cascade. Therefore, any low-order models should include these intermediate
modes in order to improve the behavior dramatically and capture the dynamic
of the full system.
Reconstruction Reynolds shear stress using the first mode
(blue), first five modes (red),
first 10 modes (green), first 25 modes
(pink), and first 50 modes (gray).
Full data statistics (black). The insets show
the reconstruction using modes 5–10 (red), 5–25
(blue), and 5–50 (green).
Barycentric map for upstream and downstream
of the considered cases. The small triangle is a color map key for ease of
interpretation.
Extracted power of the wind turbine at different angular velocities for four different
cases C6×3 (blue □), C3×3
(red ○), C3×1.5 (black ◇), and
C6×1.5 (pink △).
Reconstruction of averaged profile
Combining the POD modes with the corresponding time coefficient gives these
modes physical interpretation and shows the contribution of the modes to
the overall flow behavior. A reduced degree of the turbulence kinetic energy
is considered using only a few modes to reconstruct the streamwise-averaged
profiles of Reynolds shear stress. Reconstructions are made using either a
single mode or the first 5, 10, 25, or 50 modes to represent the stress,
shown in Fig. . Inset figures present the Reynolds shear stress
construction using modes 5–10, 5–25, and 5–50, excluding
the first four modes to isolate contributions from intermediate modes. The
black lines are the streamwise-averaged stresses from the full data in Fig. b.
Using an equal number of modes, case C6×1.5 rebuilds
the profiles of the Reynolds shear stress faster than the other cases. Case
C6×3 also shows a faster reconstruction and dissimilarity to case
C6×1.5, mainly arising from the profile of the first mode (red line).
Cases C3×3 and C3×1.5 show approximately the same trends
in reconstruction profiles. Below hub height, the four cases show the same
trend of the first-mode profiles, where the contribution in the
reconstruction profiles is zero. The maximum difference between the
successive reconstruction profiles occurs between the first mode and the
first five modes. Cases C6×3, C3×3, and
C3×1.5 show moderate variation between the profiles of the
reconstructed stress resulting from the first five and first 10 modes (red and
green lines, respectively). After mode 10, contributions by each additional
mode are quite small, shown by pink and gray lines.
The maximum difference between the full data and the reconstructed profiles
is located at y/D≈0.75 and y/D≈1.4, where the extrema in
〈-uv‾〉x are located. Generally, faster
reconstruction implies that the flow possesses coherent structures with a
greater portion of the total kinetic energy. Consequently, the flow
characterized with greater coherence in cases C6×3 and
C6×1.5. In cases C3×3 and C3×1.5, fewer
energetic features arise from the reduced spacing effect, which leads to a
reduction of the mean velocities within the canopy and an increase in lateral
wake interactions. These interactions, which become larger as a result of the
accumulated wakes, expand downstream of the rotor. Thus, the streamwise
spacing allows for the flow to recover and therefore produce larger, more
coherent structures within the domain, which in comparison eclipses
variations produced by the spanwise spacing. Also, the large spacing offers a
larger frontal area to the wind coming from above the lateral sides.
To quantify the contribution of the moderately scaled structures, the Reynolds
shear stress is reconstructed using the intermediate modes. As can be shown
in the insets of Fig. , the full data profile (black line) is
compared with profiles reconstructed from modes 5–10 (red line), 5–25 (blue
line), and 5–50 (green lines). The intermediate modes in each case
approximately take the form of the full data profiles below the hub height,
although the magnitudes of the reconstructions are smaller than those of the
full data statistics. Reconstruction Reynolds shear stress in cases
C6×3 and C3×1.5 shows minute variations between the
reconstructed profiles and is essentially a vertical line above the hub
height. This trend is opposite that shown by the profile of the first mode
alone, indicating that the most energetic modes selectively reconstruct
turbulence above hub height. Cases C3×3 and C3×1.5 show a
difference between the successive profiles above the hub height. The maximum
difference is observed between the reconstructed profiles from modes 5 to 10 and
from 5 to 25 due to the turbulence kinetic energy contained within these modes.
Reynolds stress anisotropy
To examine the dynamics and energy transfer in the wind turbine arrays with
different streamwise and spanwise spacings, a description of the anisotropy upstream and downstream of the wind turbines is presented in Fig. .
A visualization of the turbulence state is obtained via the color
map representing the barycentric map as described in Sect. .
Turbulence anisotropy effectively distinguishes the cases in terms of wake
propagation and wake interaction. The variation in the spacings changes the
background turbulence structure. The upstream window of cases C6×3
and C6×1.5 shows that the turbulence field is close to the
isotropic limit, especially in hub height region, as a result of the wake
recovery occurring under a relatively large spacing distance. Below the bottom
tip, these cases show pancake-like turbulence due to the surface effect that
appears, deeming the perturbation of the turbines virtually negligible. Near
the
top of the tip, the flow shows a turbulence of axisymmetric state (between the
pancake-like and cigar-like turbulence). With this representation, the
spacing variation leads to a changed state of the turbulence and between the
developed and developing flow conditions can be discernible. The upstream of
case C3×3 shows a pancake-like turbulence state. However, the hub
height and bottom tip regions show an isotropic and axisymmetric turbulence,
respectively. Upstream of case C3×1.5 exhibits axisymmetric and
cigar-like turbulence in most of the upstream domain, although the hub
height region continues to show isotropic turbulence.
Past the turbine, the four cases exhibit the turbulence of isotropic state in
the hub height region. The top tip region of all four cases shows
axisymmetric turbulence, although case C3×3 tends toward cigar-like
turbulence. Below hub height, the turbulence is pancake-like and the
difference amongst the cases is the covered area, where it is maximum at
C6×3 and minimum at C3×3. The longest extension is found
in case C6×3 and the lowest in case C3×3. Comparing to
C6×3, the change seen in the turbulence states is starker in
C3×3 than in C6×1.5, confirming that the impact of
reducing streamwise spacing is greater than changing the spanwise spacing.
However, the impact of the spanwise spacing is noticeable when Sx=3D.
The ability to identify the turbulence structure allows for identification of
its influence on subsequent turbines in terms of fatigue loads
. Further, regions of the flow that are
characterized by highly anisotropic turbulence are those in which one is
likely to find large-scale, coherent turbulence structures. These structures
impart the greatest axial and bending loads onto subsequent turbine rotors,
leading to accelerated fatigue and increased operational and maintenance
costs for wind farms. In addition, regions of high anisotropy correlate with
gradients in the mean flow and turbulence .
These quantities are of particular interest in wind farm modeling and design.
Accordingly, the accurate representation of gradients in wind farm design
modeling is a necessary check in accurately representing production of and
flux by turbulence kinetic energy, wake interaction, and structural loading
on constituent turbines. Finally, the stress tensor invariants, by
definition, do not depend on reflection or rotation of the coordinate system,
meaning that they are unbiased descriptors for the turbulent flow
.
Conclusions
Insight into the behavior of the flow in a wind turbine array is useful in
determining how to highlight the overall power extraction with the variation
in spacing between the turbines. The work above quantifies effects of tightly
spaced wind turbine configurations on the flow behavior. The findings of this
study have a number of important implications, especially regarding the cost
of a wind farm or when large areas are not available. Stereoscopic PIV data
are used to assess characteristic quantities of the flow field in a wind
turbine array with varied streamwise and spanwise spacing. Four cases of
different streamwise and spanwise spacings are examined; the streamwise
spacing being 6 and 3D and spanwise spacing being 3 and 1.5D. The
flow fields are analyzed and compared statistically and by snapshot proper
orthogonal decomposition.
The streamwise mean velocity and Reynolds shear stress are quantified
upstream and downstream of the wind turbine in the considered cases. In the
inflow measurement window, higher velocities are observed in cases
C6×3 and C6×1.5 compared to the other two cases whose
inflows are unrecovered wakes from preceding rows. In contrast, cases
C3×3 and C3×1.5 show higher Reynolds shear stress. The
notable differences between the cases are found above the top tip and below
the bottom tip downstream the turbines, whereas the core of the wakes shows
fewer discrepancies. The streamwise and spanwise spacings have a concerted
effect on the flow, where the degree of the impact of one change highly
depends on the other. This relationship is shown in all statistical
quantities discussed here; for example, reducing the streamwise spacing by 50 %
leads to increases in the averaged Reynolds shear stress by 16 % when
Sz=3D. According to current statistical quantities, one can infer that the
higher influence of streamwise spacing is shown when the spanwise spacing is
Sz=3D, and the significant effect of the spanwise spacing is observed when
the streamwise spacing is Sx=3D. Averaged profiles of the velocity follow
the order of higher velocity seen in the contour plots in case C6×3
and lowest velocity in case C3×1.5. The maximum and minimum
differences are observed between case C6×3 and case
C3×1.5 and case C3×3 and case C3×1.5. The result
also reveals that the streamwise spacing is more impactful than the spanwise
spacing. Spatially averaged profiles of Reynolds shear stress show that the
maximum and minimum values occur in cases C3×3 and
C6×1.5, respectively.
According to the POD analysis, the upstream measurement plane of the four
cases converges faster than the downstream window. Cases C6×3 and
C6×1.5 show rapid convergence in cumulative energy content upstream
of the turbine, but C6×3 remains behind case C6×1.5 in
the wake. The first mode of case C6×1.5 carries the maximum
turbulent kinetic energy content compared to the first mode of the other
cases. No significant difference in energy content is observed after mode 10
between the four cases. The streamwise-averaged profiles of the Reynolds
shear stress are reconstructed by back-projecting coefficients onto the set
of eigenfunctions. Low modes are used individually to demonstrate their
contributions to the overall flow. Cases C6×1.5 and C6×3
converge to their respective spatially averaged profile faster than the other two
cases. The discrepancies in reconstruction are mainly observed in profiles
using only the first five modes. The same trend in reconstruction is observed
in cases C3×3 and C3×1.5. Reconstructed profiles display
the effects of the spacing, where the array of large streamwise spacing
reconstructs faster than the other cases due to the coherent structures
embedded in the flow.
Based on the Reynolds stress anisotropy tensor and color map visualization,
the spacing modifies the anisotropic character of the turbulence. Increased
turbine spacing allows the turbulent flow to recover between devices, leading
to increasingly isotropic flow incident to the rotors. The hub height region
of the wake shows isotropic turbulence regardless of the spacing. The
differences in the color map visualization between the downstream locations
of the four cases show some structural dependency on the spacing between
turbine rotors.
Power production by the turbines is measured directly using a torque sensing
system. The power curves follow the same trend as the velocity profiles. The
maximum power extracted is at the normalized angular velocity of 15.8 ± 1
and it is harvested in case C6×3. The small difference in harvested
power is observed between cases C3×3 and C3×1.5. The
current work demonstrates that wake statistics and power produced by a wind
turbine depend more on streamwise spacing than spanwise spacing. However, the
results above pertain only to a fixed inflow direction. In the case in which the
bulk flow orientation changes, spacing in both the streamwise and spanwise
directions will be important to the optimal power production in a wind
turbine array. Continued efforts are required to understand the impact of
streamwise and spanwise spacing in infinite array flow under realistic
flow conditions, including Coriolis forcing and under different
stratification conditions.