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Analysis of Turbine Wake Characteristics using Proper
Orthogonal Decomposition (POD) and Triple Decomposition
Pavithra Premaratne1, Wei Tian
2, and Hui Hu
3()
Iowa State University, Ames, Iowa, 50010, USA.
We examine the flow characteristics behind a commonly-used three-bladed horizontal-axis wind
turbine via experiments in a large-scale wind tunnel with a scaled model placed in a typical
Atmospheric Boundary Layer (ABL) wind under neutral stability conditions. A high-resolution digital
particle image velocimetry (PIV) system was used to study the detailed flow field. Besides analyzing
average statistics of the flow quantities such as mean velocity, Reynolds stress, and vorticity
distributions in the wake, ‘‘phase-locked’’ PIV measurements also elucidated further details of the
wake vortex structures for a frozen position of the blades. Proper Orthogonal Decomposition (POD)
method was employed to identify the high energy modes that dominate the turbulent kinetic energy
distributions in the turbine wakes. Triple Decomposition (TD) was used to elucidate the underlying
physics of the intensive turbulent mixing process in the wake flow, promoting the downward transport
of high-speed flow. We managed to identify the principal coherent structures in the flow and derived
the physics behind vortex break-down and the propagation.
Nomenclature
𝑎𝑖 = Modal Coefficients for POD
a = Amplitude of Kelvin Waves
𝑨𝒊 = Eigenvector matrix
ABL = Atmospheric boundary layer
CP = Power coefficient
CT = Thrust coefficient
D = Diameter of the rotor
H = Hub height
HAWT = Horizontal axis wind turbine
k = Wave number
L = Length from the tower to measurement location
P = Power output of the wind turbine
PIV = Particle image velocimetry
𝑟𝑖𝑗 = Reynolds shear stress
TKE = Turbulent kinetic energy
TSR = Tip speed ratio
U = Velocity Magnitude/Snapshot Matrix
u,v = Axial and vertical velocity components
UH = Inflow velocity at hub height
ω = Vorticity
Z,Zr = Vertical axis location/Reference location
𝛼 = Power law exponent ∅ = POD modes
Γ = Circulation Strength
𝜑 = Stream function
1 Graduate Student, Department of Aerospace Engineering. 2 Postdoctoral Research Associate, Department of Aerospace Engineering.
3 Professor, Department of Aerospace Engineering, AIAA Associate Fellow, Email: huhui@iastate.edu
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46th AIAA Fluid Dynamics Conference
13-17 June 2016, Washington, D.C.
AIAA 2016-3780
Copyright © 2016 by Pavithra Premaratne, Wei Tian, and Hui Hu. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
AIAA AVIATION Forum
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I. Introduction
Wind energy has played a predominant role as a renewable energy source in recent history. The contribution of
wind energy towards the national and the global power grids has grown exponentially in the past decade1. Wind
power generation is considered a viable solution to the environmental and economic issues that stem from heavy
dependency on petroleum products.
Studying the aerodynamic characteristics of wind turbines is crucial to the development of this technology.
Such studies are conducted to identify the necessary design parameters that increase the power output and optimize
the siting of a wind farm. The manuscript investigates wind tunnel experiment data related to an offshore boundary
layer with lower levels of ambient turbulence (10% Intensity).
The wind turbine wake is divided into two regions, near wake and far wake. The near wake refers to the region
from the turbine to approximately one rotor diameter downstream, and the far wake is the region beyond the near
wake. While the near wake flow analysis provides the designer with crucial performance factors of turbine rotor
blades, the far wake analysis determines the turbulence-induced fatigue of the downstream wind turbines as well as
the changes in the power outputs of the downstream turbines2.
As depicted in Figure (1), once the flow passes through the turbine rotor, a reduction in momentum occurs,
causing a region of energy deficit in the wake. The deficit results in a shear layer, where high velocity components
outside the wake regions are mixed with the low velocity components in the near. As the wake progresses axially, it
undergoes expansion. The mixing process creates turbulent eddies which result in the wake recovery3. Higher levels
of turbulence intensity increase the mixing efficiency, thus decreasing the wake recovery distance. However, such
levels lead to blade fatigue in the downstream turbine, which may lead to structural failure.
Figure 1: Wake characteristics
The Particle Image Velocimetry (PIV) technique was employed in this study to obtain instantaneous velocity
measurements of the turbulent wake flow4. These velocity distributions were analyzed through statistical means as
well as principal component analysis and flow field decomposition methods. Flow solutions can be decomposed to
time averaged components, coherent structures and random fluctuating components 5. Coherent structures are
defined as connected turbulent fluid masses with instantaneous phase correlated vorticity over its spatial extent.
Turbulent mixing is an important phenomenon in wind turbine applications as the output of the downstream turbine
strictly depends on it. The POD method is used to determine the dominant flow characteristics in the wake. Studying
such structures allow designers to quantify the impact on performance, thus leading to optimized future designs.
Shear
Layer
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II. Experimental Setup
The wake flow measurement experiments were conducted at the Atmospheric Boundary Layer (ABL) wind tunnel
at Iowa State University. The wind tunnel has a 20 m long test section with a 2.4 m width and a 2.3 m height. The
experiments were focused on wake profiles pertaining to offshore wind turbine applications. Therefore, an offshore
boundary layer was introduced with several rows of a metal chain upwind of the model turbine location. Prior to
installing the turbine, velocity and turbulence intensity profiles were obtained using a COBRA probe (flow
measurement devices). The profiles are shown in Figure 2.
Figure 2: Measured velocity (left) and turbulence intensity (right) profiles
The measured velocity profile was compared with a known power law relationship which describes the
boundary layer.
𝑈
𝑈𝑟= [
𝑍
𝑍𝑟]𝛼
(1)
Where 𝛼 is assumed to be 0.116. The reference Z location (Zr) and reference velocity (Ur) were taken as the hub
height and the velocity at the hub height. The velocity profile was normalized to the velocity at hub height. Upon
observation, it is understood that the recorded velocity measurements are in agreement with the power function. The
ambient turbulence intensity at the hub height was 10%.
A three bladed horizontal axis wind turbine model (HAWT) scaled to 1:350 ratio to a 2MW industrial wind turbine
was used for the purpose of this experiment7. The rotor and nacelle assembly was constructed using a hard plastic
material in a rapid prototyping machine. A metallic rod was used as the tower and the hub height was set to 23 cm
from the floor of the test section. The airfoil information across the blade is shown in Figure 3. The blade design was
based on ERS-100 prototype turbine blades developed by TPI Composites, Inc. Parameters of the model wind
turbine are given in Table (1).
Table 1: Turbine model parameters
Parameter R
(mm)
H
(mm)
d pole
(mm)
d nacelle
(mm)
(deg.)
a
(mm)
a1
(mm)
A2
(mm)
Dimension 140 226 18 18 5o 68 20 35
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Figure 3: Wind turbine schematic and airfoil information
A high resolution PIV system was used to record information of the ZX plane at Y=0 station (symmetrical
plane). This plane was illuminated using a ND-YaG laser that emitted two consecutive pulses of 200mJ with a
wavelength of 532nm. The air flow was seeded with ~ 1μm diameter oil droplets using a smoke generator. Two 16
bit CCD cameras were used to acquire the necessary images both near field and far field. Both cameras and the laser
were synced using a delay generator in order to acquire consecutive image pairs. The user was able to adjust the
delay between pulses in order to control the shutters of each camera. Acquired image pairs were stored in a work
station. Using this high resolution PIV system, two types of experiments were conducted. A “free run” case
demanded the acquisition of 1000+ PIV measurements for the purpose of ensemble-averaged flow statistics while
the “phase locked” experiment yielded measurements at different phase angles of the blade. Phase locked tests were
conducted using an external triggering mechanism and a secondary delay generator. A tachometer measured the
RPM of the rotor and acted as the triggering device. Figure 4 depicts a schematic of the experiment setup.
Acquired image pairs were subjected to frame-to-frame cross-correlation in order to obtain the instantaneous
velocity distributions. A commercial software package, Insight 3G, was used for this purpose, wherein a 32 x 32
pixel interrogation window was used with a 50% effective overlap between the windows. The resulting velocity
field was then subjected to further refinement by removing bad information due to burnt or bad pixels. An in-house
algorithm was employed for this purpose which also calculated flow parameters such as turbulent kinetic energy
(T.K.E), Reynolds Stress, vorticity and velocity magnitude.
Figure 4: Experiment Setup
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III. Results and Discussion 3.1 PIV measurement results
The post-processed instantaneous velocity fields were statistically averaged and the contour solution is shown in
Figure 5. A region of velocity deficit due to kinetic energy harvesting can be observed aft of the wind turbine. The
vertical velocity gradient depicts the shear layer at a contour level 0.9. Increased contour density near the nacelle
and the hub can be attributed to the highly chaotic nature of the fluid. High fluctuations in the flow can be observed
closer to the nacelle, blade tips and the tower. The grey area contains erroneous information due to the shadow of
the turbine assembly upon laser illumination.
A vorticity solution for phase angle of 0 degrees is shown in Figure 6. Vortex shedding can be observed
downstream of the rotor8. Two distinct vortex cores emanating from the tip and the mid span location break down
into smaller eddies at roughly ½ rotor diameters from the tower. The magnitude and the size of the vortex cores
decrease as the wake propagates. A smaller vortex sheet is present at the root of the blade.
Figure 5: Ensemble-averaged velocity distribution
Figure 6: Normalized Vorticity (Phase Angle = 0 deg.)
3.2 Proper Orthogonal Decomposition
POD has been an effective method to identify principal structures embedded in turbulent flows by linear
decomposition and reconstruction. All of POD modes, which are spatial orthogonal, are ranked by their kinetic
energy. Thus, if predominant large-scale structures exist in the turbine wake, they can be extracted by POD and be
represented in the first few modes. Singular Value Decomposition (SVD) can be utilized if the number of degrees
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(rows) is smaller than the number of snapshots (columns). For the purpose of this manuscript, linear decomposition
method is discussed.
A step-by-step construction of the POD algorithm is provided in this section. The subscript denotes the degree
of freedom and the superscript defines the image number9,10
.
All the fluctuating velocity components (time averaged components removed) are arranged in a matrix U as:
𝑼 = [𝒖𝟏 𝒖𝟐 …… … 𝒖𝑵] =
[ 𝑢1
1 𝑢12 …
⋮ ⋮ ⋱𝑢𝑀
1 𝑢𝑀2 …
𝑣11 𝑣1
2 …⋮ ⋮ ⋱
𝑣𝑀1 𝑣𝑀
2 …
𝑢1𝑁
⋮𝑢𝑀
𝑁
𝑣1𝑁
⋮𝑣𝑀
𝑁 ]
(8)
where M is the number of spatial discrete points and N is the number of the PIV snapshots, which represent the
spatial and temporal resolutions of the PIV data respectively.
The eigenvalues and eigenvectors of the auto-covariance matrix are calculated as:
�̃� ∙ 𝑨𝒊 = 𝜎𝑖 ∙ 𝑨𝒊 (9) where, �̃� = 𝑼𝑻 ∙ 𝑼
and the eigenvalues 𝜎 are ranked in a descending order.
Each eigenmode is obtained by projecting matrix U onto each eigenvector and then normalized by its norm as:
∅𝑖 = ∑ (𝐴𝑛
𝑖 ∙ 𝒖𝒏)𝑁𝑛=1
‖∑ (𝐴𝑛𝑖 ∙ 𝒖𝒏)𝑁
𝑛=1 ‖ , 𝑖 = 1,… . , 𝑁 (10)
where ∅𝒊 = [∅𝟏 ∅𝟐 …… . ∅𝑵 ]. The coefficients of each mode can be obtained as
𝒂𝒏 = ∅𝑻 ∙ 𝒖𝒏 (11)
Reconstructing the fluctuations of nth
instantaneous solution can be performed by the summation of each mode
vector multiplied by the corresponding modal coefficient.
𝒖𝒏 = ∑𝑎𝑖𝑛∅𝑖 = ∅ ∙ 𝒂𝒏
𝐿
𝑖=1
(12)
The user has the ability to determine the order of reconstruction (L) based on the modal energy.
An Lth
order POD reconstruction of the nth
instantaneous solution can be obtained by adding the ensemble
averaged velocity components:
𝑼𝒏 = �̅� + 𝒖𝒏 (13)
where �̅� is the ensemble-averaged velocity.
POD analysis was performed on a selected region that encompasses the shear layer. Selection of a proper ROI
in the wake eliminates noisy and erroneous measurements, thus increasing the accuracy of reconstructions. The
selected region is depicted in Figure 7.
Phase averaged PIV measurements were utilized for this analysis to create a free run sequence. Snapshots for a
given phase were divided into groups of 10, where the average of each group was considered as a snapshot. This
approach reduced the measurement noise while preserving the unsteady flow characteristics. The post-processed
velocity distributions were properly sequenced to resemble a free-run experiment, thus providing a data set for the
POD analysis. The velocity magnitude and the vorticity solutions for the selected ROI are presented in Figure 8.
The velocity magnitude is normalized to velocity at the hub height and the vorticity is normalized to the rotor
diameter and the hub velocity.
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Figure 7: Region of Interest (ROI)
(a) (b)
Figure 8: Ensemble Averaged Solution (Mode 0)
It can be deduced that the ROI has successfully encompassed the turbulent shear layer (velocity gradient) and
the vortex street. Ensemble averaged solution is considered Mode 0 for the purpose of this analysis. Low order
reconstruction of an arbitrary snapshot requires the addition of this mode. As for the initial results, Figure 9 and
Table 3 present the modal energy contained within each Eigen-mode obtained from the analysis.
Figure 9: Modal energy vs. mode number (left) Table 2: Modal Energy
Mode
Number
Energy
%
1 35.6
2 10.3
3 7.0
4 5.3
5 3.6
6 3.2
7 1.9
8 1.7
9 1.3
10 1.2
Z/D
Z/D
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According to Table 2, the first few modes contain the highest amount of energy. These modes correspond to
dominant flow features buried within the flow solution. The less dominant modes may correspond to measurement
noise and smaller eddies.
Modal reconstructions elucidate high kinetic energy flow components that are dominant, while filtering the
aforementioned measurement noise and low energy turbulent components. Normalized vorticity is the parameter of
interest, as vorticity serves as a quantifiable representation of dominant structures. Velocity fluctuation
reconstructions (𝑎𝑖∅𝑖) for an arbitrary snapshot, representing stream-wise vorticity (ω), are depicted in Figure 11.
The vorticity solutions are normalized to the ratio of diameter (D) and hub velocity.
(a) Recon: 𝒂𝟏∅𝟏 (b) Recon:𝒂𝟏∅𝟏 + 𝒂𝟐∅𝟐
(c) Recon:𝒂𝟏∅𝟏 + ⋯ + 𝒂𝟓∅𝟓 (d) Recon:𝒂𝟏∅𝟏 + ⋯ + 𝒂𝟏𝟎∅𝟏𝟎
Figure 10: Fluctuation Reconstructions (Cumulative Eigen-modes)
Summation of the fluctuation modes (eg. 𝑎1∅1 + 𝑎2∅2 + ⋯+ 𝑎𝑛∅𝑛 ) results in more pronounced flow
features. The magnitude of the vorticity increases with the addition of each fluctuation mode. The alternating
direction of the vortices indicates the periodic nature of the phenomenon. The first reconstruction encompasses most
of the kinetic energy (~35%), and some vortices buried in the wake region are observed. As the number of
cumulative modes increase, the shape and the formation of the coherent structures become apparent. However, no
significant changes were observed between Figure 10 (c) and (d), thus proving the convergence of the
reconstructions.
Addition of the time-averaged components to the fluctuations according to Eq. (13) results in a low-order
reconstruction of the original instantaneous solution preserving the buried dominant coherent structures. Summation
of all individual fluctuation modes and the time-averaged solution will result in the original velocity distribution.
The step-by-step low order reconstruction and a comparison to the original image are illustrated in Figure 11.
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(a) Reconstructed using Mode 1 + 𝑈 (b) Reconstructed using Mode 1 + Mode 2 + 𝑈
(c) Reconstructed using Mode 1 to Mode 5 + 𝑈 (d) Reconstructed using Mode 1 to Mode 10 + 𝑈
(e) Instantaneous measurement
Figure 11: Low order reconstructions of the instantaneous image
The increase of the vorticity magnitude and in the number of flow features can be observed in Figure 11, further
proving the trends witnessed and elaborated in Figure 10. A comparison between modal reconstructions and the
instantaneous solutions clarifies the capability of POD to extract underlying dominant structures in a turbulent flow
field contaminated by noise.
The amplitude of each modal coefficient series is proportional to the fluctuation kinetic energy embedded in
the corresponding mode. The amplitude of the modal coefficients decreases as the mode number increases (e.g., the
amplitude of the first modal series is higher than the second one) as anticipated. This is further observed by
computing the standard deviation of each coefficient time series11.
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Figure 12: Modal coefficient time series solutions
POD provides crucial information about the helical nature of the vortex street. The tip vortex street has a
higher propagation velocity compared to the mid-span vortex line. Presence of a highly turbulent shear layer results
in the break-down of tip vortices before the mid-span vortex street. According to Figure 11, the vortex lines are 0.3D
vertically apart from each other while the horizontal distance between the core centers is 0.27~0.24D. Both vortex
lines can be represented as a Kelvin wave as shown in,
𝑥 = 𝑎 ∗ 𝑐𝑜𝑠(𝑘𝑥 − 𝜔𝑡)
𝑦 = 𝑎 ∗ 𝑠𝑖𝑛(𝑘𝑥 − 𝜔𝑡)
where, 𝑎 stands for the amplitude, k for wave number and 𝜔 for angular velocity12
. An initial wave solution (t = 0)
was constructed as shown in Figure 13.
Figure 13: Kelvin wave representation of vortex lines
Both vortex lines have the same sign in vorticity (shown in blue). The waves were modeled preserving the
amplitude and the phase information. The mid-span vortex line emanates from a radial station located at 60% of the
radius from the hub. A slight phase shift between the two waves was also added to the model upon observing the
POD reconstructions. The shift occurs due to the differences in wave propagation speeds. The tip vortex travels
faster than the mid-span vortex wave as the blade speed is proportional to the radial location (𝑣 = 𝑟𝜔). Even though
the kelvin wave assumption predicts a continuous wave behind the rotor that extends below the hub height, the
presence of a tower during the experiment creates significant disturbances in the axial propagation of the wake.
However, for the purpose of this analysis we only considered the vortex behavior and breakdown in the shear layer.
Mode Std. Dev
1 1.34
2 0.72
3 0.59
Table 3: Standard deviation of modes
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The POD reconstructions of the fluctuations confirm the presence of two alternating vortex filaments originating
from each location (tip or mid-span). A corresponding solution has been derived using the Kelvin wave assumption
as shown in Figure 14. The fluctuation vortex filament with –Y vorticity nullifies as the time averaged mean is
added to the flow for the final reconstructions.
Figure 14: Kelvin wave representation of fluctuation vortex lines
The vortex breakdown happens during the first diameter. This is caused by the perturbations induced by the
proximity between each helical turn and the phase offsets. The distance between the mid-span vortex filament and
the tip vortex filament also contributes the breakdown of the vortices. The filament breaks down in the highly
turbulent near wake (X/D <1) region suggesting short wave instability13
. These modes of instability occur due to the
perturbations in the vortex cores. Presence of an external strain field such as a secondary vortex filament contributes
to the growth of the short wave instability14,15
. In order to identify such influences in the wake, Biot-Savart law was
employed. The velocity perturbation (𝑑𝑉̅̅̅̅ ) induced by a single vortex filament at a given point (P) is given by the
Biot-Savart law16
,
𝑑𝑉̅̅̅̅ =Γ
2𝜋
𝑑𝑠̅̅̅̅ ×�̅�
‖𝑟3‖ (14)
where, Γ stands for the circulation strength of the vortex. The position vector between the point of interest and the
filament location is given by �̅�. The 𝑑𝑠̅̅ ̅ is a tangential vector along the filament and the direction is determined by
the right hand rule. Integrating along the filament will provide the total velocity induced at P. This expression can be
further simplified to accommodate 2D flow scenarios or a planar projection of the helical filament. The decay rate
changes from ‖𝑟3‖ to ‖𝑟2‖ and the interactions from a N number of vortices at a given point in space (x) is given
by,
�̅�(𝑥) = ∑ 𝜔(𝑥𝑖)�̅�×(�̅�−𝑥𝑖̅̅ ̅)
2𝜋|�̅�−𝑥𝑖̅̅ ̅|2𝑁𝑖=1 (15)
𝜑(𝑥) = ∑ −𝜔(𝑥𝑖)𝑙𝑛|�̅�− 𝑥𝑖̅̅ ̅|
2𝜋
𝑁𝑖=1 (16)
where, the stream function and the velocity vector are denoted by 𝜑 and u17
. The directional vector of vorticity is
given by �̅� and the vorticity magnitude of the ith
vortex is given by 𝜔(𝑥𝑖). A sample numerical solution based on
Eq:14 can be obtained to describe the 2D PIV measurements. The solution resembles the propagating vortex cores
(mid-span and tip) towards downstream locations with decaying magnitudes. A stream function solution for this
case is shown in Figure 15.
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Figure 15: Vortex Interactions
Upon observing the stream functions contours, it can be concluded that most of the flow complexities exist
between each vortex along both longitudinal and lateral directions. Monitoring the transport phenomenon in these
regions may provide a comprehensive picture on the levels of turbulence and flow characteristics affected by the
vorticity.
Observing the momentum transfer through the shear layer provides how the helical vortex filaments affect the
said process. Reynolds shear stress provides a quantifiable parameter to estimate the efficiency of vertical
momentum transfer18
.
3.3 Triple Decomposition (TD)
The instantaneous velocity can be presented as a combination of mean value, contribution of organized wave
and random fluctuating components given by
𝑢 = �̅� + �̃� + 𝑢′ (17)
The method was presented by Hussain and Reynolds in 19865. The phase averaged velocity can be obtained by
averaging velocity measurements for each individual phase19
. The Reynolds stress can be calculated by determining
the contributions from coherent flow structures (unsteady) and turbulent flow artifacts. The turbulent velocity
components can be calculated by
𝑢′ = < 𝑢 > − 𝑢 (18)
where <u> denotes phase averaged velocities. Coherent velocity components can be calculated using
�̃� =< 𝑢 > − �̅� (19)
where the time-averaged flow field is denoted by �̅� . Based on the definitions developed for the turbulent and
coherent velocity components, Reynolds stress can be calculated as10
𝑟𝑖𝑗
−𝜌= (𝑢′ + �̃�) ∙ (𝑣′ + �̃�)̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ = 𝑢′𝑣′ + 𝑢′�̃� + 𝑣′�̃� + �̃� �̃�̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅ (20)
where u and v denote velocity components on x and y directions with ensemble averaging. As coherent flow
components are uncorrelated to the turbulent components it can be concluded that 𝑢′�̃� = 𝑣′�̃� = 0. Therefore the
normalized Reynolds stress can be calculated as:
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𝑟𝑖𝑗 =−(�̃��̃�̅̅̅̅ + 𝑢′𝑣′)̅̅ ̅̅ ̅̅ ̅
𝑈ℎ𝑢𝑏2 (21)
The contributions of both coherent and random fluctuations can be visualized along with the final solution as
depicted in Figure 16. By utilizing the phase locked PIV data obtained earlier, a phase averaged velocity distribution
was obtained for each phase angle. The phase averaged solutions along with an ensemble averaged solution from the
“free run” were used to extract the random and coherent components of the flow, which yielded phase decomposed
Reynolds stresses. Statistical averaging of the phase decomposed solutions will result in the ensemble averaged
solution depicted in Figure 16.
(a) −�̃��̃�̅̅ ̅̅ ̅̅ (b) −𝒖′𝒗′̅̅ ̅̅ ̅̅
(c) Ensemble-averaged Reynolds stress −(𝒖′𝒗′ + �̃��̃�)̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅
Figure 16: Reynolds Stress Constructions
The contribution from−�̃��̃�̅̅ ̅̅ was calculated to be 1.5% of −𝒖′𝒗′̅̅ ̅̅ ̅̅ in the ROI and concentrated near the turbine
blade and the nacelle. Therefore, the −�̃��̃�̅̅ ̅̅ can be neglected from future analysis. The vortex street in the wake
region contains Reynolds stress concentrations (magnitude of 0.004) that dissipate into a broader shear layer starting
at X/D = 0.8. The Reynolds stresses available aft of the nacelle and tower assembly correspond to the presence of
mechanical turbulence.
In Figure 17, Reynolds stresses decomposed to phase angles, are plotted for angles ranging from 0 to 90 deg. to
identify changes in turbulence at different blade positions. The wake region that encompasses the shear layer has
been highlighted, where a comparison between decomposed Reynolds stress and vorticity has been drawn.
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A discrepancy between the locations of vortices shed and the locations of stress concentrations was observed.
This observation suggests that the presence of vortices has hindered the vertical momentum transfer in the near wake
region. As the vortices propagate axially, they break down into turbulent eddies and dissipate into a broader shear
region, further proving the ensemble-averaged results depicted in Figure 17. Downwind wind turbines may be
adversely affected by the shear layer as the turbulent eddies may induce blade fatigue20
, thus causing structural
failures. Highly concentrated Reynolds stresses can also be observed near the nacelle and the tower due to high
levels of mechanical turbulence.
Phase Angle = 0 deg.
Phase Angle = 30 deg.
Phase Angle = 60 deg.
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. Phase Angle = 90 deg.
Figure 17: (−< 𝒖′𝒗′ >)(left) and a comparison with vorticity contours (right) at different phase angles for the
region that encompasses the shear layer
IV. Conclusion
Understanding the unsteady aerodynamics related to the operation of a wind turbine is a crucial aspect when
attempting to increase its efficiency. The flow phenomenon is highly complex with embedded coherent structures
and randomly fluctuating fluid masses. A comprehensive study to identify and quantify such structures can provide
researchers with insights to the recharging of the flow via momentum transport. Wind tunnel experiments are critical
when studying the near wake flow, as utilizing computational fluid dynamics is time-expensive, and often leads to
inaccuracies due to the assumptions made. Near wake flow is highly dynamic, thus requiring advanced modeling of
rotational flow with turbulence. Particle Imaging Velocimetry (PIV) is a reliable flow measurement technique which
provides high resolution data of an illuminated plane of interest. Processing the measurements using a commercial
cross-correlation algorithm and an in-house algorithm provided velocity distributions with minimal error. Increasing
the number of realizations will result in better ensemble averaged solutions.
Dominant structures buried within a complex flow account for most of the fluctuation kinetic energy in the
flow, and it has been proven that they play a major role in wake recovery. The Proper Orthogonal Decomposition
(POD) method is a successful approach to identify the behavior and energy of the dominant energy components. A
POD analysis was conducted to identify the coherent structures in the near wake region. The analysis isolated
principal flow components that account for 68% of the kinetic energy spread between the first 10 Eigen-modes.
Reconstructions of an instantaneous solution with POD modes clearly showed the dominant vortex roll-ups in the
wake. The existence of these roll-ups is further proven by the phase averaged vorticity distribution. However, there
were no observable modal dependencies observed between the modal coefficients. Taking the low order
reconstructions, it can be concluded that these coherent structures role affect the recharging the flow for the
secondary harvesting of the downstream turbine. A Kelvin wave assumption along with the Biot-Savart law was
employed to understand the perturbations induced by the helical vortex filaments on each other as well as the rest of
the shear layer. A stream function derivation prompted us to investigate the vertical momentum transport in the near
wake shear layer between the tip and mid-span vortex filaments.
Vertical momentum transport is another method where high velocity elements outside the wake region transfer
momentum to the velocity deficit. As shown in Figure 1, the shear layer that lies between the wake region and high
velocity flow outside the wake is directly responsible for the wake recovery. The turbulence present in this region
governs the energy harvest of the downstream wind turbine as well as the blade fatigue. Triple decomposition
method was employed to decompose the turbulent velocity field to time averaged, coherent, and incoherent flow
structures. Coherent and random fluctuating components were used in calculating the Reynolds shear stress
distributions in the near wake region. Reynolds stress contribution due to coherent flow structures was considered
minimal compared to the contribution from the turbulent fluctuating quantities. The Reynolds stresses were
predominantly concentrated between the vortices shed by the turbine, suggesting that the presence of a vortex street
may act as a barrier for the process of vertical momentum transfer. The magnitude of Reynolds stresses dissipates
along the axial direction, suggesting the presence of a larger shear layer that extends to the far-field where turbulent
mixing occurs. Mechanical turbulence at the nacelle and tower assembly also leads to high Reynolds stress
concentrations.
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Using tools such as POD and TD allow the industry professionals not only to observe wake flows, but also to
implement critical design changes for existing designs to optimize the output. Wind farm designers can also use
these tools to determine the siting patterns and the optimal distance between each wind turbine, which will pave the
way for more economical wind power generation schemes.
Acknowledgment
The authors thank Mr. Bill Rickard of Iowa State University for his help in conducting the wind-tunnel experiments.
The support from the National Science Foundation (NSF) with Grant Number CBET-1133751 and CBET-1438099,
and the Iowa Energy Center with Grant Number of 14-008-OG are gratefully acknowledged.
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