1

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

2

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.

References

1. Eia. Electric Power Monthly: with data for January 2015. U.S. Energy Inf. Adm. (2015).

2. Adaramola, M. S. & Krogstad, P. Å. Experimental investigation of wake effects on wind turbine performance. Renew.

Energy 36, 2078–2086 (2011).

3. Sanderse, B. Aerodynamics of wind turbine wakes: Literature review. 1–46 (2009). doi:10.1002/we

4. Westerweel, J. Fundamentals of digital particle image velocimetry. Meas. Sci. Technol. 8, 1379–1392 (1997).

5. Hussain, a. K. M. F. & Reynolds, W. C. The mechanics of an organized wave in turbulent shear flow. Part 2. Experimental

results. J. Fluid Mech. 54, 241 (1972).

6. Tian, W., Ozbay, A. & Hu, H. Effects of incoming surface wind conditions on the wake characteristics and dynamic wind

loads acting on a wind turbine model. Phys. Fluids 26, 125108 (2014).

7. Hu, H., Yang, Z. & Sarkar, P. Dynamic wind loads and wake characteristics of a wind turbine model in an atmospheric

boundary layer wind. Exp. Fluids 52, 1277–1294 (2012).

8. Lignarolo, L. E. M. et al. Experimental analysis of the wake of a horizontal-axis wind-turbine model. Renew. Energy 70,

31–46 (2014).

9. Bernero, S. & Fiedler, H. E. Application of particle image velocimetry and proper orthogonal decomposition to the study of

a jet in a counterflow. Exp. Fluids 29, 274–281 (2000).

10. Ma, X., Geisler, R., Agocs, J. & Schröder, A. Investigation of coherent structures generated by acoustic tube in turbulent

flow separation control. Exp. Fluids 56, 46 (2015).

11. Cammilleri, a et al. POD-spectral decomposition for fluid flow analysis and model reduction. Theor. Comput. Fluid Dyn.

27, 787–815 (2013).

12. Sonin, E. B. Dynamics of helical vortices and helical-vortex rings. EPL (Europhysics Lett. 97, 46002 (2012).

13. Leweke, T., Quaranta, H. U., Bolnot, H., Blanco-Rodríguez, F. J. & Dizès, S. Le. Long- and short-wave instabilities in

helical vortices. J. Phys. Conf. Ser. 524, 012154 (2014).

14. Quaranta, H. U., Bolnot, H. & Leweke, T. Long-wave instability of a helical vortex. J. Fluid Mech. 780, 687–716 (2015).

15. Hattori, Y. & Fukumoto, Y. Effects of axial flow on the stability of a helical vortex tube. Phys. Fluids 24, 222–249 (2012).

16. Szymendera, C. J. Computaitonal Free Wake Analysis of a Helicopter Rotor. (2002).

17. Chapter 2D Vortex Methods. http://www.rsmas.miami.edu/personal/miskandarani/Courses/MSC321/lectvortexmethods.pdf

18. Bernard, P. S. & Handler, R. a. Reynolds stress and the physics of turbulent momentum transport. J. Fluid Mech. 220, 99

(1990).

19. Lignarolo, L. E., Ragni, D., Simao Ferreira, C. & van Bussel, G. G. Experimental quantification of the entrainment of

kinetic energy and production of turbulence in the wake of a wind turbine with Particle Image Velocimetry. 32nd ASME

Wind Energy Symp. 1–12 (2014). doi:10.2514/6.2014-0706

20. Troldborg, N. & Sørensen, J. A simple atmospheric boundary layer model applied to large eddy simulations of wind turbine

wakes. Wind Energy 17, 657–669 (2014).

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