Wake Interaction Modeling Using a Parallelized Free Vortex Wake Model

Wake Interaction Modeling Using A Parallelized Free Vortex Wake Model

Dissertation

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

Kelsey Shaler, M.S.

Graduate Program in Mechanical Engineering

The Ohio State University 2020

Dissertation Committee: Mei Zhuang, Co-advisor Krista M. Kecskemety, Co-advisor Datta V. Gaitonde Jen-Ping Chen Copyright by

Kelsey Shaler

2020 Abstract

Turbine-wake interactions pose signiﬁcant challenges in the development of wind farms. These interactions can lead to an increase in wind energy cost through reduction in wind farm power efﬁciency as well as a reduction of functional turbine lifetime. The overall objective of this work is to extend and assess a moderate-

ﬁdelity free vortex wake (FVW) model to capture turbine-wake interactions between multiple turbines. Speciﬁc focus areas include: (1) analyzing the effects of turbine-wake interaction; (2) benchmarking of the model against experimental wind farm measurements; and (3) comparing wake interaction effects between the

FVW model and a dynamic wake meandering (DWM) model. Results show that

FVW produces an increased dynamic response in wake-inﬂuenced turbines than

FAST.Farm, which is an important factor in fatigue life of turbine blades. Param- eter studies for various operating and layout conditions are performed. Analysis focuses on impact of wake interaction on wake structure, rotor power, and blade root bending moments. The parameter study shows expected power trends for all tested parameters. The effects of turbine-wake interactions are analyzed in terms

ii of wake structure, rotor power, and structural response. The FVW model predicts increased unsteadiness in wake-inﬂuenced turbine rotor power and out-of-plane blade root bending moment. This could have implications for prediction of turbine life and suggests that the transient as well as average response of turbines should be considered to fully capture the effects of wake interaction. Comparisons between the FVW predictions and experimental measurements of relative rotor power are made over varying yaw angle and freestream velocity. Overall trends are predicted by the FVW approach, with less than 13% error on average when compared to wind farm measurements. These results indicate the FVW method is a useful tool for carrying out improved optimization of wind farms.

iii Dedication

To my parents, Steve and Robin, for their unconditional love, support, and encouragement. I could not have hoped for better parents or role models.

iv Acknowledgments

I would like to thank my advisor, Prof. Mei Zhuang, for her guidance and support on this project. I would also like to thank my co-advisor, Dr. Krista Kecskemety, for her time and assistance with this work. I would also like to thank Profs. Datta

Gaitonde, Mo Samimy, and Jim Gregory for serving on my candidacy committee and Profs. Datta Gaitonde and Jen-Ping Chen for serving on my dissertation committee. The results in Chapter 2 were produced with Dr. Krista Kecskemety and

Dr. Jack McNamara as senior authors and is summarized in Shaler, Kecskemety, and McNamara, “Benchmarking of a Free Vortex Wake Model for Prediction of

Wake Interactions,” Renewable Energy, V136, June 2019.

I am deeply grateful for the ﬁnancial support I have received for this work. The support from the National Science Foundation Graduate Research Fellowship Pro- gramming allowed me the freedom to conduct this research. This work was also supported in part by an allocation of computing time from the Ohio Supercomput- ing Center.

I am grateful to all of my lab mates. Thank you to Emily Dreyer, Drs. Brent

v Miller, Zach Riley, Rohit Deshmukh, Ben Grier, Steve Nogar, and everyone else.

You were all always there to help and support me, and we had a few good times along the way.

To everyone at NREL, thank you for the support and encouragement throughout this process. Without your kind words I’m not sure I would have had the resolve and conﬁdence to push through.

Thank you to my entire support system at Ohio State. To the WEGC ladies, thank you for helping me grow as a professional, as a person, and as a feminist.

To the squad: Drs. Caroline, Elena, Rachel, and Sarahx2. I’m not sure I could have made it without you ladies. The workouts, wine nights, and laughter helped me through the hard times and I will always appreciate everything you’ve done for me. Andrew, thank you for the mid-day walks, singing emo songs with me, and bringing a bit of sparkle into my day.

Kona and Olaf, thank you for the cuddles and unconditional love.

Aaron, there are no words to describe how you have helped and supported me.

Washing almost every pan I’ve used over the years; making me laugh; and always reminding me that I’m good enough, I’m smart enough, and gosh darn it people like me. For this and so much more, I am so thankful to have you in my life.

Thank you to my family. To my mother, for encouraging me in everything

I’ve ever wanted to do and being my best friend. To my father, for knowing me better than I know myself and insisting I would like physics and coding, as well as providing valuable insight throughout this whole journey. To Ross, nam ens

vi optimum fuisse umquam fratrem me.

And ﬁnally, I would like to thank every teacher who has ever inﬂuenced me. I have been truly blessed to have phenomenal educators through every step of my education. From preschool through graduate school, each of you has helped to shape me into the person I am. I know I would not be where I am today without you and I will forever be thankful for what you have given me.

vii Vita

2011 ...... B.S. Mechanical Engineering, University of Maine

2016 ...... M.S. Mechanical Engineering, The Ohio State Univer-

sity

2011 - 2016 ...... NSF GRFP Fellowship, The Ohio State University

2011 - 2012 ...... Graduate Research Assistant, The Ohio State Univer-

sity

2014 - 2015 ...... Graduate Research Assistant, The Ohio State Univer-

sity

2016 - 2017 ...... FAST Fellowship, The Ohio State University

2017 - 2018 ...... Graduate Research Assistant, The Ohio State Univer-

sity

2018 - 2019 ...... Post-Doctoral Researcher, National Renewable Energy

Laboratory

2019 - present ...... Research Engineer, National Renewable Energy Labo-

ratory viii Publications

Journal Publications

Shaler, K., Jonkman, J., “FAST.Farm Development and Veriﬁcation of Struc- tural Load Prediction Against Large Eddy Simulations”. Wind Energy, 2020. Sub- mitted.

Doubrawa, P., Quon, E., Martinez-Tossas, L. A., Shaler, K., Debnath, M., Hamil- ton, N., et al., “Multi-model Validation of Single Wakes in Neutral and Stratiﬁed

Atmospheric Conditions”. Wind Energy, 2020.

Shaler, K., Kecskemety, K.M., and McNamara, J. J., “Benchmarking of a Free

Vortex Wake Model for Prediction of Wake Interactions”. Renewable Energy, 2019.

Robertson, A. N., Shaler, K., Sethuraman, L., Jonkman, J., “Sensitivity Analysis of Wind Characteristics and Turbine Properties on Wind Turbine Loads”. Wind

Energy Science, 2019.

ix Shaler, K., Debnath, M., Jonkman, J., Brugger, P., and Porte-Agel, F., “Vali- dation of FAST.Farm Against Full-Scale Turbine SCADA Data and LiDAR Wake

Measurements for a Small Wind Farm”. Journal of Physics: Conference Series,

2020. Accepted.

Martinez-Tossas, L. A., Branlard, E., Shaler, K., Vijayakumar, G., Ananthan,

S., Sakievich, P., Jonkman. J., “Wind Turbine Wakes: High-Thrust Coefﬁcient”.

Journal of Physics: Conference Series, 2020. Accepted.

Shaler, K., Jonkman, J., and Hamilton, N., “Spatial and Temporal Discretization

Studies of Wake Meandering and Turbine Structural Response Using FAST.Farm”.

Journal of Physics: Conference Series, 2019.

Shaler, K., Branlard, E., Platt, A., and Jonkman, J., “Preliminary Introduction

of a Free Vortex Wake Method Into OpenFAST”. Journal of Physics: Conference

Series, 2019.

Technical Reports

Jonkman, J. and Shaler, K., “FAST.Farm User’s Guide and Theory Manual”

NREL Technical Report, 2020. Submitted.

x Shaler, K., Branlard, E., and Platt, A., “OLAF User’s Guide and Theory Man-

ual”, NREL Technical Report, 2020. Submitted.

Conference Publications and Presentations

Shaler, K., Jonkman, J., Quon, E., and Hamilton, N., “Veriﬁcation of FAST.Farm

Structural Load Prediction Against Large Eddy Simulations”, Wind Energy Science

Conference, June 17-20, Cork, Ireland, 2019.

Shaler, K., Jonkman, J., Doubrawa, P., Hamilton, N., “FAST.Farm Response to

Varying Inﬂow Conditions”, AIAA SciTech Wind Energy Conference, January 7-11,

San Diego, CA, 2019.

Shaler, K., Robertson, A. N., Sethuraman, L., Jonkman, J., “Assessment of

Airfoil Property Sensitivity on Wind Turbine Extreme and Fatigue Loads”, AIAA

SciTech Wind Energy Conference, January 7-11, San Diego, CA, 2019.

Shaler, K., Kecskemety, K. M., Gogulapati, A., and McNamara, J. J., “Wind

Farm Optimization Using a Free Vortex Wake Model”, AIAA SciTech 36th ASME

Wind Energy Symposium, January 9-13, Kissimmee, FL, 2018.

Shaler, K., Kecskemety, K. M., and McNamara, J. J., “Benchmarking of a Free

xi Vortex Wake Model for Prediction of Wake Interactions”, AIAA SciTech 35th ASME

Wind Energy Symposium, January 9-13, Grapevine, TX, 2017.

Shaler, K., Kecskemety, K. M., and McNamara, J. J., “Wake Interaction Effects

Using a Parallelized Free Vortex Wake Model”, AIAA SciTech 34rd ASME Wind

Energy Symposium, January 4-8, San Diego, CA, AIAA2016-1520, 2016.

Shaler, K., Kecskemety, K. M., and McNamara, J. J., “Preliminary Study of

Wake Interaction Effects Using a Free Vortex Wake Model”, AIAA SciTech 33rd

ASME Wind Energy Symposium, January 5-9, Kissimmee, FL, AIAA2015-0686, 2015.

Shaler, K. and Gaitonde, D. V., “Flow Control of a Retreating Airfoil via NS-

DBD Actuators Using Large Eddy Simulations”, 4th ASME Joint US-European Flu-

ids Engineering Summer Meeting, August 3-7, Chicago, IL, 2014.

Shaler, K., “Characterization of Sharp-Edged Airfoils Using Large Eddy Simu- lations”, 21st AIAA Computational Fluid Dynamics Conference, June 24-27, San Diego,

CA, 2013.

Gaitonde, D. V., Sahin, M., Shaler, K., Glaz, B., and Dinavahi, S. P. G., “High-

Fidelity Simulations of NS-DBD-Based Control of a Stalled NACA0015 Airfoil”,

51st AIAA Aerospace Sciences Meeting, January 7-10, Grapevine, TX, 2013.

xii Shaler, K. and Glaz, B., “Determining Validity of Isotropic Eddy Viscosity Tur- bulence Models for Nanosecond-Pulsed Plasma Flow Control Applications”, 8th

Annual Dayton Engineering Sciences Symposium (DESS), October 29, Dayton, OH,

2012.

Gaitonde, D. V., McCrink, M. H., and Shaler, K., “A Semi-Empirical Model of a Nanosecond-Pulsed Plasma Actuator for Flow Control Simulations Using LES”,

37th Dayton-Cincinnati Aerospace Sciences Symposium (DCASS), November 30, Day- ton, OH, 2012.

Shaler, K. and Lewendowski, E., “Advanced Stirling Convertor (ASC) Perfor- mance Analysis Using Phasor Diagrams”, 1st Annual Nuclear and Emerging Tech- nologies for Space, February 7-10, Albuquerque, NM, 2011.

Fields of Study

Major Field: Mechanical Engineering

xiii Table of Contents

Abstract ...... ii

Dedication ...... iv

Acknowledgments ...... v

Vita ...... viii

List of Tables ...... xvii

List of Figures ...... xix

List of Symbols ...... xxvi

Chapters

1 Introduction and Objectives ...... 1 1.1 Introduction ...... 1 1.2 Cost of Energy ...... 5 1.3 Wake Structure and Turbine-Wake Interaction ...... 7 1.4 Literature Review ...... 10 1.4.1 Experimental Methods ...... 11 1.4.2 Computational Methods ...... 14 1.4.2.1 Wake Interaction Superposition ...... 15 1.4.2.2 FAST.Farm Model ...... 16

xiv 1.4.2.3 Computational Fluid Dynamics ...... 28 1.4.2.4 Vortex Wake Models ...... 33 1.5 Dissertation Objectives ...... 37 1.6 Key Novel Contributions of this Dissertation ...... 38

2 Free Vortex Wake Model ...... 40 2.1 Free Vortex Wake Model ...... 40 2.1.1 Modeling Improvements ...... 40 2.1.2 Model Formulation ...... 41 2.1.2.1 Vortex Convection ...... 42 2.1.2.2 Predictor-Corrector Time-Marching Scheme . . . 44 2.1.2.3 Induced Velocity ...... 48 2.1.2.4 Circulation ...... 50 2.1.3 FAST-Free Vortex Wake Model Formulations ...... 52 2.1.4 Multi-Turbine Free Vortex Wake Interaction ...... 54 2.2 Convergence Studies ...... 57 2.2.1 Wake Cutoff Length ...... 58 2.2.2 Wake Discretization ...... 65 2.3 Multi-Turbine Free Vortex Wake Model Validation ...... 69 2.3.1 Wind Farm Experiment and Simulation Cases ...... 69 2.3.2 Wake Interaction Effects ...... 72 2.3.3 Varying Freestream Velocity ...... 79 2.3.3.1 Rotor Power ...... 80 2.3.3.2 Structural Response ...... 83 2.3.4 Varying Inﬂow Yaw Angle ...... 85

3 Characterization of Wake Interaction Effects Using FAST-FVW and FAST.Farm ...... 90 3.1 Wake Structure ...... 92

xv 3.2 Rotor Power ...... 94 3.2.1 Varying Freestream Velocity ...... 95 3.2.2 Varying Yaw Angle ...... 102 3.2.3 Varying Offset Distance ...... 103 3.2.4 Varying Separation Distance ...... 106 3.2.5 Varying Turbulence Intensity ...... 108 3.3 Structural Response ...... 112 3.3.1 Varying Freestream Velocity ...... 112 3.3.2 Varying Yaw Angle ...... 124 3.3.3 Varying Offset Distance ...... 127 3.3.4 Fatigue Loads ...... 129

4 Concluding Remarks ...... 132 4.1 Principal Conclusions Obtained in This Study ...... 132 4.2 Recommendations for Future Research ...... 135 4.2.1 Further Veriﬁcation and Validation ...... 135 4.2.2 Code Improvements and Applications ...... 136

Bibliography ...... 139

A Multi-Variate Taylor Series Expansion ...... 151

B Representative V90 Wind Turbine ...... 152

C Turbulent Inﬂow Modeling ...... 154

D Fatigue Load Modeling ...... 156

xvi List of Tables

1.1 Overview of scale ranges involved in wind farm simulations. [80] 28

2.1 Nominal wind turbine parameters for the representative Vestas V90 turbine...... 58 2.2 Computational time for representative Vestas V90 two-turbine simulations with various wake cutoff lengths with 6 m/s inflow. Com- puted using Intel Xeon E5-2680 v4 CPUs with a single processor. . 60 2.3 Wake cutoff lengths for three-turbine wake cutoff length study. . . 62 2.4 Average percent error of rotor power for different inflow averaging techniques...... 81 2.5 Average percent error of minimum, mean, maximum, and range of relative flapwise bending moment between computational results and experimental measurements for non-waked (T1) and waked (T2) turbines across all freestream velocities ...... 84 2.6 Average percent error of relative rotor power between computational results and experimental measurements for varying freestream velocities and yaw angles...... 86

3.1 Ranges of inﬂow conditions and turbine locations...... 91 3.2 Main parameters of the UAE Phase-VI wind turbine...... 92 3.3 Average percent error of rotor power for different inﬂow averaging techniques...... 96

xvii 3.4 Average percent standard deviation (σ) of average rotor power for varying freestream velocities for FAST-FVW and FAST.Farm results with uniform and turbulent inflow conditions...... 101 3.5 Average percent standard deviation (σ) of average rotor power for varying offset distances for FAST-FVW and FAST.Farm results with turbulent inflow conditions...... 105 3.6 Average percent error of minimum, mean, maximum, and range of relative flapwise bending moment between computational results and experimental measurements for non-waked and waked turbines...... 118 3.7 Average percent difference of flapwise and edgewise blade root bending moments between FAST-FVW and FAST.Farm results for varying freestream velocities...... 121 3.8 Average standard deviations of flapwise and edgewise blade root bending moments as a percentage of the mean for FAST-FVW and FAST.Farm results for varying freestream velocities with turbulent inflow...... 123

C.1 Summary of input parameters used by TurbSim for turbulent in- ﬂow generation...... 155

D.1 Sample chart of amplitude and count of each cycle in a non-uniform signal...... 159

xviii List of Figures

1.1 Schematic of wake structure behind a turbine. Adapted from Sanderse. [19] 8 1.2 Axisymmetric wake deﬁcit and meandering evolution. [54] . . . . 18 1.3 FAST.Farm submodel hierarchy. [54] ...... 19 1.4 Schematic of FAST.Farm domains and wake trajectories for a two- turbine wind farm with a yawed upstream turbine. The low-resolution domain is represented by the yellow points. The high-resolution domains surrounding each turbine are represented by the green points. The wake plane centers and orientations are represented by the blue points and arrows. Instantaneous wake plane trajectories are represented by the blue curved lines. The wake volumes connecting each plane are represented by the grey dashes, with zones of wake overlap. [70] ...... 23 1.5 Schematic of the actuator line model. [86] ...... 31

2.1 Solution process for the FAST-FVW model. [104] ...... 42 2.2 Evolution of blade tip vortices and Lagrangian markers. [113] . . 43 2.3 Constant and variable rotor speed stencils used in time-marching predictor-corrector scheme...... 45 2.4 Schematics of Weissinger-L lifting line method. Adapted from Gupta. [103] 51 2.5 Flowchart of FASTv8/OpenFAST modularization. [120] ...... 53 2.6 Flowchart of the solution process for the integrated FAST-FVW model...... 53

xix 2.7 Flowchart of the parallelization process for a multi-turbine wake interaction model...... 55 2.8 Schematic of process used to determine if a downstream turbine could potentially be waked by an upstream turbine...... 56 2.9 (a) Average rotor power and (b) percent difference for single-turbine wake convergence study...... 59 2.10 (a) Relative rotor power and (b) percent difference for two-turbine wake convergence study...... 60 2.11 Wake cutoff length convergence study for NREL UAE Phase VI turbine with uniform inflow...... 61 2.12 Rotor power results for wake cutoff length study involving three aligned turbines. Results include (a) rotor power for each turbine and (b) standard deviation of rotor power for each turbine. . . . . 63 2.13 Relative rotor power results for wake cutoff length study involving three aligned turbines...... 64 2.14 Sample time-series of rotor speed for upstream and downstream turbines at various uniform freestream velocities...... 66 2.15 Discrepancies between specified and realized discretization levels for T1 in single-turbine simulations...... 67 2.16 Discrepancies between specified and realized discretization levels for T2 in two-turbine simulations...... 67 2.17 Comparison of relative rotor power for varying discretization levels. 69 2.18 Egmond aan Zee windfarm layout. Adapted from Churchfield. [11] 70 2.19 Wake structure for two-turbine simulations using the FAST-FVW ◦ method. Simulations use turbulent inflow at U∞ = 8 m/s at (a) 0 , (b) 5◦, and (c) 10◦ yaw angles...... 73 2.20 Rotor power results for single- and two-turbine simulations in turbulent 8 m/s wind inflow. (a) Normalized average rotor power and (b) standard deviation of average rotor power...... 75

xx 2.21 Standard deviation of out-of-plane blade root bending moment results for single- and two-turbine simulations in turbulent 8 m/s wind inﬂow...... 77 2.22 Relative rotor power for computational results versus varying freestream turbulence intensities...... 78 2.23 (a) Rotor power standard deviations of average rotor power for computational results. (b) Percent change of computational results relative to computational 1% TI results. Cases include varying freestream turbulence intensities...... 79 2.24 (a) Relative rotor power and (b) percent error for FAST-FVW and experimental [35] results. Cases include varying turbulent freestream velocities...... 80 2.25 Relative ﬂapwise bending moment comparisons between computational and experimental results [35] for (a) upstream and (c) downstream turbines at varying freestream velocities. Percent error between computational and experimental results are shown for the (b) upstream and (d) downstream turbines...... 83 2.26 (a) Relative rotor power for computational results and experimental measurements; [35] and (b) percent error for computational results relative to experimental measurements. Cases include vary-

ing yaw angle with U∞ = 6 m/s...... 86 2.27 (a) Relative rotor power for computational results and experimental measurements; [35] and (b) percent error for computational results relative to experimental measurements. Cases include vary-

ing yaw angle with U∞ = 8 m/s...... 87

3.1 Two-turbine layout for parametric studies. Deﬁnitions of various operating conditions and spacing deﬁnitions are included. . . . . 91

xxi 3.2 Top view of wake tip vortex structure for offset distances of (a) 0 m, (b) 5 m, and (c) 10 m. Wakes are shown together (i) and separately (ii and iii). All cases are for uniform freestream velocity at 10 m/s with turbines separated by 2D. Solid black and red lines represent T1 wake structure. Dashed blue and green lines represent T2 wake structure...... 93 3.3 FAST-FVW and FAST.Farm rotor power time series for T1 (left) and T2 (right) for uniform inflow conditions for varying steady and turbulent freestream velocities. For each color group, darkest color represents FAST.Farm results and the lighter colors represent FAST-FVW results. When three lines are present in a color group, the lightest color is for T0 FAST-FVW results...... 95 3.4 (a) Relative rotor power and (b) percent error for FAST-FVW, FAST.Farm, and experimental [35] results. Cases include varying steady and turbulent freestream velocities...... 97 3.5 Comparison of average turbine rotor power of (a) T1 and (b) T2 for FAST-FVW and FAST.Farm model results, including percent difference between model results. Cases include varying steady freestream velocities...... 98 3.6 Comparison of non-dimensionalized average wake velocity profiles for FAST-FVW and FAST.Farm model results. Cases include varying steady freestream velocities and downstream distances. . 99 3.7 Standard deviation of rotor power for FAST-FVW and FAST.Farm results with uniform and turbulent inflow conditions...... 100 3.8 Relative rotor power for computational results from the (a) FAST- FVW and (b) FAST.Farm methods. Cases include varying yaw angle with varying turbulent freestream velocities...... 102

xxii 3.9 Average relative (top) and standard deviation (bottom) of rotor power for varying offset distances for FAST-FVW and FAST.Farm results. Percent difference between computational results are included (left)...... 104 3.10 Average relative and standard deviation of rotor power for varying separation distances for FAST-FVW and FAST.Farm results. Per- cent difference between computational results are included. . . . 107 3.11 Relative rotor power for computational and experimental results versus varying freestream turbulence intensities. Simulations were

performed for U∞ = 6 − 12 m/s at each TI. Bars represent the minimum and maximum values (unavailable for experimental results), and the symbols represent the mean value...... 109 3.12 Rotor power standard deviations of average rotor power for computational results (left) and percent difference of computational results relative to computational 3% TI results (right) for FAST-FVW and FAST.Farm computational results. Cases include varying turbulence intensities for 6 and 8 m/s inflow velocities...... 111 3.13 FAST-FVW and FAST.Farm blade-root flapwise bending moment time series for T1 (left) and T2 (right) for varying steady and turbulent freestream velocities. For each color group, darkest color represents FAST.Farm results and the lighter colors represent FAST- FVW results. When three lines are present in a color group, the lightest color is for T0 FAST-FVW results...... 113 3.14 FAST-FVW and FAST.Farm blade-root flapwise bending moment PDF for T1 (left) and T2 (right) for varying steady and turbulent freestream velocities...... 114 3.15 FAST-FVW (left) and FAST.Farm (right) blade-root flapwise bending moment PSD for T1 and T2 for steady and turbulent 6 m/s freestream velocities...... 115

xxiii 3.16 FAST-FVW and FAST.Farm blade-root flapwise bending moment PSD for T1 (left) and T2 (right) for steady and turbulent freestream velocities...... 117 3.17 FAST-FVW and FAST.Farm blade-root edgewise bending moment PDF for T1 (left) and T2 (right) for varying turbulent freestream velocities...... 117 3.18 Relative flapwise bending moment comparisons between computational and experimental results [35] for (a) T1 and (c) T2 at varying freestream velocities. Cumulative percent error between computational and experimental results are shown for the (b) T1 and (d) T2...... 119 3.19 RMS of flapwise (top) and edgewise (bottom) blade root bending moments for FAST-FVW and FAST.Farm results at varying uniform and turbulent freestream velocities...... 120 3.20 Standard deviations of flapwise and edgewise root-blade bending moments for T1 (left) and T2 (right) results at varying freestream velocities ...... 122 3.21 RMS of flapwise (top) and edgewise (bottom) blade root bending moments and standard deviations for FAST-FVW and FAST.Farm results with varying yaw angles...... 124 3.22 Exaggerated edgewise load differences due to different waked turbine scenarios, resulting in different combinations of aerodynamic and gravitational loads. Freestream conditions are shown as well as two partially-waked turbine conditions. Blade positions 1 and 2 are labeled on the turbine figures, which correspond to the numbered points in the loading figures on the right. Reproduced from [127] with permission...... 126

xxiv 3.23 RMS of ﬂapwise (top) and edgewise (bottom) blade root bending moments and standard deviations for FAST-FVW and FAST.Farm results with varying offset distance...... 128 3.24 DELs of FAST-FVW and FAST.Farm simulations for varying offset distance...... 130

B.1 Single-turbine power comparison of expected representative V90 turbine and computational FVW results...... 153

D.1 Example time-series plots of uniform and non-uniform loading. . 157 D.2 Sample qualitative rainﬂow analysis for non-uniform loading. . . 158

xxv List of Symbols

− a1 = numerical constant= 2 × 10 4 ABL = atmospheric boundary layer AEP = Annual Energy Production c = chord CapEx = ﬁxed wind farm costs CFD = computational ﬂuid dynamics COE = cost of energy D = rotor diameter DEL = damage equivalent load DWM = Dynamic Wake Meandering

Fc = core radius factor FLORIS = FLOw Redirection and Induction in Steady State FVW = Free Vortex Wake HRTF = High-Reynolds number Test Facility IEC = International Electrotechnical Commission l = ﬁlament length LCOE = levelized cost of energy LES = large eddy simulation LiDAR = Light Detection and Ranging

MEdge = blade-root edgewise bending moment

MFlap = bade-root ﬂapwise bending moment

xxvi N = number of rotor revolutions before wake cutoff condition NREL = National Renewable Energy Laboratory NS = Navier-Stokes OpEx = variable wind farm costs OWEZ = Egmond aan Zee P = turbine rotor power PDF = probability distribution function PSD = power spectral density R = rotor radius ~r = vector between point of interest and vortex segment ~r(ψ, ζ) = position vector of Lagrangian markers

rc = core radius

r1, r2 = vector between point of interest and end point of vortex RANS = Reynolds-Averaged Navier-Stokes RMS = Root Mean Square RSS = root-sum-squared SCADA = supervisory control and data acquisition SOWFA = Simulator fOr Wind Farm Applications SWiFT = Scaled Wind Farm Technology T0 = single turbine T1 = upstream turbine T2 = downstream turbine t = time TI = turbulence intensity

U∞ = freestream velocity V~ = velocity vector ~ Vind = induced velocity vector x¯ = average value of x

xxvii y = lateral coordinate variable z = downstream coordinate variable α = numerical constant= 1.25643 Γ = circulation strength δ = measure of viscous diffusion = measure of strain ζ = vortex wake age

ζ0 = vortex wake age offset ∆ζ = step size for wake age θ = blade pitch angle ν = kinematic viscosity σ = standard deviation ψ = azimuth blade position ∆ψ = step size for blade rotation Ω = rotational speed of wind turbine

Subscripts FAST.Farm = results from FAST.Farm simulation FAST-FVW = results from FAST-FVW simulation T0 = results from single-turbine simulation T1 = results from upstream turbine in two-turbine simulation T2 = reuslts from downstream turbine in two-turbine simulation

T2/T1 = relative value, e.g., xT2/xT1

xxviii Chapter 1

Introduction and Objectives

1.1 Introduction

Now more than ever, it is imperative for the sustainability of the planet to research and invest in renewable energy sources. The burning of fossil fuels, the primary source of energy in most of the industrialized world, emits greenhouse gases such as nitrous oxide, methane, and carbon dioxide (CO2). [1] While all of these sub- stances occur naturally, the burning of fossil fuels has caused CO2 concentration levels to increase by more than a third since the beginning of the industrial revolution. [1] Increased CO2 levels have repercussions such as rising ocean levels, increasing extreme weather events, shrinking ice caps, ocean acidiﬁcation, and rising average global temperature. [2] Despite overwhelming evidence that CO2 emissions are reaching dangerous levels, the use of carbon fuels is continuing to rise worldwide. [3] Fueled by the need to decrease reliance on fossil fuels, a 2008

Department of Energy report outlined a plan in which 20% of the nation’s energy

1 could be produced by wind energy by 2030. [4] Since 2006, U.S. wind energy generation has grown to an estimated 12, 000 GW and 4, 150 GW for land-based and offshore sites, respectively. [5, 6] This corresponds to a 400% increase and this number only continues to grow. [7] In 2016 alone, the generating capacity of wind energy increased 30%. [7] This demonstrates the desire as well as the ability for increased reliance on wind energy within the United States.

Wind energy has been a focus in the renewable energy sector because of its availability as well as room for improvement in the industry. Since 2008, the United

States has increased wind energy capacity from 11.6 GW to 80 GW. [4, 7] In addition to increased funding and focus on renewable energy, technology advancements were necessary to reach this capacity level. Such technology advancements include taller hub heights, larger turbine rotors, improved turbine blade design, and improved turbine controls. Increasing turbine height allows for higher mean wind speeds and larger rotor size increases the amount of potential energy that can be produced by individual turbines. Turbine blades have become significantly more flexible, allowing for greater deformation and reduced risk of blade damage. Blade optimization also allows for improved blade aerodynamics. Significant changes have been made to wind turbine controls, including blade pitching, or feathering, which is used to control the turbine rotor speed in high wind speeds and to stop the turbine in dangerous wind speeds. [8] Turbine yaw control allows turbines to be directed into the primary wind direction at optimal wind speeds and out of the wind at dangerous wind speeds. Turbine wake steering is a form

2 of wind farm optimization in which the wakes of upstream turbines are deﬂected

away from downstream turbines for the purpose of reducing turbine-wake inter-

actions and increasing the overall wind farm performance. These advancements,

along with many others, have allowed for increased turbine power production as

well as turbine lifetime, thereby reducing the cost of wind energy and increasing

its potential.

Despite signiﬁcant advancements, numerous challenges and areas of improve-

ment remain. Setting aside blade design improvements and the potential for fur-

ther increase to turbine hub height and rotor diameter – which have signiﬁcant

associated beneﬁts and challenges – there is substantial potential for improvement

to turbine performance simply through a better understanding of the environment

in which wind turbines operate. Namely, wind turbines typically operate at ∼ 40%

capacity factor, [9] which is a ratio of the actual power generated over a period of

time to the potential, i.e., rated, power generation. [10]

Actual P ower CapacityF actor = (1.1) Rated P ower × Operating T ime

A primary reason for sub-optimal turbine power production is the operation of wind turbines in varied inﬂow conditions, such as atmospheric stability, turbulence intensity (TI), and in the wakes of upstream turbines. Such wake interaction typically leads to power reduction, as well as an increase in turbine fatigue loading and thus reduce turbine functional lifetime. [11] Both wind farm power pro-

3 duction and turbine lifetime are primary components that determine the overall cost of energy of wind farms, and therefore the overall viability of wind energy.

While varying atmospheric conditions will always exist, a better understanding of turbine-wake interactions in wind farm settings can better inform how such interactions impact wake-affected turbines. This information can potentially be used to reduce wind farm cost of energy through increasing turbine performance and lifetime by improving turbine operating conditions in preexisting wind farms, e.g., turbine yaw control and wake steering, and better informing turbine placement in future wind farms.

While environmental concerns are of great importance, it is necessary for renewable energy costs to be competitive with traditional, non-renewable energy costs. In fact, increasing reliability and decreasing cost relative to standard energy sources are key for wind turbines to reach full potential as a signiﬁcant worldwide energy producer. To achieve this goal, wind farm layouts and operating conditions must be optimized. Thus, an improved understanding of wake interactions and their role in turbine performance and longevity will lead to more productive wind turbine layouts. Given the complexity of wake structures and their impact on wind farm operation, a better understanding of turbine-wake interactions could result in more precise layout standards for more efﬁciently organized wind farms. [12]

4 1.2 Cost of Energy

While there are several ways in which energy costs can be measured, the most widely-used metric is Cost of Energy (COE). [13] This metric varies depending on the form of energy being considered, but for wind energy the simplest form is expressed as Cost COE = Farm (1.2) AEPFarm

where CostFarm is the total cost of installing and operating the wind farm and

AEPFarm is the annual energy production of the wind farm. [13] This metric represents the break-even price of a wind farm and is typically expressed in [$/kWh].

Speciﬁc COE deﬁnitions vary in how wind farm cost is estimated. A more compre- hensive measure of wind farm cost is employed in the Levelized Cost of Energy

(LCOE) formulation. [9]

(CapEx × FCR) + OpEx LCOE = (1.3) AEPnet

Here, CapEx, FCR, and OpEx represent the capital expenditures, fixed charge rate, and operational and maintenance expenditures, respectively. Based on the COE definition, this method distinguishes between fixed (CapEx) and variable (OpEx) costs. [9, 13] Capital expenditures represent the initial costs associated with a wind farm, such as electrical infrastructure, assembly and installation, construction fi-

5 nance, and the turbines themselves. [9] Operational and maintenance expenditures include land lease costs and turbine repair costs. Fixed charge rates are used as an economic evaluation of wind farm investments. They represents the annual amount needed to pay fixed investment charges such as construction financing, depreciation, and income tax. The LCOE is typically used by corporations and in- vestors to assess the financial viability of proposed wind farms. [14, 15] As COE models represent wind farm cost relative to produced power, lower COE values represent a better financial investment.

LCOE cost models include a wide range of financial and operational components to determine the financial viability of wind farms. Excluding purely financial considerations, there are many ways to boost the financial viability of wind farms.

Modifying the turbines themselves could lead to reduced costs. Reducing production costs and component weight while increasing material durability would lead to reduced wind farm costs. For offshore wind farms, foundation costs could be reduced by considering water depth when positioning the turbines. Operation and maintenance costs could be reduced by considering turbine component fatigue in layout design. Turbine-wake interactions impact downstream turbines by increasing turbulent loading, thereby increasing fatigue loads and reducing turbine longevity. [16] Increasing wind farm power production would offset the costs in- curred by the wind farm. This can be done by increasing the number of turbines in a wind farm, though this would incur additional costs. Power production of individual turbines could be increased by increasing turbine height or rotor diameter.

6 Because of increased blade ﬂexibility, would likely increase blade fatigue loads as

well as incur additional material costs. Optimizing wind turbine layouts to reduce

turbine-wake interactions would increase wind farm power production. Such in-

teractions can reduce downstream turbine power output by 5 − 8% and 30 − 40%

when turbines are in partial and full wake ﬂow, respectively. [17] Reducing such

interactions would increase the power production of the wind farm.

1.3 Wake Structure and Turbine-Wake Interaction

As freestream air passes over a wind turbine, the pressure gradients in the ﬂow

are relaxed because of the extraction of energy from the ﬂow. [18] This causes the

mean wake velocity to drop as the wake rapidly expands downstream of the rotor

plane, [18] as depicted in Figure 1.1. This rapid expansion typically ends ∼ 2 rotor diameters (D) downstream of the turbine. [20] As the rapid expansion ends, the maximum velocity deficit is reached. At this point, turbulent mixing between the wake and freestream begins, which reduces the velocity deficit. In addition to a strong velocity deficit, this region is dominated by helical tip vortices, increased

TI, and a shear layer. [19, 21] The shear layer begins to form at the edge of the wake because of the velocity difference between the wake and freestream air, and thickens as it moves downstream. [19] This region of the ﬂow is referred to as the “near wake” and continues until the expanding shear layer reaches the wake centerline. [18] This typically occurs ∼ 3 − 5D downstream of the turbine and

7 Atmospheric Boundary Layer Wake Weakened Minimal Maximum Shear Layer Velocity Deficit Velocity Deficit Velocity Deficit

Near Wake Far Wake

Figure 1.1: Schematic of wake structure behind a turbine. Adapted from Sanderse. [19] is mostly influenced by the turbine geometry, as opposed to atmospheric conditions. [19] As the wake extends downstream, mixing occurs between the wake and freestream flow conditions, causing the tip vortices to dissipate and wake properties to approach those of freestream wind conditions. Here, the velocity deficit decays monotonically, with the rate of decay highly dependent on atmospheric

TI. [18] This region is classified as the “far wake” and is mostly influenced by atmospheric conditions. [22] In this region, the velocity deficit becomes minimal after 10D [23] while increased TI can still be observed after 15D. [22]

In a wind farm setting, turbine wakes will interact with downstream turbines.

These turbine-wake interactions often exhibit negative effects on downstream turbine performance and longevity. In particular, the wake velocity deﬁcit is the main

8 driver behind the power reduction observed in turbine-wake interactions. [19, 24–

26]. Such interactions can reduce downstream turbine power output by 5 − 8% and 30 − 40% when turbines are in partial and full wake ﬂow, respectively. [17]

Additionally, the increased TI can lead to increased structural loading of 45% and

10% at turbine spacing of 5D and 9.5D, respectively. [27]

The extent of such interactions is largely dependent on three variables: 1) turbine location; 2) freestream velocity; and 3) freestream TI. [19, 28] The dependence on turbine location is clear. If a turbine is fully waked by downstream of another turbine relative to the primary flow direction (i.e., fully waked), the frequency of turbine-wake interactions is increased more than if the turbine was located else- where, thus reducing downstream turbine power generation. Alternatively, partially waked turbines may produce more power but are subject to higher load fluctuations which can affect turbine longevity. The separation distance between the turbines is also a key factor. If a downstream turbine is located in the near wake of another turbine, it will operate in a significant velocity deficit with increased TI levels. With a larger separation distance, the downstream turbine experiences a weakened velocity deficit and relaxed TI, leading to lessened effects on the downstream turbine power production. Freestream velocity and TI affect the extent of turbine- wake interactions in a similar manner. Wake growth and recovery determine the range and strength of a wake. Lower freestream velocity and TI reduce the level of mixing that occurs between the freestream and the wake shear layer, thereby allowing the wake to grow larger and remain coherent further downstream. Con-

9 versely, high freestream velocity and TI results in rapid entrainment of the wake into the freestream, thereby increasing the rate of velocity recovery and dissipating the wake at a faster rate.

The impact of velocity deﬁcit on power production is taken into consideration when contructing wind farms, with a standard spacing of 6 − 10D in the prominent wind direction and 1.5 − 3D in crosswind directions. [19] However, this standard spacing is too close to alleviate all effects of increased TI. Additionally, these standards are too general to result in optimized wind farms as they can lead to unnecessarily sparse layouts. [19] The effect of waking on turbine fatigue loads is not currently a part of wind farm design. An increased understanding of the afore- mentioned effects of turbine waking will help to better inform wind farm design standards.

The remainder of this dissertation will ﬁrst describe state-of-the-art experimental and computational wind turbine methods, focusing on wakes interaction studies. Next, the scope and objectives of this research are presented followed by key novel contributions. A methodology to implement this work is then discussed and

ﬁnally results are presented.

1.4 Literature Review

This dissertation focuses on wake interaction modeling of wind turbines, in relation to aerodynamic and structural response. The current state-of-the-art for this

10 problem is established by reviewing experimental and computational methods for studying turbine-wake interactions, as well as previous studies performed with such methods.

1.4.1 Experimental Methods

The evolution of a wake, especially when wake interactions are present, is a complex process that involves many aspects, including mean wind speed, TI, turbulent length scale, and wind shear. [16] While wind farm experiments have limitations, they provide a realistic basis for these complex flows. Experimental data on wake interactions typically comes from either large, full-scale wind farms or smaller, research-based wind farms. Full-scale wind farms give insight into the operation of turbines in actual conditions. [28, 29] Most data from wind farm field studies comes from nacelle or meteorological mast anemometry measurements of the flow

ﬁeld as well as sensors for structural loading. [26, 29, 30] Supervisory control and data acquisition (SCADA) data collected from individual turbines is also used to compare power performance, pitch angles, rotor speeds, and other turbine operating parameters to study the effects of wake interactions. [11, 31–33] Effects of wind direction and yaw misalignment on turbine loads and power deﬁcit can also be studied with large wind farms. [32] Two such wind farms that are often studied for wake interactions are the OWEZ [11, 34, 35] and Horns Rev [17, 28, 29, 36, 37] offshore wind farms located off the shore of Norway. Wake interaction studies

11 have been performed for various wind farms by numerous sources [11, 17, 28,

29, 34, 36, 38–42] and all studies found reduced power and rotor speed of wake- impacted turbines based on turbine location and operation as well as atmospheric conditions.

In addition to turbine response, wind farm experiments can provide insight into wake development and meandering, which are important to understanding downstream turbine response as well as potential methods for enhanced wake breakdown, such as active control, or interaction mitigation, such as wake steering. [32] For such studies, Light Detection and Ranging (LiDAR) techniques can be used for wake visualization as well as wake position and velocity measurements. [31] These studies have been used to aid in validation of wake steering methods, as well as characterize how wake meandering can affect downstream turbine power. [31]

While full-scale field studies are necessary, they are limited by few measuring stations to describe the entire flow field, and structural data is regularly collected from a few select turbines. As such, there is a high likelihood of some instrumentation malfunctions during such long time frames. [29] The data represents months to years of operation and great care must be taken to average and filter the results properly to allow for comparisons and trends to be elicited; [29] this is a subjec- tive process and accurately representing such data sets is difficult. Furthermore, the farms are subject to uncontrollable wind conditions that are difficult to characterize. Wind direction studies are subject to measurement uncertainty. [29, 37]

12 In particular, the measurement of the wind farm reference wind direction is determined from a yaw position sensor located on a single upstream wind turbine. [29]

This process can result in an uncertainty of more than 7◦ because of yaw misalignment of the reference turbine. [29] As wind is stochastic in nature, there is spatial variability in wind direction within a wind farm. [37] This makes it difﬁcult to look into the effects of turbine layout based on speciﬁc conditions. [43]

Smaller-scale research wind farms offer greater variability in wind turbine location and have been used to study the effects of turbine separation distance, offset distance, [26] and yaw angle. [21] Smaller turbines also allow for more complex instrumentation to be used, particularly for wake ﬂow visualization and velocity measurement. The Scaled Wind Farm Technology (SWiFT) Facility at Sandia has been under development since 2013 and uses LiDAR techniques to provide highly accurate wake measurements. [44, 45] This technology has recently been used for single-turbine code validation. [46] Despite the increased variability in layout and measurement abilities of smaller-scale wind farms, they remain subject to uncontrollable wind conditions. It is postulated that reduced turbine hub heights yield different results than in full-scale wind farms as a result of wake-generated turbulence interacting within the atmospheric boundary layer. [47]

Wind tunnel experiments are another means of studying wake interactions, as they provide more control over inflow conditions and turbine farm layout. Such studies have shown the potential for increased downstream turbine production by varity turbine configuration, [48] as well as the effects of a turbine wake on floating

13 turbines subject to pitch and tilt. [49] To accurately capture relevant wake characteristics and ensure that results will scale to full-scale applications, the freestream

Reynolds number and tip speed ratio must be matched simultaneously. [50] Until recently this was not possible. Thus far, it has only been achieved by the High-

Reynods number Test Facility (HRTF) at Princeton University. [51] As such, it is evident that computational analysis has a critical role in the improved understanding and mitigation of wake interactions in wind farms.

1.4.2 Computational Methods

Because of the uncontrollability and limited scope of wind turbine experimental methods, turbine wake models are of great importance and have been studied extensively. Simulating wind turbines is a multifaceted problem with various components requiring different modeling tools. Different computational models are used for structural response, blade aerodynamics, and turbine wakes. Modeling components range in complexity and computational cost. Such models can be used in various combinations. For example, a simple, inexpensive blade aerodynamic model can be used with a complicated, expensive wake model. As the proposed research focuses on wake interaction, wake models are the focus of this section.

There are two main components to a wake model: wake development and wake interaction. These components can either be modeled separately, as they typically are with the Dynamic Wake Meandering (DWM) models, or together, as they are

14 with Large Eddy Simulations (LES). A simple wake interaction method is ﬁrst de-

tailed, followed by DWM and computational ﬂuid dynamics (CFD) methods, and

lastly vortex wake methods

1.4.2.1 Wake Interaction Superposition

The simplest and most computationally efﬁcient wake interaction models are su-

perposition models. These models use results of single-wake calculations and ac-

count for the combined effect of different wakes on downstream turbines using

superposition assumptions. [24, 35, 52–54] Many models make up this category

and are based on a variety of underlying theories and assumptions. Wake super-

position was ﬁrst developed by Lissaman in 1979, where linear superposition of

single-turbine velocity deﬁcits was assumed. [55] This assumption fails for large

perturbations as it overestimates velocity deﬁcits and may lead to negative ve-

locities when many wakes are superimposd. [57] The most commonly used su-

perposition method was developed by Katic` et al. in 1986, who assumed linear superposition of the squares of velocity deﬁcits. [56]

2 2 2 V V1 V2 1 − = 1 − + 1 − (1.4) U∞ U∞ U∞

Here, V1 and V2 are the two original wake velocities and V is the resulting total wake velocity. In general, this assumption provides better agreement with experimental results than Lissaman’s method. [57]

15 Superposition methods are used because of their low computational cost, com- pleting wind farm-scale simulations in a matter of minutes. Such a method is used in FLOw Redirection and Induction in Steady State (FLORIS), an engineering model typically used for wind plant controls studies such as wake steering. [58]

This model has been used extensively to study the effects of different controls al- gorithms and optimization techniques on wind farm power production. [32, 59]

Despite their popularity for wake interaction modeling, such methods have multiple limitations. For example, they may provide poor approximations of the ﬂow

ﬁeld. [60, 61] They do not capture the turbulent nature of the wakes and can result in incorrect power and load predictions unless empirical corrections are employed. [24, 62] Superposition methods mostly assume linear wake expansion [24,

56, 63] which is not consistent with actual wake expansion. [64, 65] They are particularly inaccurate for highly three-dimensional ﬂow, such as yawed ﬂow, because of the numerous assumptions required by said models. [21] Thus, superposition methods are inadequate for capturing a number of important aspects of wake interactions.

1.4.2.2 FAST.Farm Model

Wake meandering is an observed unsteady phenomena that describes the large scale movement of the entire wake. [64, 65] It can cause power fluctuations and significant unsteady mechanical loads in downstream turbines. [11] This wake movement was first documented by Taylor et al. in 1985 [66] by studying experimental

16 wake measurements. This study found that the variability of wind inﬂow direction shifted the wake trajectory, moving it along the area of the downstream rotor. [66]

This movement increased the average rotor power of the downstream turbine, because of the reduced time the downstream rotor was impacted by the upstream wake. [66]

Wake meandering was not considered in early turbine wake models and did not become a point of focus until Larsen et al. began development of the DWM model in 2008. [67] DWM models aim to capture key wake features such as wake- deficit and wake meandering, important for turbine performance and loads, respectively. These features are necessary to accurately predict of wind farm power performance and wind turbine loads. Such wake features are captured by modeling the lateral bulk wake movement of a turbine wake in combination with the wake deficit evolution and wake-added turbulence. The wake deficit evolution and wake meandering are illustrated in Figure 1.2.

DWM methods model the wake meandering using a passive tracer method.

The wake deﬁcit is modeled via the thin shear-layer approximation of the Reynolds-

Averaged Navier-Stokes (RANS) equations under quasi-steady-state conditions in axisymmetric coordinates, with turbulence closure captured by using an eddy- viscosity formulation. [54] The wake deﬁcit is then superimposed onto the wake meandering component.

Recently, a DWM model has been incorprated with the National Renewable

Energy Laboratory (NREL) code OpenFAST, which solves the aero-hydro-servo-

17 (a) Wake Deﬁcit

(b) Wake Meandering

Figure 1.2: Axisymmetric wake deficit and meandering evolution. [54] elasto dynamics of individual turbines. FAST.Farm is a multiphysics engineering tool that accounts for wake interaction effects on turbine performance and structural loading in wind farm applications. [54] This method accounts for wake deficits, advection, deflection, meandering, and merging of wind turbines in a wind farm setting. FAST.Farm expands on the DWM model [67] to address many limitations of past implementations. The development and mathematical details of FAST.Farm were originally presented in Jonkman et al. 2017, [54] with some recent updates provided in Jonkman et al. 2018 [68] and Shaler and Jonkman. [69]

This section summarizes the key principles of how the model works.

FAST.Farm accounts for various physics domains of a wind farm. Each domain is represented by a different submodel within FAST.Farm. The FAST.Farm submodel structure is illustrated in Figure 1.3. There is one instance of the OpenFAST and wake dynamics modules for each simulated wind turbine. Each component

18 Figure 1.3: FAST.Farm submodel hierarchy. [54]

19 of the code is described in the subsections below. These descriptions are largely taken from Shaler and Jonkman [69] with additional details provided in Jonkman and Shaler. [70]

OpenFAST OpenFAST models the loads and motions of individual turbines, capturing the effects of wind inﬂow and response of the rotor, drivetrain, nacelle, tower, and controller. OpenFAST receives disturbed wind information across each high-resolution wind domain. In response, OpenFAST outputs the individual turbine response.

Wake Dynamics Wake evolution is calculated for an individual rotors. This evolution includes wake advection, deﬂection, and meandering; a wake deﬁcit; and a near-wake correction.

The wake dynamics calculations involve many user-speciﬁed parameters that can be calibrated to better match experimental data or high-ﬁdelity modeling. De- fault values have been derived for each calibrated parameter based on LES simulations using the NREL code Simulator fOr Wind Farm Applications (SOWFA). [71]

An axisymmetric grid it used to compute the wake deficit evolution. This grid consists a fixed radial grid on a fixed number of 2D rotor wake planes. [54] At each FAST.Farm time step, the wake dynamics module uses the rotor position, ori- entation, and radially dependent thrust distribution for an individual rotor from

OpenFAST. It also uses the wake plane advection, deﬂection, and meandering ve-

20 locity for the rotor, the rotor-disk-averaged ambient wind speed normal to the disk, and the ambient wind TI at the rotor. [54] Wake dynamics calculations consist of wake plane positions and orientations, as well as the radially-distributed axial and radial wake velocity deﬁcits at the wake planes. These quantities are are needed for disturbed wind calculations.

FAST.Farm models the wake deﬁcit evolution via the thin shear-layer approximation of the RANS equations under quasi-steady-state conditions in axisymmetric coordinates with turbulence closure captured by using an eddy-viscosity formulation. [18] The thin shear layer approximation neglects the pressure term and assumes that the velocity gradients are larger in the radial direction than in the axial direction. [54] From these simpliﬁcations, analytical expressions for the conservation of momentum (Equation 1.5) and conservation of mass (Equation 1.6) are as follows:

∂Vx ∂Vx 1 ∂ ∂Vx Vx + Vr = (rνT ) (1.5) ∂x ∂r r ∂r ∂r

∂Vx 1 ∂ + (rVr) = 0 (1.6) ∂x r ∂r

Here, νT is the eddy viscosity.

The axial and radial wake velocity deﬁcits at the rotor disk are computed via the near-wake correction. [54] This correction is necessary as the pressure gradient behind the rotor is neglected. In this region, the near-wake correction accounts for the drop in wind speed and radial expansion of the wake. [54]

The axial and radially-distributed wake velocity deﬁcits – Vx(r) and Vr(r), re-

21 spectively – are derived using Equation 1.7. [54]

Plane Rel Vx(r ) = −Vx CNearWakea(r) (1.7)

v u r u Z 1 − a(r0) rPlane = u2 r0 dr0 t 0 (1.8) 1 − CNearWakea(r ) 0

Rel Here, Vx is the low-pass time-ﬁltered rotor-disk-averaged relative wind speed

normal to the disk; a(r) is the radially-distributed axial induction; rPlane is the ra-

0 dial expansion of the wake associated with r; r is a dummy variable; and CNearWake

is a user-speciﬁed calibration parameter. The right-hand side of Equation 1.7 rep-

resents the axial induced velocity at the end of the pressure-gradient zone. [54] The

negative sign is due to the axial wake deﬁcit acting opposite the axial freestream

wind. [54] The wake deﬁcit evolution is not expected to be accurate for downwind

distances within the ﬁrst few rotor diameters past the rotor. As such, modiﬁcations

are needed to accurately model closely-spaced wind turbine situations.

Ambient Wind and Array Effects Ambient wind and wake interactions are computed across the wind farm. For these computations, wake volumes a used, as depicted in Figure 1.4. The module uses the positions, orientations, and wake deﬁcits for each wake plane as computed by the wake dynamics module for each individual wind turbine. For each individual turbine, the module computes the disturbed wind needed for the calculation of turbine response in OpenFAST. Additional calculations are the velocity of each wake plane, the ambient inﬂow, and the ambient

22 Figure 1.4: Schematic of FAST.Farm domains and wake trajectories for a two- turbine wind farm with a yawed upstream turbine. The low-resolution domain is represented by the yellow points. The high-resolution domains surrounding each turbine are represented by the green points. The wake plane centers and orientations are represented by the blue points and arrows. Instantaneous wake plane trajectories are represented by the blue curved lines. The wake volumes connecting each plane are represented by the grey dashes, with zones of wake overlap. [70]

TI at the rotor needed to calculate the wake dynamics for each individual wind turbine.

The TI calculated in FAST.Farm is based on a uniform spatial average of the three vector components. This is in contrast to the common deﬁnition of TI used in the wind industry, which consists of a time-averaged quantity. It it important to note that wake-added turbulence has not yet been incorporated into FAST.Farm.

This is especially important in low TI atmospheric conditions, as shown in Jonkman et al. 2018 [68] and Shaler and Jonkman. [31]

Zones of wake overlap are identified by finding wake volumes that overlap in space. Axial wake deficits are superimposed based on the root-sum-squared (RSS)

23 method, [56] whereas transverse deﬁcits are superimposed using a vector sum.

~ FAST.Farm computes the velocity of each wake plane, Vnp , for each turbine as the weighted spatial average of the disturbed wind velocity on the wake plane, using Equation 1.9.

N P np w V~ ~ n=1 n Distn Vnp = (1.9) PNnp n=1 wn

Equation 1.9 includes a spatial weighting factor, wn, dependent on the radial distance of point n from the center of the wake plane. Larsen et al. [67] proposed a uni-

Wake form spatial average where all points within a circle of diameter 2Dnp are given equal weight. Two additional weighted spatial averaging methods have been implemented in FAST.Farm.

Past Work In the context of wake interaction studies, ample work has been done in the past few years to verify and validate FAST.Farm. FAST.Farm has been verified against SOWFA simulations for rigid turbine response using both LES precorsor-genearated turbulent inflow and TurbSim-genearated synthetic turbulent inflow. Cases included single-turbine and small wind farm simulations subject to various inflow and control conditions. FAST.Farm and LES results were compared in terms of thrust and power for individual turbines; wake meandering behavior across different atmospheric conditions; and averaged wake-deficit advection, evolution, and merging effects. [68, 74] Comparisons using LES precursor-

24 generated inflow (i.e., both FAST.Farm and LES simulations used the same inflow) showed that FAST.Farm can reasonably predict the statistical distribution of generator power, generator torque, rotor thrust, and rotor speed for turbines both in isolation and down a row. [68] These results captured losses due to wake interaction. Good agreement was also seen for time series and statistical distribution of horizontal and vertical wake meandering, as well as azimuthally and tempo- rally averaged wake-deficit advection, evolution, and merging, especially in the far wake. [68] Comparisons to LES results were also made against FAST.Farm simulations that used TurbSim-generated synthetic turbulent inflow. [74] These comparisons showed reasonable to strong agreement with the results from Jonkman et al. 2018 [68] and demonstrated that, with proper setup, it is possible to use

TurbSim-generated inﬂow with FAST.Farm to accurately predict turbine response and wake development in a wind farm setting. [74]

Flexible turbine response was studied by comparing FAST.Farm and SOWFA-

OpenFAST models for predicting turbine structural response in a wind farm setting. [69] This was achieved by comparing FAST.Farm and SOWFA-OpenFAST structural results of 15 quantities of interest for simulations of a small wind farm with ﬂow down the row of three structurally ﬂexible turbines. [69] For all considered structural components, strong agreement was seen between time series results for upstream turbines, with increased differences for downstream turbines. [69]

Statistical results compared well for most structural components in all turbines and cases, with most average percent differences of the mean value remaining around

25 5%. [69] Overall, percent differences of the standard deviation show better agreement between methods, with mean average percent differences for each component remaining below 20% and maximum values remaining below 59%. [69] Over-

all, higher differences were seen for cases with lower ambient TI and yawed tur-

bines due the lack of wake-added turbulence and curled wake in FAST.Farm. [69]

For all cases and turbines, small differences were seen between mean and standard

deviations of blade deﬂections and bending moments. [69] Comparatively, larger

differences were seen for the tower-top and tower-base bending moments, related

to more high frequency energy content in the wake of LES. [69] Overall, it was con-

cluded that ﬂexible turbine structural results are comparable between FAST.Farm

and SOWFA-OpenFAST simulations, especially for blade loads. [69]

FAST.Farm has also been used to study the effects of atmospheric lateral coher-

ence and wake meandering on downstream turbine performance [74] and fatigue

loads. [75, 76] These studies found that wake meandering has a large effect on

downstream turbine power production and fatigue loads, making this an impor-

tant physical feature to accurately model.

Recently, FAST.Farm turbine and wake response were compared to multi-turbine

measurements from a subset of a full-scale wind farm for two separate valida-

tion studies. [31] In the ﬁrst study, FAST.Farm predictions of turbine generator

power, rotor speed, and blade pitch for ﬁve-turbine simulations were compared to

SCADA data. [31] For this study, FAST.Farm generally underpredicted the mean

rotor speed and overpredicted the mean blade pitch below rated wind speeds,

26 likely related to inaccuracies in the generic controller. [31] However, FAST.Farm generator power mean and standard deviation results reasonably matched measured data, as well as the mean rotor speed and blade pitch above rated wind speeds. [31] The second study is still ongoing but will compare wake evolution and meandering to experimental scanning Doppler LiDAR measurements. [31]

Though FAST.Farm results have been shown to compare well to LES and (limited) experimental results for most tested inﬂows, detailed parametric studies have yet to be performed. The model has also shown to be less accurate in some situations due to the lack of a curled wake model or wake-added turbulence, the latter of which is especially important in low-TI inﬂows. Additionally, the axisymmetric wake and velocity smoothing could lead to reduced damage equivalent loads

(DELs). The FAST.Farm model also does not capture upstream propagation of information, meaning that downstream turbines will not influence upstream turbines and wake blockage effects are ignored. Due to the lack of a pressure gradient in the near wake, DWM-type models are not considered to predict accurate near- wake solutions. Thus, small turbine spacing (e.g., less than 3D) will lead to inaccurate results. Though uncommon, there are existing wind farms that contain such wind turbine spacing, such as the Lillgrund wind farm, [77] and novel applications that require it, such as offshore floating multi-turbine platforms. [78, 79] Therefore, other computationally efficient methods for studying turbine-wake interactions in challenging inflow conditions is necessary.

27 Table 1.1: Overview of scale ranges involved in wind farm simulations. [80] Domain Length Scale (m) Velocity Scale (m/s) Time Scale (s) Airfoil Boundary Layer 10−3 102 10−5 Airfoil 10−1 102 10−2 Rotor 102 101 101 Wind Farm 104 101 103

1.4.2.3 Computational Fluid Dynamics

Computational Fluid Dynamics (CFD) methods can be used to compute the ﬂow

ﬁeld surrounding wind turbines. They provide accurate solutions to the incom-

pressible Navier-Stokes (NS) equations and can generate detailed information about

the physics of wind farm ﬂows. However, even with the incompressibility assump-

tion these methods have signiﬁcant computational costs. In particular, the turbu-

lent nature of these ﬂows results in a wide range of important length scales from

O(< 1 mm) in the blade boundary layer to O(1 km) in the atmospheric boundary layer, as summarized in Table 1.1. This necessitates a simpliﬁed representation of both the wind turbine and corresponding wake. [81]

Equations 1.10 and 1.11 are used to express mass and momentum conservation for incompressible ﬂow. [81]

∇ · u = 0 (1.10)

∂u + (u · ∇)u = −∇p + ν∇2u + f (1.11) ∂t

Here, u is the velocity vector, p is the pressure divided by density, ν is the kinematic

viscosity, and f is a body force. While ﬁne-scale solutions of these equations for

28 wind turbine applications are not possible with current computing power, approximations of the NS equations are tractable for wind turbine modeling. RANS methods simplify the wake calculations by decomposing the ﬂow quantities into mean and ﬂuctuating components (Reynolds decomposition [82]), as in Equation 1.12.

u(x, t) = ¯u(x) + u0(x, t) (1.12)

This relationship is then substituted into the NS equations, which are then averaged (Reynolds averaging). Using ¯u¯(x) = ¯u(x) and u0(x,¯ t) = 0, Equation 1.13 is obtained. ∂¯u + (¯u · ∇)¯u = −∇p + ν∇2¯u − ∇ · (u¯0u0) (1.13) ∂t

The term u0u0 is called the Reynolds stress tensor and represents the averaged momentum transfer because of turbulent ﬂuctuations. [81] To solve this nonlinear system of equations, a turbulence closure model is required to estimate the Reynolds stress tensor. Despite many limitations, the Reynolds stress tensor is widely modeled using the Boussinesq hypothesis, which relates the Reynolds stress tensor to the mean velocity gradients and a turbulent eddy viscosity, νT . Determination of the turbulent eddy viscosity requires a turbulence model, for which there are numerous options that range in complexity and computational cost.

Such methods typically require separate wake interaction models, such as the superposition method, to account for wake-wake and wake-turbine interactions.

29 Simulations typically take O(hours) to compute, but they often involve a number of empirical constants and require tuning. [81] A commonly-used simpliﬁed

RANS method for computing the quasi-steady wake deﬁcit was developed by

Ainslie. [18] This method uses the steady-state thin-shear layer approximation of the Navier-Stokes equations and is incorporated in a number of commercial applications. For example, GH WindFarmer [83] is a commonly-used RANS commercial code that performs well for small offshore wind farms [84] and has options to increase the accuracy for large wind farms. [83] Because of its empirical nature,

GH WindFarmer can model thousands of wind speed and directional scenarios needed for wind farm energy analysis in a reasonable amount of time. [38] How- ever, WindFarmer and many other RANS models use superposition methods to account for wake interactions and require empirical tuning for different ﬂow conditions. [20, 83]

LES methods are less empirical as they compute the large eddies in the flow and only the small-scale turbulence is modeled, or filtered, with subgrid-scale models. The filtered continuity and momentum equations are given in Equations 1.14 and 1.15. [81]

∇ · ˜u = 0 (1.14)

∂˜u 2 + ˜u · ∇˜u = −∇p − ∇ · τ + ν∇ ˜u + f (1.15) ∂t

Here, x˜ denotes a ﬁltered quantity, τ is the sub-grid scale stress tensor, and f is a body force used to represent the effect of a turbine on the ﬂuid. Typically, an

30 Lift Lift

Drag Drag

Figure 1.5: Schematic of the actuator line model. [86]

eddy viscosity-type model is used to model the sub-grid scale tensor. This allows

for a wider range of cases to be modeled and do not require separate wake inter-

action models. However, they are more computationally expensive than RANS,

with simulations taking O(days) to compute. One of the reasons for the dramatic increase in computational time of LES over RANS is the necessary reﬁnement in three directions needed in solid-surface boundary layers, whereas RANS only requires reﬁnement in the wall-normal direction. [81]

While LES allows for highly resolved solutions of the wind turbine wake and inﬂow conditions, further modeling is required to include the wind turbines themselves. The most common approach is to represent them as actuator disks or lines.

For wind farm computations, actuator line methods are regularly used to main- tain acceptable computational cost and accuracy. This method was introduced by

Sørensen and Shen [85] and replaces turbine blades with a set of points along the blade axis, as depicted in Figure 1.5. Each point has associated lift and drag forces which are imposed onto the ﬂuid ﬂow through the body forces in Equation 1.15.

31 These point forces are smeared using a Gaussian function to better represent dis-

tributed forces along actual turbine blades. The resulting blade loads are highly

dependent on a smearing parameter, , and providing a more physics-based guide- line for this parameter is an active area of research. [86]

Due to computational cost, LES is typically limited to modeling a single array of turbines, [25, 87] although some applications have considered entire wind farms. [11, 88] An actuator line model was coupled with the 3D ﬂow solver Ellip-

Sys3D [89, 90] to investigate the wake interaction between two turbines at various inﬂow conditions both in a full- and partial-wake conﬁguration. [25] This same actuator line method was coupled with another LES code built using the OpenFOAM

CFD toolbox. [91] This LES code computed the atmospheric boundary layer in which the wind farm operates. The LES and actuator line method was integrated with OpenFAST – an aeroelastic code detailed in Section 2.1.2 – to compute aerodynamic forces at the blades and account for dynamic stall as well as wind turbine structural response. [87] This coupling was been used to study the impact of wake interaction on rotor power [11, 69] and structural loads. [11, 69, 87]. In particular, the work of Lee et al. [87] used the SOWFA-OpenFAST method to study the effects of atmospheric stability and wake interaction on downstream turbine DELs. Over- all, moderate dependence was found on atmospheric stability, with downstream turbine out-of-plane blade loads and low-speed-shaft torque yielding lower DELs for certain atmospheric stabilities. For all considered cases, in-plane blade load

DELs were always decreased for the downstream turbine. Fore/aft and side/side

32 tower DELs and nacelle yaw bearing yaw moment DELs were always increased for downstream turbines.

There are RANS-LES hybrid wind turbine codes, such as Helios which was developed as part of the Department of Defense CREATE A/V program. It was used to model the Lillgrund [92] offshore wind farm and examined power variations because of wake interaction. [88] These methods are promising as they can retain the accuracy of LES while mitigating some of the computational cost. However, they have not been studied extensively and remain computationally demanding.

1.4.2.4 Vortex Wake Models

Vortex wake methods treat the wake explicitly by tracking the circulation and spatial locations of the vortical elements trailed from the turbine blades and convected downstream. These models are based on the assumption of the incompressible potential ﬂow, with all vorticity assumed to be concentrated within the vortex

filaments. From the vortex filament circulation, the induced velocity field is determined using the Biot-Savart law. [93] There are two general methods by which wake vortex models attempt to model these strong tip vortices: prescribed and free.

Prescribed wake methods use pre-deﬁned wake structures based on semi-empirical rules. These methods compare well with experiments. [94, 95] However, input parameters are important to obtaining accurate results [95] and experimental results are used for formulation purposes. Prescribed wake methods are therefore

33 not predictive tools, are limited in scope, and can only be solved for steady-state problems. Alternatively, free vortex wake (FVW) methods model the convection and diffusion of the wake as it is shed and trailed from the rotor blade. These models have been developed extensively for helicopter applications starting from the late 1960s [96] and in the past few years have gained popularity in the wind turbine community. [97–99] These methods are based on incompressible, inviscid, irrotational ﬂow theory. Under these assumptions, the motion of the vortex lines is described by the motion of Lagrangian ﬂuid markers, [93] which can be tracked in time-accurate solutions.

There are a number of different methods of tracking the vortex structures, including vortex particles, [100] vortex ﬁlaments, [101] and vortex lattice. [98, 99]

These methods differ in the complexity of the wake structure and thus computational expense, with vortex point methods having the lowest computational bur- den and vortex lattice methods the highest. Vortex lattice methods are beneficial in that wakes are tracked along the entire span of the turbine blades, which allows for a detailed analysis of wake structure. However, the computational expense is significant as the number of markers needed to track the solution can become quite large. While vortex lattice methods track entire sheets of Lagrangian markers, vortex filament approaches are based on the premise that the tip vortices trailed from the rotor blades are the most dominant structure in the wake [101] and thus are the only tracked feature. These tip vortices are formed by the near wake region on the blades rolling up towards the ends of the rotors, resulting in concentrated vor-

34 tices shed off the blade tips. Tracking only the tip vortices allows for a signiﬁcant reduction in Lagrangian markers while still capturing the most prominent feature of the wake.

In recent years, vortex methods have been used extensively for individual wind turbine analysis and their necessity has been established. Vortex wake methods are particularly important for studying large, ﬂexible turbines because large blade de-

ﬂections may lead to a swept area that deviates signiﬁcantly from the rotor plane.

Such deviations violate assumptions used by common aerodynamic models, such as the blade element momentum (BEM) method, which rely on actuator-disk assumptions that are only valid for axisymmetric rotor loads contained in a plane.

Large blade deﬂections may also cause the turbine near wake to diverge from a uniform helical shape. Further, interactions between turbine blades and the local near wake may increase, thus violating assumptions of models that do not account for the position and dynamics of the near wake. Additionally, highly

flexible blades will likely cause increased unsteadiness and three-dimensionality of the aerodynamic effects, increasing the importance of accurate and robust dynamic stall models. There are many other complex wind turbine situations that violate simple engineering assumptions. Such situations include obtaining accurate aerodynamic loads for non-straight blade geometries (e.g., built-in curvature or sweep); skewed flow caused by yawed inflow or turbine tilt; and large rotor motion as a result of placing the turbine atop a compliant floating platform offshore.

FVW methods have been used extensively for helicopter research. A particu-

35 larly well-known FVW method was developed by Leishman et al. [101] and later extended for use with wind turbines. [102, 103] This implementation showed good agreement to BEM method results for low tip-speed ratios (TSRs) but showed deviations at higher TSRs, with the FVW method performing better in this regime.

This method was also compared to the NREL UAE Phase-VI turbine experiments and showed good agreement in wake geometry for all measured TSRs and pitch angles, for both axial and moderately-yawed flow. This method was later implemented into the NREL code FASTv7 and showed the impact of inflow turbulence on turbine performance and loads. [104] Further examples of long-standing FVW codes in the field of wind energy include GENUVP, [100] which uses a vortex particle approach, and AWSM, [105] which uses a vortex filament approach. Both tools have been coupled to structural solvers. The method was extended by Bran- lard et al. [106] to use vortex methods to perform aero-elastic simulations of wind turbines in sheared and turbulent inflow.

To the knowledge of the author, only one other study has looked at extending a

FVW method to include multiple turbines. [107] This work used vortex particles to simulate two laterally-aligned turbines separated by 2D. Only constant inﬂow was considered, but the upstream turbine was incrementally yawed up to 30◦. This study showed expected results, with increased rotor loading of the downstream turbine. However, this method is only able to predict aerodynamic forces and has not been explored further. Therefore, at present it can not be used to study full turbine structural response.

36 1.5 Dissertation Objectives

While experimental methods provide valuable insight into the physical realities of wind farm operation, they are limited by uncontrollable inﬂow conditions and

fixed turbine layout. Given these limitations, computational analysis has a critical role in the improved understanding, modeling, and mitigation of wake interactions in wind farms. There are a variety of turbine-wake models considered for this purpose, ranging from low-fidelity models to high-fidelity CFD models. Lim- itations of these models lead to a need for moderate-fidelity approaches that better optimize model accuracy and computational cost so as to enable improved broad parametric study and optimization of wind farms. Vortex wake models potentially meet this need and have been sought for just this purpose in applications such as rotorcraft, [108] wind turbines, [103, 104] and flapping-wing MAVs. [109] In particular, FVW models have been observed to provide suitable performance for a broad range of applications. [101, 104] They are more computationally demanding than the lower-fidelity models but remain tractable for wind farm scale analysis. De- spite the promise of these models, they have not been used extensively for wake interactions. Thus, the objective of this dissertation are motivated by two over- arching goals: (1) extending a single-turbine FVW method to include wind farm simulations and (2) benchmarking the performance of this code versus experimental results and a state-of-the-art lower-fidelity model. The specific objectives are:

37 1. Extend a previously-developed single-turbine FAST-FVW code to include

multiple turbines;

2. Parallelize this process so turbine responses can be computed simultaneously

and relative to one another;

3. Benchmark FAST-FVW code performance versus the FAST.Farm method and

experimental measurements;

4. Investigate the effects of turbine-wake interaction on downstream turbine

performance and structural response using FAST-FVW and FAST.Farm mod-

els.

The remainder of this dissertation is organized as follows: An overview of the

FAST-FVW method is provided in Chapter 2, along with benchmarking of the

FAST-FVW method versus experimental measurements; characterization of turbine- wake interactions for various operating conditions and turbine layouts using FAST-

FVW and FAST.Farm models is explored in Chapter 3; and the principal conclusions and suggested future work are discussed in Chapter 4.

1.6 Key Novel Contributions of this Dissertation

The principal contributions to the state-of-the-art made in this dissertation are summarized below:

38 1. Extension and parallelization of a single-turbine FAST-FVW method to in-

clude any number of variable-speed turbines in any conﬁguration, while in-

cluding complex turbine-wake interactions.

2. Benchmarking of FAST-FVW method results against experimental measure-

ments in terms of relative rotor power for varying yaw angle and freestream

velocity.

3. Comparison of the FAST-FVW and FAST.Farm methods in a time-averaged

and transient sense.

4. Characterization of wake interaction effects of wake structure, rotor power,

and blade root bending moments for various operating conditions and two-

turbine layouts. This includes comparisons of FAST-FVW and FAST.Farm

results.

39 Chapter 2

Free Vortex Wake Model

2.1 Free Vortex Wake Model

In this section, the FAST-FVW model used in this dissertation is detailed with an emphasis on improvements made for the included studies.

2.1.1 Modeling Improvements

Modeling improvements are summarized here and described in detail in the text.

• Extension of single-turbine FAST-FVW method to allow for multi-turbine

simulations (Section 2.1.4);

• Migration of code from FASTv7 into FASTv8 (Section 2.1.3)

• Nearly halved computational time by restructuring code;

• Introduced automatic generation of initial wake conditions (Section 2.1.2.2);

40 • Modiﬁed code to accommodate variable rotor speed turbines (Section 2.1.2.2);

• Modiﬁed cutoff condition to be in terms of rotor diameter instead of number

of rotor revolutions, allowing for reduced number of Lagrangian markers at

high wind speeds (Section 2.2.1);

• Introduced method of automatically detecting whether a turbine wake will

impact a downstream turbine and modifying the cutoff condition accord-

ingly (Section 2.1.4);

• Added code output options for improved wake visualization and analysis.

2.1.2 Model Formulation

A detailed description of the FAST-FVW model used in this study is provided by

Kecskemety and McNamara. [104] FAST-FVW was compared to FAST-BEM results for single-turbine configurations. Both show good agreement in terms of rotor power and aerodynamic force coefficients with experimental results for uniform and turbulent inflow. [97] The model also demonstrated good agreement with experimental and CFD results for single turbine configurations. [104] The process of computing the wake using the free vortex wake method is depicted in Figure 2.1.

First, the bound circulation on the blade is calculated using a Weissinger-L representation [110] adapted for rotating lifting surfaces. [103, 104, 111, 112] The Biot-

Savart law, using bound circulation and self-induced wake velocities, is then applied to compute the induced velocity on each vortex ﬁlament. [104] The predictor-

41 Compute circulation Compute the Convect the Lagrangian markers Compute the on the blades using induced velocity on using a time marching scheme to induced velocity on a Weissinger-L the Lagrangian solve: the blades using the model and the flow markers using the 휕풓(휓, 휁) 휕풓(휓, 휁) 푽[풓(휓, 휁)] + = Biot-Savart Law tangency condition Biot-Savart Law 휕휓 휕휁 Ω

Advance to next time step and blade location

Figure 2.1: Solution process for the FAST-FVW model. [104]

corrector time-marching scheme then convects the vortex ﬁlaments. [104] Further,

the induced velocities on the blades are used to compute the vortex ﬁlament posi-

tions. [104] Finally, the effective angle-of-attack on the blade is calculated to obtain

the bound circulation of the blade for the next time step. [104]

2.1.2.1 Vortex Convection

The FAST-FVW model is based on a Lagrangian approach in which the tip vor-

tices are discretized into Lagrangian markers and connected by vortex ﬁlaments.

In this work, a hybrid lattice/ﬁlament method is used, as depicted in Figure 2.2.

A lattice method is used in the near wake of the blade. The near wake spans over

a user-speciﬁed angle. After this period, the wake is assumed to instantaneously

roll up into a tip vortex. The near wake is speciﬁed to enter the ﬂow tangent to the

blade surface and then convects downstream with the local freestream velocity.

The Lagrangian markers track the wake as it evolves and convects downstream.

The position of the Lagrangian markers is deﬁned in terms of wake age, ζ, and azimuthal blade location, ψ; by convention, ζ=0◦ is on the blade surface. Each

42 � z ψ Lagrangian markers Freestreamy Velocit

x Ω -y �

Straight line vortex approximation r(ψ, �)

Curved vortex filament

Figure 2.2: Evolution of blade tip vortices and Lagrangian markers. [113]

43 Lagrangian marker is joined to adjacent markers by vortex ﬁlaments, which are

approximated to second order accuracy as straight line segments. [114] After the

initial discretization of the helical wake, the vortex ﬁlaments are allowed to stretch,

rotate, and translate. The wake structure is changed by this movement as the mark-

ers convect downstream. The governing equation of motion for a vortex ﬁlament

is given by: d~r(ψ, ζ) = V~ [~r(ψ, ζ)] (2.1) dt

Using the chain rule, Equation 2.1 is rewritten as:

∂~r(ψ, ζ) ∂~r(ψ, ζ) V~ [~r(ψ, ζ)] + = (2.2) ∂ψ ∂ζ Ω

where dψ/dt = Ω and dψ = dζ. Here, ~r(ψ, ζ) is the position vector of a Lagrangian

marker and V~ [~r(ψ, ζ)] is the velocity. The assumption that dψ = dζ stems from

solving the left-hand side of Equation 2.2 as a one-dimensional wave equation and

setting the characteristic lines to be at an angle of 45◦, which forces dψ = dζ. [101]

2.1.2.2 Predictor-Corrector Time-Marching Scheme

A predictor-corrector scheme is used to numerically solve the left-hand side of

Equation 2.2 for the vortex ﬁlament location. Initially, this scheme was developed

for constant rotor speed [104] and has since been updated to accommodate variable

rotor speed. The stencil for both constant rotor speed and variable rotor speed

∂r(ψ,ζ) ∂r(ψ,ζ) is shown in Figure 2.3. The difference operators ∂ζ and ∂ψ are found by

44 Constant �� + Δ�

Δ� � + 2 Δ� = Δ� Δ� � + 2 � � − 2Δ� � − Δ� � � + Δ�

Δ� Δ� Δ� Varying �� + Δ�

Δ� � + Δ� = Δ� 2 Δ� � + 2 � � − Δ� � � + Δ� � − Δ� − Δ�

Δ� Δ� Δ�

Figure 2.3: Constant and variable rotor speed stencils used in time-marching predictor-corrector scheme.

45 ∂r(ψ,ζ) means of a Taylor series expansion about the point (ζ + ∆ζ/2, ψ + ∆ψ/2). ∂ζ is

∂r(ψ,ζ) computed using a two steps backwards method and ∂ψ by central differencing.

This results in a scheme that is second-order accurate in ζ and third-order accurate in ψ. This process is detailed in Appendix A. Items 1-4 from Appendix A are used for central differencing, which results in:

∂~r(ψ, ζ) = ∂ζ (2.3) ~r(ψ + ∆ψi, ζ + ∆ζ) − ~r(ψ + ∆ψi, ζ) + ~r(ψ, ζ + ∆ζ) − ~r(ψ, ζ) 2∆ζ

Items 1-8 from Appendix A are used for two-steps backwards, which results in:

∂~r(ψ, ζ) = ∂ψ 23~r(ψ + ∆ψi, ζ + ∆ζ) + 23~r(ψ + ∆ψi, ζ) − 21~r(ψ, ζ + ∆ζ)

− 21~r(ψ, ζ) − 3~r(ψ − ∆ψi−1, ζ + ∆ζ) − 3~r(ψ − ∆ψi−1, ζ) (2.4) + ~r(ψ − ∆ψi−1 − ∆ψi−2, ζ + ∆ζ) + ~r(ψ − ∆ψi−1 − ∆ψi−2, ζ)

−1 46∆ψi + 4∆ψi−1 − 2∆ψi−2

The right-hand side of Equation 2.2 is computed by averaging the velocities surrounding the point (ζ+∆ζ/2, ψ+∆ψ/2). The wake position at rm(ψ+∆ψ, ζ+∆ζ) is then found by substituting the difference operators and velocity averaging into

46 Equation 2.2 and rearranging to obtain:

m ~r (ψ + ∆ψi, ζ + ∆ζ) = V~ 1 23 1 21 − − + ~rm(ψ + ∆ψ , ζ) − − ~rm(ψ, ζ + ∆ζ) Ω 2∆ζ φ i 2∆ζ φ 1 21 3 3 + + ~rm(ψ, ζ) + ~rm(ψ − ∆ψ , ζ + ∆ζ) + ~rm(ψ − ∆ψ , ζ) 2∆ζ φ φ i−1 φ i−1 1 1 1 23−1 − ~rm(ψ − ∆ψ − ∆ψ , ζ + ∆ζ) − ~rm(ψ − ∆ψ − ∆ψ , ζ) + φ i−1 i−2 φ i−1 i−2 2∆ζ φ (2.5)

where ~ m−1 m−1 V = 4V∞ + Vind(~r (ψ, ζ) + Vind(~r (ψ + ∆ψ, ζ) (2.6) m−1 m−1 + Vind(~r (ψ, ζ + ∆ζ) + Vind(~r (ψ + ∆ψ, ζ + ∆ζ)

φ = 46∆ψi + 4∆ψi−1 − 2∆ψi−2 (2.7)

Equation 2.5 is the general form of the predictor and corrector equations, indicated

by the superscript m. It is ﬁrst used in the predictive step to compute the predicted

wake positions for all Lagrangian markers using initial guess values for the wake

positions (~rm) and velocity values at wake positions from the previous time step

(~rm−1). The resulting wake positions are then used as the m time step in the cor-

rector equation to compute the corrected wake position at the current time step

(~rm+1). This process iterates until a converged wake locations are reached. Wake

location is assumed to be converged when the difference in wake position between

iterations reaches a value of less than 0.001 m Root Mean Square (RMS). [104] This

is typically achieved in two to three iterations.

47 Computational expense increases as n2 with the number of Lagrangian markers

(n). To limit computational expense, the tip vortex is truncated after a speciﬁed

number of rotor diameters (D) downstream of the turbine, which results in N wake

revolutions. Because induced velocity information is a necessary wake boundary

condition, an additional rotation (N + 1) is deﬁned at the end of the wake and

set to be identical to the previous N revolution. In this way, truncation error is minimized. [101] An analysis of how the wake cutoff length criteria was selected is detailed in Section 2.2.1.

To accommodate the two-steps backwards method in azimuthal angle, a helical initial wake is speciﬁed. In past versions of the code, it was necessary for the user to manually provide these initial wake conditions via many individual text ﬁles.

To improve ease of use, the code has since been modiﬁed to automatically generate the initial wake conditions.

2.1.2.3 Induced Velocity

The velocity term on the right-hand side of Equation 2.2 is a nonlinear function of the vortex position, representing a combination of the freestream and induced velocities. [115] The induced velocities at point k, caused by each straight-line ﬁla-

ment, are computed using the Biot-Savart law, which considers the locations of the

Lagrangian markers and the intensity of the vortex elements: [101]

~ ~ Γv dl × ~r dVk = Fc (2.8) 4π |~r|3

48 ~ Here, Γv is the circulation strength of the ﬁlament, l is the vector connecting the

ﬁlament endpoint locations, and ~r is the perpendicular distance between ~l and

point k. This can be rewritten as a function of the distances, r1 and r2, between a

point of interest and the vortex ﬁlament endpoints (e.g., point k and segment AB,

respectively, in Figure 2.2):

~ Γv 1 1 1 dV = Fc (r1 × r2) + (2.9) 4π |r1| |r2| |r1||r2| + r1 · r2

This computation is performed for each Lagrangian marker relative to every vortex segment, making it an expensive component of the solution. The factor Fc, deﬁned in Equation 2.10, is introduced because of the singularity that occurs in

Equation 2.8 at the ﬁlament location. [103]

|~r|2 Fc = (2.10) p 4 4 |~r| + rc

Viscous effects prevent this singularity from occurring and diffuse the vortex strength

with time. The circular zone where the velocity drops to zero around the vortex is

referred to as the vortex core. An increase of length of the vortex segment will re-

sult in a decrease of the vortex core radius, and conversely for a decrease of length.

Diffusion, on the other hand, continually spreads the vortex radially. To account

for these effects, the correction Fc from the Vatistas model [116] is used. rc is the

49 viscous core radius of a vortex ﬁlament, modeled by Equation 2.11.

s Z ζ 2 4αδνζ −1 rc(ζ, ) = rc0 + (1 + ) dζ (2.11) Ω 0

4αδνζ0 r2 = (2.12) c0 Ω

α1Γν δ = 1 + (2.13) ν

∆l = (2.14) l

Here, α = 1.25643, δ is viscous diffusion, ν is kinematic viscosity, is the vortex

−4 ﬁlament strain, a1 = 2 × 10 , l is the ﬁlament length, and ζ0 is the vortex wake age

offset. The integral in Equation 2.11 represents strain effects, while the remainder

of the equation accounts for viscous effects. A value of a1 is chosen based on empir-

◦ ical results for a rotating blade, [103] and ζ0 is nominally set to ζ0 = 30 . [108, 117]

2.1.2.4 Circulation

The blade circulation, computed using a Weissinger-L [110] based representation of the lifting surface, [103, 112, 118] is coupled to the wake through dependence on the induced velocities. [104] In lifting line theory, bound circulation is placed on the blade at 1/4-chord and collocation points at 3/4-chord, at which a ﬂow tangency

condition is enforced, as depicted in Figure 2.4. The blade is discretized into a ﬁnite

number of segments and a horseshoe vortex is used at each segment to compute

the bound and trailed circulation. The trailed vortex circulation is computed using

50 Bound circulation Γ'|$ Γ Γ'|$%& '|$(& z distribution Ω y

Bound vortex segement at ¼ chord Γ"|$%& Γ"|$ Control point at 3/4 chord Trailed circulation in near wake

Figure 2.4: Schematics of Weissinger-L lifting line method. Adapted from Gupta. [103]

Equation 2.15.

Γt|i = Γb|i − Γb|i+1 (2.15)

The ﬂow tangency constraint on the blade [112] is enforced using Equation 2.16.

(Vbi + VNWi + VFWi + V∞i ) · nˆ = 0 (2.16)

The bound circulation is then found by solving a linear system of equations, Equa- tion 2.17, formed by combining Equations 2.8 and 2.15 along with the ﬂow tan-

51 gency condition. [112]

! Γb|i d~l × ~r Γb|i − Γb|i+1 d~l × ~r + · nˆ = −(VFW + V∞ ) · nˆ (2.17) 4π |~r|3 4π |~r|3 i i

After computing the bound circulation, the circulation of the tip vortices is set to the maximum bound circulation. This process is computed at each time step. Note that the Weissinger-L approach is valid for near-wake calculations. As previously mentioned, the near wake is assumed to extend 30◦ behind the blade, at which point it instantaneously rolls up into a tip vortex, representing the far wake.

2.1.3 FAST-Free Vortex Wake Model Formulations

FAST is an open-source code maintained by NREL. [119] It is a multi-physics, multi-ﬁdelity tool designed to simulate the aero-hydro-servo-elastic response of individual horizontal-axis wind turbines, as depicted in Figure 2.5. For this research, the aerodynamics, structural, and turbine controller modules are used. The

FAST-FVW model was originally integrated with FASTv7 to allow for aeroelastic response predictions and turbine operation analysis [97] and has since been integrated with FASTv8. The integration of the wake model with FAST is shown schematically in Figure 2.6. Note that the FVW model is primarily integrated into

AeroDyn, which is the aerodynamic component of the FAST code. Induced velocities are computed in the FVW portion of the code and then input into AeroDyn to compute the forces, moments, and effective angle-of-attack on the blades.

52 External Applied Wind Turbine Conditions Loads Control System & Actuators

InflowWind AeroDyn Aero- Rotor Drivetrain Power Wind-Inflow dynamics Dynamics Dynamics Generation

Nacelle Dynamics

ServoDyn

Tower Dynamics

HydroDyn Waves & Hydro- Platform Dynamics Currents dynamics ElastoDyn

Mooring Dynamics MAP++, MoorDyn, or FEAMooring

Figure 2.5: Flowchart of FASTv8/OpenFAST modularization. [120]

AeroDyn ElastoDyn Blade positions BEM and velocities Blade forces, Induced blade moments, velocities Wind inflow and αeff velocities Free Vortex Code InflowWind

Figure 2.6: Flowchart of the solution process for the integrated FAST-FVW model.

53 2.1.4 Multi-Turbine Free Vortex Wake Interaction

Extension of the above FAST-FVW model to incorporate interactions with other turbines and wakes is theoretically a straightforward process of applying the Biot-

Savart law (Equation 2.9) to the near wakes and tip vortices of every turbine. How- ever, given the associated computational costs, this is best carried out by parallelizing the analysis across multiple compute nodes. The developed parallelization structure is shown in Figure 2.7. The model is set up such that any number of turbines in any conﬁguration can be considered. After the turbine layout is set, a global domain is established and FAST is initiated for each turbine. Note that the domain of the upstream turbine is extended to the end of the downstream turbine domain. This increases the number of Lagrangian markers in the upstream turbine, which does not change the turbine power for single-turbine simulations.

Each instance of FAST is assigned to a separate processor, which allows each turbine to progress in time simultaneously. The beneﬁt of this model is that the wake position and structural response for each turbine are computed simultaneously and relative to each other. This allows for a complete interaction between the upstream and downstream turbines. One component of parallelization is the merging of the branches at the predictor-corrector step, which is also where the bulk of the Biot-Savart calculations take place. Next, the individual turbines proceed in a parallelized manner and the process iterates with time.

An additional complication for multi-turbine simulations is determining proper

54 Figure 2.7: Flowchart of the parallelization process for a multi-turbine wake interaction model.

55 Non-waked turbine

Waked turbine

� = 20°

Figure 2.8: Schematic of process used to determine if a downstream turbine could potentially be waked by an upstream turbine. wake distance. For computational cost considerations, it is impractical to specify all turbines wakes to be the same wake length and therefore the same number of

Lagrangian markers. Instead, a method was introduced to determine when the wake of a turbine will impact a downstream turbine. This method consists of or- dering the turbines by streamwise location. If a turbine is downstream of another turbine, it is considered to be at chance of being waked if it lies within a ±20◦ sector emanating from the upstream turbine blade tips and aligned with the streamwise direction, as shown in Figure 2.8. This range was chosen to accommodate a practical range of inﬂow yaw angles as well as wake meandering. When it is determined that a downstream turbine could be waked by a given turbine, the wake length of the upstream turbine speciﬁed to be the streamwise distance between the turbines

56 plus the speciﬁed wake cutoff condition (e.g. 4D). If a given turbine will not wake a downstream turbine, or if there are no downstream turbines, the cutoff condition of that turbine is simply the speciﬁed cutoff condition. While further improvements could be made to this method to potentially further reduce the number of

Lagrangian markers contained in this system, this method is effective for the purposes of this work and allow for easy wind farm simulations with minimal user input.

2.2 Convergence Studies

In this section, wake cutoff length and discretization convergence studies are performed to determine appropriate parameters that balance model accuracy and cost. Wake cutoff length studies are performed for constant and variable rotor speed two-turbine simulations. As the goal of this research is to study FAST-

FVW simulations of wind farms, three-turbine wake cutoff length convergence was studied as well. A discretization convergence study is performed for variable- speed two-turbine simulations. A representative Vestas V90 turbine is used for the variable rotor speed turbine simulations, as characterized by the parameters listed in Table 2.1. Further details for this turbine are given in Appendix B. Simulations are performed for uniform inﬂow conditions and turbine separation distance of

7D. Freestream velocities for two-turbine simulations range from 6 to 20 m/s and

8 m/s for three-turbine simulations.

57 Table 2.1: Nominal wind turbine parameters for the representative Vestas V90 turbine. Parameter Value Rotor Blade Airfoil FFAW3 and NACA63 Rotor Radius (R) 45 m Hub Height 70 m Number of Blades 3 Rotor Blade Pitch (θ) Variable Rotor Speed (rpm) Variable

2.2.1 Wake Cutoff Length

A cutoff condition is included in the FAST-FVW method to limit the number of

Lagrangian markers used in a simulation. This condition is typically speciﬁed in

terms of the number of downstream rotor revolutions to be included in the simula-

tion. [104, 114] Each time a turbine completes one rotor revolution it releases a set

number of Lagrangian markers into the ﬂow ﬁeld. For example, a cutoff condition

set at N = 6 would track six rotor revolutions worth of Lagrangian markers shed from the turbine. However, the number of markers shed per revolution varies with rotor speed. This presents a challenge when simulating a variable-speed turbine, because final rotor speed is not known a priori, especially in multi-turbine simulations. Therefore, it is preferable to use a cutoff condition that does not vary with rotor speed when using variable-speed turbines. For this, a cutoff condition is instead specified in terms of wake distance downstream, which does not depend on rotor speed. Additionally, a distance-based cutoff condition can limit the number of trivial markers tracked in the simulation for high-velocity inflow conditions.

58 a) b) 06ms 00 30 07ms 15 08ms 00 09ms 25 10ms 11ms 00 10 20 12ms 13ms 00 15 14ms 15ms 5 20ms 1000

Avg. Rotor Power (kW) Power Avg. Rotor 5 0 0 1 5 9 13 1 5 9 13 Cutoff Distance (D) (%) Difference Percent Pwr. Rot. Cutoff Distance (D)

Figure 2.9: (a) Average rotor power and (b) percent difference for single-turbine wake convergence study.

In this study, this method of deﬁning wake cutoff length is investigated to identify

the adequate cutoff length to capture wake interaction for the representative Vestas

V90 turbine.

As shown in Figure 2.9, single-turbine cutoff results reach an average percent

difference of 0.25% at 4D. However, two-turbine results do not clearly converge, as

shown in Figure 2.10. Consistent relative rotor power values, P T2/T1 = P T2/P T1, for

varying cutoff distances are shown in Figure 2.10(a). Variable percent difference

values are demonstrated in Figure 2.10(b). Results largely converge around 4D with an average percent difference of 2.32% for freestream velocities above rated power. While there is not a clear convergence of P T2/T1 for freestream velocities

below rated power, the percent difference remains bounded.

Shown in Figure 2.11 are normalized wake velocity proﬁles at varying distances

59 Figure 2.10: (a) Relative rotor power and (b) percent difference for two-turbine wake convergence study.

Table 2.2: Computational time for representative Vestas V90 two-turbine simulations with various wake cutoff lengths with 6 m/s inﬂow. Computed using Intel Xeon E5-2680 v4 CPUs with a single processor. Wake Cutoff Length Computational Time 1D 68.36 hours 4D 194.19 hours 9D 319.49 hours

aft of the upstream turbine at a freestream velocity of 6 m/s.

Computational times for two-turbine simulations with various wake cutoff lengths at 6 m/s inﬂow are given in Table 2.2. These computational times equate to a nearly

65% increase in computational time when cutoff length increases from 4D to 9D.

Thus, a cutoff length of 4D was selected because of bounded two-turbine results and tractable computational cost. This cutoff length is used for the remainder of the variable-speed studies presented in this research.

60 z = 3D4D z = 5D

0.8 0.8

Normalized Wake Velocity (•) Wake Velocity Normalized 0 0.5 1 1.5 2 (•) Wake Velocity Normalized 0 0.5 1 1.5 2 y/R (•) y/R (•) z = 6D z = 7D

1D 2D 3D 4D 5D 0.8 0.8 6D 7D 8D 9D 10D 12D

Normalized Wake Velocity (•) Wake Velocity Normalized 0 0.5 1 1.5 2 (•) Wake Velocity Normalized 0 0.5 1 1.5 2 y/R (•) y/R (•)

Figure 2.11: Wake cutoff length convergence study for NREL UAE Phase VI turbine with uniform inﬂow.

61 Table 2.3: Wake cutoff lengths for three-turbine wake cutoff length study. Turbine Number Wake Cutoff Length T1 14.2D±{4, 3, 2, 1, 0}D T2 7.1D+4D T3 4D

As the goal of this research is to study FAST-FVW simulations of wind farms,

a study was performed to determine wake cutoff length convergence for three tur-

bines. This study was performed to determine the impact of the wake from the

ﬁrst turbine (T1) on more downstream turbines (T3). Thus, wake cutoff lengths for

T2 and T3 remained constant at 4D while the wake cutoff length of T1 was varied.

This cutoff length was speciﬁed to vary from −1D to 4D, relative to the position

of T3, i.e., a cutoff length of −1D ends the wake approximately 1D upstream of

T3. Turbine cutoff lengths are detailed in Table 2.3. Results for each turbine with

varying T1 cutoff distance (l) are depicted in Figures 2.12(a) and (b) for average

rotor power, P l, and standard deviation, σ(Pl). Here, σ(Pl) is computed following

Equation 2.18. There is minimal change in P T1 and σ(PT1) with varying T1 cutoff

distance. However, there is signiﬁcant variation of these quantities for T2 and T3

with respect to changing T1 cutoff distance. When the cutoff distance of T1 is less

than 0D, the wake of the upstream turbine does not impact the most downstream

turbine. This results in the most downstream turbine producing more power than

T2, with comparable σ(PT urb) values. When the cutoff distance of T1 is greater than

0D, the wake of the upstream turbine impacts the most downstream. In this case,

P T3 drops below P T2 while σ(PT3) increase above σ(PT2). Thus, when the wake of

62 d)c)a) •4DT1P2/P1 b) 10001.4 •3DT2P3/P1 20 1.2 •2DT3P3/P2 1.2 •1D 0D 15 800201 CX 1D 1 2D X 3D 0.8 C 4D 10 6000.8 10 0.6 C 0.6 X 5 C Rotor Power (kW) Power Rotor 400

Percent Difference (%) Difference Percent 0.4 X Relative Rotor Power (•) Power Rotor Relative

Rotor Power Std. Power Dev. (%) Rotor 0 1 •4 •2 02 2 4 3 •4 •2 0 2 4 T1 TurbineCutoff Distance Number (D) T1 Cutoff Distance (D)

Figure 2.12: Rotor power results for wake cutoff length study involving three aligned turbines. Results include (a) rotor power for each turbine and (b) standard deviation of rotor power for each turbine.

T1 is interacting with T3, it has a signiﬁcant impact on reducing rotor power and increasing unsteadiness.

Shown in Figure 2.13 is the average rotor power of each turbine relative to T1.

There is some variation in the relative rotor power of T2, with relative rotor power increasing with increasing T1 cutoff length. Alternatively, relative rotor power for

T3 reduces with increasing T1 cutoff length. Again, a clear reduction in P T3 is seen when the wake from T1 is long enough to interact with T3. As shown by experimental measurements, turbines beyond the second row are less affected by wake interactions that those in the second row, with P T3+/P T1 typically higher than

P T2/P T1. [11] This reduced impact is because of the diffusion of turbine wakes, resulting in a reduced effect on interior wind farm turbines. [11] This relationship

63 •4D •3D 1.2 •2D •1D 0D 1 CX 1D 2D X 3D 0.8 C 4D

0.6 CX

0.4 X Relative Rotor Power (•) Power Rotor Relative 1 2 3 Turbine Number

Figure 2.13: Relative rotor power results for wake cutoff length study involving three aligned turbines.

64 is produced by the FAST-FVW simulations when the cutoff length of T1 does not

reach T3. It is likely that in a wind farm setting the wake from T1 diffuses to

a non-impactful level before reaching T3. When the cutoff length is speciﬁed as

being greater than 0D, it is artiﬁcially forcing the wake to remain intact for longer

than it would in a wind farm setting, thereby increasing the impact on the most

downstream turbine and reducing P T3/P T1 below expected levels.

2.2.2 Wake Discretization

Resolution in wake discretization is a key parameter for model accuracy. In past studies, 10◦ was determined to be sufﬁciently accurate for FAST-FVW methods. [101,

103] However, as these studies were for constant rotor speed single-turbine simulations, it is necessary to re-evaluate discretization of variable-speed multi-turbine simulations. Wake discretization (dψ) is deﬁned by the computational model in terms of time step (dt) and rotor speed (Ω), such that dψ = dt × Ω. Because rotor

speed is a turbine characteristic, dψ is deﬁned using dt, which remains constant

throughout the simulation. For a turbine with constant rotor speed, it is therefore

straightforward to exactly specify dψ. However, when a variable-speed turbine

is used, as in this work, this is more difﬁcult because rotor speed can vary with

freestream velocity as well as throughout the simulation. Additionally, average

rotor speed can be signiﬁcantly reduced in wake-impacted turbines while the un-

steadiness in rotor speed can be signiﬁcantly increased, as shown in Figure 2.14.

65 6 m/s_T1 20 6 m/s_T2 8 m/s_T1 8 m/s_T2 10 m/s_T1 10 m/s_T2 15 14 m/s_T1 14 m/s_T2

10 Rotor Speed (RPM) Rotor

5 300 400 500 600 700 800 Time (seconds)

Figure 2.14: Sample time-series of rotor speed for upstream and downstream turbines at various uniform freestream velocities.

This causes dψ to vary between and throughout each simulation. Therefore, it is

important to characterize the sensitivity of the solution to variable dψ. Speciﬁed wake discretization is varied from 5◦ to 15◦.

Upstream turbine results indicate negligible sensitivity to wake discretization level. This is demonstrated by the low percent error between theoretical and actual discretization levels in Figure 2.15b. Greater sensitivity was seen in downstream turbine results. Discrepancies between actual and speciﬁed wake discretization levels for a range of freestream velocities are depicted in Figure 2.16 in terms of both actual discretization level and percent difference from the speciﬁed value.

Large percent differences between actual and speciﬁed discretization levels are

66 a) a) 15 00 30

00 25 10 00 20

00 5 1 5

1000 Rotor Power (kW) Power Rotor 00 5 0 0

6 8 10 12 14 (%) Error Percent Power Rotor 6 8 10 12 14 Wake Discretization (degrees) Wake Discretization (degrees) a)b) b) 6 m/s 15 10 m/s ) 15 12 m/s

deg 15 m/s

10 10

5 5

Actual Discretization Discretization Actual( 0 0 6 8 10 12 14 6 8 10 12 14 Discretization Percent Error (%) Error Percent Discretization (%) Error Percent Discretization TheoreticalWake Discretization Discretization (degrees) (deg) TheoreticalWake Discretization Discretization (degrees) (deg)

Figure 2.15: Discrepancies between speciﬁed and realized discretization levels for T1 in single-turbine simulations.

a) b) 15 0

•10 10

•20

5 06 m/s 10 m/s •30 12 m/s 15 m/s (%) Difference Percent Actual Discretization (deg) Actual Discretization 0 •40 5 10 15 5 10 15 Specified Discretization (deg) Specified Discretization (deg)

Figure 2.16: Discrepancies between speciﬁed and realized discretization levels for T2 in two-turbine simulations.

67 present for lower freestream velocities. These discrepancies result from the wake

deﬁcit experienced by T2, which leads to a reduced rotor speed. This results in

ﬁner wake discretization of affected downstream turbine wakes because of the

proportional relationship between wake discretization and rotor speed. Since the

wake is reﬁned, this does not degrade the solution quality but does lead to in-

creased computational cost because of the increase in number of shed Lagrangian

markers. When the wake coarseness becomes too high, stability issues are en-

countered. This provides an upper limit to wake discretization. For this turbine,

solution instability was encountered when the wake resolution reached 12◦. Note

that this behavior was not encountered for single turbine simulations, stressing

the importance of performing a multi-turbine wake discretization study. Changes

in relative rotor power are investigated to determine the appropriate discretiza-

tion level. As shown in Figure 2.17, there is minor variation in P T2/T1 for varying

discretization levels.

When comparing to the 5◦ discretization level, average percent difference values do not exceed 12%. Due to variable rotor speeds, there can be discrepancies in actual versus specified resolution. However, the actual discretization will not exceed the specified discretization. Since there is not a clear convergence that would point to an ideal wake discretization, the decision must be made based on bal- ancing accuracy and computational cost. Specified wake discretization of 10◦ was determined to meet this balance for the purposes of this research.

68 06ms 15 10ms 12ms 1 15ms

10 0.8

0.6 5

0.4 Percent Difference (%) Difference Percent Relative Rotor Power (•) Power Rotor Relative 0 5 10 15 5 10 15 Wake Discretization (degrees) Wake Discretization (degrees)

Figure 2.17: Comparison of relative rotor power for varying discretization levels.

2.3 Multi-Turbine Free Vortex Wake Model Validation

2.3.1 Wind Farm Experiment and Simulation Cases

Wind farm measurements chosen for validation are from the Egmond aan Zee

(OWEZ) offshore wind farm, located off the coast of The Netherlands. OWEZ con-

sists of 36 generator-torque and blade-pitch controlled Vestas V90 3.0 MW wind turbines, described in Table 2.1. As depicted in Figure 2.18, they are positioned in four rows of varying sizes with a spacing of ∼ 7.1D in the streamwise direction. Rotor power measurements are from Turbines 11 and 12 where a wind direction of 0◦ corresponds to column-aligned inﬂow. [35] These turbines were selected to minimize the inﬂuence of other turbines in the wind farm to better match the two-turbine simulations performed in this work. Turbine 12 is designated as the

69 36 35 X12 29 34 X11 28 21 10 33 27 20 9 26 32 19 8 18 25 31 7 17 30 6 24 5 16 23 4 15 22 3 14 2 13 1

Figure 2.18: Egmond aan Zee windfarm layout. Adapted from Churchﬁeld. [11]

upstream turbine (T1) and Turbine 11 as the downstream turbine (T2).

Assessment of the FAST-FVW method is carried out by comparing simulations representing turbines 11 and 12, which are laterally aligned and separated by 7.1D.

The effects of wake interaction are investigated through wake structure; average normalized rotor power; and standard deviations of average rotor power and out- of-plane blade root bending moment. To further investigate the effects of inﬂow conditions on wind farm power, P T2/T1 and standard deviation of PTurb (σ(P )) are

examined relative to average TI values. Inﬂow conditions for a wind farm envi-

ronment are computed in TurbSim [121] using the IEC Kaimal turbulence spectra,

as this spectra yields the most realistic inﬂow conditions. [10] Unless otherwise

stated, all cases are carried out with an average TI of ∼ 5%.

70 For benchmarking against measurements, considered inﬂow conditions include

freestream velocities ranging from 6 m/s to 14 m/s and yaw angles ranging from

−17◦ to 18◦. Assessing the performance of the FAST-FVW method includes analyzing mean relative rotor power (P T2/T1 = P T2/P T1) and associated percent error for a range of turbulent inﬂow conditions. The method of including turbulent inﬂow conditions is described in Appendix C

In this section, the effects of two-turbine wake interaction are studied in terms of wake structure, average rotor power (P ), rotor power standard deviations (σ(P )),

and blade root bending moment standard deviations (σ(M)). These results are

compared to single-turbine simulations to study the effects of wake interaction on

upstream turbines. Next, the effects of TI level are studied in terms of P T2/T1 and

σ(P ).

Benchmarking of the FAST-FVW approach is achieved by comparing P T2/T1

for varying freestream velocity and yaw angle to experimental results. Varying

freestream velocity results are ﬁrst considered for different inﬂow conditions and

averaging techniques. Next, P T2/T1 is investigated as it relates to varying inﬂow

yaw angles. Each range of inﬂow yaw angles is considered for varying freestream

velocity, using the inﬂow averaging technique described in Section 2.3.3.

For varying freestream velocity, unaveraged results are shown along with aver-

aged results. Unaveraged results are provided for a single FAST-FVW simulation

at the speciﬁed freestream velocity and yaw inﬂow angle. In order to provide a

consistent comparison between experimental data and computational predictions,

71 rotor power is averaged over yaw inflow angles of ±2◦ (Averaging I). [11, 35] Sep- arately, results are ensemble-averaged for varying turbulence inflow files (Averag- ing II). This is done by manipulating the pseudorandom number generator used by TurbSim, which results in different realizations of turbulence between the wind inflow files. Using this method, TI can vary ∼ 1% from the specified TI level. In this way, the variability of the inflow is better captured.

2.3.2 Wake Interaction Effects

Wake structures from two-turbine simulations are shown in Figure 2.19. All simulations were performed for turbine separation distance of 7.1D with 8 m/s turbulent inﬂow wind conditions. Shown in Figure 2.19(a) is the wake structure resulting from 0◦ yaw angle. The near wake of the upstream wake is mostly helical. As the wake transitions into the far wake region, the helical structure breaks down and turbulent wake behavior occurs. This far wake region is typically reached around 3D downstream. [19] Further downstream, wake meandering occurs. Properly capturing wake meandering is an important component to accurately predict downstream turbine power and structural loads for two reasons. First, turbine power production and loads are affected by how strongly the upstream wake is interacting with the turbine. Second, wake meandering causes additional unsteadiness in power and load ﬂuctuations. When the upstream wake reaches the downstream turbine it appears to be fully turbulent. At 0◦ yaw an-

72 Figure 2.19: Wake structure for two-turbine simulations using the FAST-FVW ◦ ◦ method. Simulations use turbulent inﬂow at U∞ = 8 m/s at (a) 0 , (b) 5 , and (c) 10◦ yaw angles.

73 gle, the downstream turbine is “fully waked”, meaning that the upstream wake

contacts the entire downstream turbine. There is also signiﬁcant distortion of the

downstream wake. Shown in Figure 2.19(b) is the wake structure resulting from 5◦ yaw angle. Here, the near wake is comprised of a mostly helical wake that begins to degenerate further downstream than the unyawed wake. Again, the wake transitions to turbulence and is fully turbulent before it reaches T2. Wake meandering is no longer clearly present. At 5◦ yaw angle, the downstream turbine is “partially waked”, meaning that there are periods where only part of T2 is in contact with the upstream turbine. While there is still distortion of the downstream wake, it is less than that of the fully waked turbine. The wake structure resulting from 10◦ yaw

angle is shown in Figure 2.19(c). Here, wake transition occurs further downstream

than the lower yaw angles. Though the wake is fully turbulent before it reaches T2,

there is no visible physical contact of the upstream wake with T2. This likely re-

sults in a reduced impact on downstream turbine power and loads. Additionally,

the near wake of both turbines is longer than in lower yaw angle cases because of

the lack of wake interaction. Downstream turbine wake structure is similar to that

of T1 and is less distorted than wakes produced by wake-inﬂuenced turbines.

P and σ(P ) are investigated for single- and two-turbine simulations for a range of yaw angles. Shown in Figure 2.20(a) are normalized average rotor power results for the single, upstream, and downstream turbines. All values are normalized by the single turbine value in 0◦ yawed ﬂow. As expected, single turbine rotor power reduces as the yaw angle increases. Upstream rotor power does not signiﬁcantly

74 X Single Turbine 25 Upstream Turbine X X X X X 1 X X X X X X X X X X Downstream Turbine X X X X X X X X X X X X X X X X 0.9 X X X X 20 X

0.8 )*100 (%))*100 15 Avg 0.7 )/(P 10 0.6 Turb X X X X X X X X X X X X X X X X X X X X X X X XX X X X X X X X (P X X X X Normalized Rotor Power Rotor Normalized 0.5 5 •10 0 10 •10 0 10 Yaw Angle (degrees) Yaw Angle (degrees)

Figure 2.20: Rotor power results for single- and two-turbine simulations in turbulent 8 m/s wind inﬂow. (a) Normalized average rotor power and (b) standard deviation of average rotor power.

change over the range of yaw angles, though the curve is ﬂatter in the lower yaw

angle range than single-turbine simulations. This demonstrates the ability of a

downstream turbine to affect the performance of an upstream turbine. This in-

teraction is likely because of the blockage effect of a wind turbine, in which the

pressure drop across a turbine causes a reduction in wind speed more than 3D upstream of the turbine. [122, 123] LES simulations using EllipSys and OpenFoam have shown a similar response for two turbines operating in steady 10 m/s inflow wind. [124] Though turbine power of operational turbines without the presence of downstream turbines is unavailable, experimental results from the Lillgrund offshore wind farm show that reduced distance between two turbines reduces the mean power generated by the upstream turbine at inflow conditions comparable to those used in this work. [124] Note that wake superposition methods would not be able to capture this influence. As expected, the downstream turbine experiences

75 a signiﬁcant power deﬁcit at low yaw angles. As the yaw angle increases from 0◦ to

8◦, the rotor power increases until it is approximately the same as the upstream turbine. At this point, the upstream wake is minimally affecting the downstream turbine. As yaw angles increase further, the downstream rotor power decreases along with the upstream rotor power and there are minimal differences between turbine behavior. Rotor power standard deviations, σ(P ), are shown in Figure 2.20(b) for

varying yaw angles. Here, σ(P ) is computed following Equation 2.18.

σ(vali) σ(val) = × 100 (2.18) val

Demonstrated in this plot is the effect of turbine-wake interactions on rotor power

unsteadiness. Again, the effects on an upstream turbine because of a downstream

turbine are displayed, with comparable response seen in LES and experimental

results. [124] Wake superposition methods would not capture this deviation from

single-turbine performance. Additionally, σ(P ) of the downstream turbine reaches

up to 13% higher than the upstream turbine when turbine-wake interaction is

strong. This increased unsteadiness is because of the turbulent nature of the wake

from the upstream turbine. When this turbulent wake comes into contact with the

downstream turbine it modifies the inflow conditions and produces greater fluc-

tuations in turbine rotor power. When the ﬂow is slightly yawed, rotor power

unsteadiness further increases because of partial waking of the downstream tur-

bine.

76 X Single Turbine 25 Upstream Turbine Downstream Turbine

20 )*100 (%)

Avg 15 )/(M 10 Turb X X X X X X X X X X X X X X X X

(M X X X X X X X X X X X X X X X XX X X X 5 •10 0 10 Yaw Angle (degrees)

Figure 2.21: Standard deviation of out-of-plane blade root bending moment results for single- and two-turbine simulations in turbulent 8 m/s wind inﬂow.

Similar trends for standard deviations of out-of-plane blade root bending moments are shown in Figure 2.21. Here, σ(M) is computed in the same manner as

Equation 2.18. As with σ(P ), upstream turbine bending moment standard deviations are increased above those of a single turbine. As before, this is a result of increased turbulence in the wake of the upstream turbine, which modifies inflow conditions of the downstream turbine. As load unsteadiness is a main component of blade fatigue failure, it is necessary to accurately capture this increased unsteadiness because of turbine-wake interactions. As wake superposition models typically require smoothing of the wake velocity field and TI, [125] the turbulent nature of the inflow conditions experienced by wake-influenced turbines is not properly captured and could lead to inaccurate blade failure prediction.

77 6 m/s 0.7 8 m/s

0.6

0.5

0.4 Relative Rotor Power (•) Power Rotor Relative 0.3 0 2 4 6 8 10 Turbulence Intensity (%)

Figure 2.22: Relative rotor power for computational results versus varying freestream turbulence intensities.

The effects of varying TI on P T2/T1 and σ(P ) are investigated for U∞ = 6 and

8 m/s. Relative rotor power results are shown in Figure 2.22 for varying freestream turbulence intensities. As expected, P T2/T1 increases with increasing TI. This is because of the increased wake mixing which facilitates wake velocity recovery and thus an increased P T2. P T2/T1 values range 0.095 and 0.16 units for U∞ = 6 and

8 m/s, respectively. This indicates signiﬁcant dependence on TI. Given the uncertainty in wind farm measurements, this sensitivity should be considered when determining the validity of computational wake models.

Rotor power standard deviations, σ(P ), and percent difference of σ(P ) values are shown in Figure 2.23 for varying turbulence intensities. Here, σ(P ) is computed following Equation 2.18 and the percent difference is relative to the TI= 1%

78 6 m/s, T 200 up 6 m/s, Tdown 8 m/s, T 30 up 8 m/s, Tdown 150

)*100 (%) 20 100 Avg )/(P 10 50 Turb (P

Percent Difference (%) Difference Percent 0 0 0 2 4 6 8 10 0 2 4 6 8 10 Turbulence Intensity (%) Turbulence Intensity (%)

Figure 2.23: (a) Rotor power standard deviations of average rotor power for computational results. (b) Percent change of computational results relative to computational 1% TI results. Cases include varying freestream turbulence intensities.

case. As expected, σ(P ) increases with increasing TI. Additionally, σ(PT2) is greater than σ(PT1). This is captured without any additional models or empirical tuning.

The percent change depicted in Figure 2.23(b) demonstrates that the upstream turbine is more sensitive to changing TI than the downstream turbine. This is likely because of the already turbulent inﬂow experienced by T2.

2.3.3 Varying Freestream Velocity

For varying freestream velocity, unaveraged results are shown along with averaged results. Unaveraged results are provided for a single FAST-FVW simulation at the specified freestream velocity and yaw inflow angle. In order to provide a consistent comparison between experimental data and computational predictions, rotor power is averaged over yaw inflow angles of ±1◦ and freestream velocities of ±1 m/s (Averaging I). [11, 35] Separately, results are ensemble averaged for

79 a) b) Averaging I 1.1 40 No Averaging Experimental 1 Averaging II 0.9 30 0.8 20 0.7

0.6 10

0.5 (%) Error Percent Relative Rotor Power (•) Power Rotor Relative 0.4 0 6 8 10 12 14 6 8 10 12 14 Freestream Velocity (m/s) Freestream Velocity (m/s)

Figure 2.24: (a) Relative rotor power and (b) percent error for FAST-FVW and experimental [35] results. Cases include varying turbulent freestream velocities.

varying turbulence inﬂow ﬁles (Averaging II). This is done by manipulating the

pseudorandom number generator used by TurbSim, which results in different re-

alizations of turbulence between the wind inﬂow ﬁles. Using this method, TI can

vary ∼ 1% from the speciﬁed TI level. In this way, the variability of the inﬂow condition is better captured.

2.3.3.1 Rotor Power

It is commonly observed that wake interaction reduces the power in wake-inﬂuenced turbines, with lessening effects as freestream velocity increases. Both of these trends are captured by the FAST-FVW method, as shown in Figure 2.24. Changes in relative rotor power with varying freestream velocity are shown in Figure 2.24(a).

Here, signiﬁcant reduction in P T2/T1 is seen for low freestream velocities. FAST-

FVW results for P T2/T1 are under-predicted for U∞ < 10 m/s, with close agreement

80 Table 2.4: Average percent error of rotor power for different inﬂow averaging techniques. Average Percent Error No Averaging 11.87% Averaging I 12.64% Averaging II 14.84%

at U∞ = 10 m/s. Both the experimental measurements and FAST-FVW predictions exhibit increasing P T2/T1 with increasing freestream velocity, albeit at different rates. The lessened effect of wake interaction with increased freestream velocity is partly due to reaching rated rotor power, which reduces the variance in turbine power. It is likely that the FAST-FVW simulation results approach P T2/T1 = 1 at a lower freestream velocity than experimental measurements because the simulated Vestas turbine reaches rated power at a lower freestream velocity than the experimental turbine. Percent errors relative to measurements are shown in Fig- ure 2.24(b), with average percent errors summarized in Table 2.4. For all turbulent inﬂow cases, the average percent error is less than 15%.

One possible source of discrepancy between FAST-FVW predictions and experimental measurements is the limited number of turbines included in the simulations. As has been shown, downstream turbines can affect rotor power of an upstream turbine. It is possible that if more turbines were included in the simulations, better agreement would be reached. Another potential source of discrepancy is the extent of wake meandering. If wake meandering is not properly captured, the amount of time a downstream turbine is impacted by the wake of an upstream

81 turbine could be overpredicted, thus resulting in underpredicting downstream tur-

bine rotor power. [11] Wake meandering is brieﬂy discussed in Section 3.3.4 as it

relates to fatigue loads, but further research is needed to quantify its importance

in more detail. Despite possible sources of error, these results indicate acceptable

agreement of relative rotor power when turbines are aligned.

There are minor differences between non-averaged and averaged results for the

full range of freestream velocities. However, there is an appreciable range in results

between the varying TI inﬂow conditions used for Averaging II, as shown by in-

dividual circles in Figure 2.24. Each circle represents a different wind inﬂow ﬁle

and each diamond represents the average of all points at that freestream velocity.

For these results, P T2/T1 and percent error vary as much as 0.2 and 13.1%, respectively. Recall that TI varies by < 1%. While the average value differs only slightly from the other results, this shows a clear dependence on the wind input ﬁle used in the simulation. Comparison to the steady inﬂow simulations strengthens this, with P T2/T1 reaching as low as 0.37 and linearly increasing with freestream velocity.

Turbulent inﬂow leads to a marked reduction in downstream turbine rotor power

deﬁcit, as indicated by differences between steady and turbulent inﬂows. This is

expected, since turbulent inﬂow increases mixing between the turbine wake and

surrounding freestream conditions, leading to faster recovery of the wake velocity

deﬁcit. [29] This trend is shown through the comparison in Figure 2.24, as well as

the ability of the code to capture strong power deﬁcits and thus wake interaction.

Unless stated otherwise, averaging over inﬂow conditions (Averaging I) is used

82 ] s / 400 m

FVW

0 1.5

1 Exp 300

@ min

a 1.0 mean m 200 max M / 0.5 range M [

t 100 n Percent Error [%] e 0.0 m

o 0 Relative Flapwise Bending

M 6 8 10 12 14 16 18 20 6 8 10 12 14 16 18 20 Uinf [m/s] Uinf [m/s]

(a) Upstream Turbine Relative Flapwise Bend- (b) Percent Error of Upstream Turbine Relative ing Moment Flapwise Bending Moment ] s / 400 m

FVW

0 1.5

1 Exp 300

@ min

a 1.0 mean m 200 max M / 0.5 range M [

t 100 n Percent Error [%] e 0.0 m

o 0 Relative Flapwise Bending

M 6 8 10 12 14 16 18 20 6 8 10 12 14 16 18 20 Uinf [m/s] Uinf [m/s]

(c) Downstream Turbine Relative Flapwise (d) Percent Error of Downstream Turbine Rela- Bending Moment tive Flapwise Bending Moment

Figure 2.25: Relative ﬂapwise bending moment comparisons between computational and experimental results [35] for (a) upstream and (c) downstream turbines at varying freestream velocities. Percent error between computational and experimental results are shown for the (b) upstream and (d) downstream turbines. for all remaining results.

2.3.3.2 Structural Response

Minimum, mean, and maximum ﬂapwise bending moments, relative to the mean at 10 m/s of the non-waked turbine, are shown in Figure 2.25(a) and (c) for experimental results as well as FAST-FVW simulations. Here, the mean value is denoted by the symbol while the minimum and maximum values are denoted by

83 Table 2.5: Average percent error of minimum, mean, maximum, and range of relative ﬂapwise bending moment between computational results and experimental measurements for non-waked (T1) and waked (T2) turbines across all freestream velocities . Min. Mean Max. Range MFlap,T1 22.10 18.44 18.97 26.68 MFlap,T2 46.81 24.13 24.79 16.82

the limits of the vertical bars. The percent error of the minimum, mean, max-

imum, and values for the simulation results relative to experimental results are

shown in Figures 2.25(b) and (d). Non-waked upstream turbine results are shown

in Figures 2.25(a) and (b), while waked downstream turbine results are shown

in Figures 2.25(c) and (d). Cumulative percent error of minimum, maximum,

mean, and range values between FAST-FVW and experimental results are shown

in Figures 2.25(b) and (d). Average percent error results are summarized in Ta-

ble 2.5. Minimal differences are seen for upstream turbine response with rated

wind speeds. With the exception of U∞ = 12 m/s, FAST-FVW is shown to under- predict ﬂapwise bending moment loads at below-rated wind speeds. However, the load range is overpredicted for U∞ ≥ 8 m/s. Overall, percent error between

FAST-FVW and experimental load results decreased with increased freestream velocity. With an average percent error of mean load of 18.4%, it can be determined that FAST-FVW results compare reasonable well to experimental results for upstream turbine results. Larger discrepancies are seen for downstream turbine results. Most notably, the load range is overpredicted for all inﬂow conditions and percent error does not decrease until U∞ = 14 m/s, above which experimental

84 results are not available. This range over-prediction is mostly due to the under-

estimate of minimum ﬂapwise bending moment, with an average percent error

of 46.8%. There are a number of sources of error from which these discrepancies

could arise, all of which were discussed in the previous section. Though there

are large discrepancies is actual values, overall load trends compare well for both

upstream and downstream turbines. Speciﬁcally, for both FAST-FVW and experi-

mental results, relative ﬂapwise loads increase until 12−14 m/s, at which point the relative load decreases. Additionally, for upstream turbine results, the load range increases with increasing freestream velocity. A similar trend is seen for downstream turbine results, as well as an increased range compared to upstream turbine results. Given the assumptions and limitations required to make this comparison, an overall agreement of trends is considered promising and most important.

2.3.4 Varying Inﬂow Yaw Angle

To further demonstrate how computational results compare to experimental measurements, P T2/T1 is compared for a range of inﬂow yaw angles for each freestream velocity considered in Section 2.3.3. Average percent errors for each freestream velocity are summarized in Table 2.6. These results are below 15% and indicate the

ability of the FAST-FVW method to predict relative turbine rotor power values

and trends to within acceptable agreement with experimental data. Relative ro-

tor power results for varying yaw angle at U∞ = 6 m/s are shown in Figure 2.26.

85 Table 2.6: Average percent error of relative rotor power between computational results and experimental measurements for varying freestream velocities and yaw angles. −17◦ to 18◦ (%) −10◦ to 10◦ (%) 6 m/s 13.57 18.54 8 m/s 9.75 13.42 10 m/s 6.85 9.62 12 m/s 7.96 12.20

a) Experimental b) % Error_Avging 1.1 FVW 40 % Error_NoAvging FVW_No Averaging 1 0.9 30 0.8 20 0.7

0.6 10

0.5 (%) Error Percent Relative Rotor Power (•) Power Rotor Relative 0.4 0 •20 •10 0 10 20 •20 •10 0 10 20 Yaw Angle (degrees) Yaw Angle (degrees)

Figure 2.26: (a) Relative rotor power for computational results and experimental measurements; [35] and (b) percent error for computational results relative to experimental measurements. Cases include varying yaw angle with U∞ = 6 m/s.

FAST-FVW turbulent inﬂow results compare reasonably well with experimental measurements, with an average percent error of 13.57%. As yaw angle increases, turbine-wake interaction reduces and P T2/T1 approaches 1. While the rate of recovery is increased with the FAST-FVW model, this trend is captured by the FAST-

FVW method.

As previously discussed, inﬂow averaging is employed when post-processing the computational results. While this averaging did not yield signiﬁcant improvement for 0◦ yaw angle results, unaveraged computational results are shown in

86 a) Experimental b) Percent Error 1.1 FVW 40 1 30 0.9

0.8 20 0.7

0.6 10 Percent Error (%) Error Percent

Relative Rotor Power (•) Power Rotor Relative 0.5 0 •20 •10 0 10 20 •20 •10 0 10 20 Yaw Angle (degrees) Yaw Angle (degrees)

Figure 2.27: (a) Relative rotor power for computational results and experimental measurements; [35] and (b) percent error for computational results relative to experimental measurements. Cases include varying yaw angle with U∞ = 8 m/s.

Figure 2.26 to demonstrate the full impact of this averaging. There is increased

rotor power variance in the unaveraged results, while the averaged results yield

a smoother curve that closer matches measured data. As experimental results use

similar ﬁltering techniques, [11, 35] it is expected that such averaging yields a bet-

ter comparison. [37]

Similar trends are evident for varying yaw angles at U∞ = 8 m/s, depicted in

Figure 2.27, where the average percent error is 9.75%. As with U∞ = 6 m/s, P T2/T1

approaches 1 as yaw angle reaches ±10◦. Computational results exhibit a steady

increase in P T2/T1 before reaching unity. This trend is comparable to experimental trends. For U∞ = 6 and 8 m/s, computational P T2/T1 results are underpredicted for

◦ ◦ yaw angles between −3 and 2 . For higher yaw angles, P T2/T1 values are higher

in computational results. At these freestream velocities, P T2/T1 is predicted an av-

erage of 0.08 units higher between −10◦ and 10◦ yaw. Comparable trends are seen

87 a) Experimental b) Percent Error 1.1 FVW 40 1 30 0.9

0.8 20 0.7

0.6 10 Percent Error (%) Error Percent

Relative Rotor Power (•) Power Rotor Relative 0.5 0 •20 •10 0 10 20 •20 •10 0 10 20 Yaw Angle (degrees) Yaw Angle (degrees)

Figure 2.28: (a) Relative rotor power for computational results and experimental measurements; [35] and (b) percent error for computational results relative to experimental measurements. Cases include varying yaw angle with U∞ = 10 m/s.

for U∞ = 10 m/s and 12 m/s (see Figures 2.28 and ??, respectively). For these

higher freestream velocities, P T2/T1 values are always higher than experimental results, though on average yield lower percent errors. At these freestream veloci-

◦ ◦ ties, P T2/T1 is predicted an average of 0.09 units higher between −10 and 10 yaw.

While not shown, U∞ = 14 m/s yields P T2/T1 ∼ 1.0 for all yaw angles.

There are a few possible reasons why the FAST-FVW simulations reach P T2/T1 =

1 at a faster rate than experimental data. In addition to the sources of discrepancies

already introduced in this chapter, a major source of uncertainty is the necessary

use of representative V90 turbine parameters. It is possible that if real controller

parameters are attained, computational results would more closely match experi-

mental measurements. Despite the noted discrepancies, these results indicate that

a FVW model is a promising approach for capturing relative turbine rotor power

values and trends when wake interactions are present in wind farms.

88 a) Experimental b) Percent Error 1.1 FVW 40 1 30 0.9

0.8 20 0.7

0.6 10 Percent Error (%) Error Percent

Relative Rotor Power (•) Power Rotor Relative 0.5 0 •20 •10 0 10 20 •20 •10 0 10 20 Yaw Angle (degrees) Yaw Angle (degrees)

Figure 2.29: (a) Relative rotor power for computational results and experimental measurements; [35] and (b) percent error for computational results relative to experimental measurements. Cases include varying yaw angle with U∞ = 12 m/s.

89 Chapter 3

Characterization of Wake Interaction Effects

Using FAST-FVW and FAST.Farm

To highlight the differences and similarities of a FVW method versus a DWM-type method for multi-turbine simulations, a comparison is made to the FAST.Farm model, described in Section 1.4.2.2. For both models, a study is performed to characterize the effects of turbine-wake interactions at various operating conditions and turbine layouts. Here, steady inflow wind conditions are used to highlight the effects of turbine-wake interactions without introducing the unsteadiness of turbulent inflow. Additionally, turbulent inflow is used to study the effects of more realistic turbine operational conditions. The parameters varied are separation distance, offset distance (lateral distance between the turbine rotors), freestream velocity, and yaw angle. See Table 3.1 for nominal and range values. See Figure 3.1 for a turbine layout schematic. Unless otherwise stated, all simulations employed laterally-aligned turbines with a 7.1D separation distance and freestream velocity

90 Table 3.1: Ranges of inﬂow conditions and turbine locations. Parameter Nominal of Interest Value Values Increment Freestream Velocity 8 m/s 6 to 25 m/s 1 m/s Yaw Angle 0◦ −20◦ to 20◦ 2◦ Offset Distance 0D −2D to +2D 0.1 or 0.5D Separation Distance 7.1D 3 to 20D 1D

Top View

U∞ � Offset Distance

Yaw Angle DD == 10.058 90 m m

Separation Distance

Figure 3.1: Two-turbine layout for parametric studies. Deﬁnitions of various operating conditions and spacing deﬁnitions are included.

91 Table 3.2: Main parameters of the UAE Phase-VI wind turbine. Parameter UAE Phase-VI Rotor Position (-) upwind Number of Blades (-) 2 Rotor Diameter (D) 10.058 Rated Blade Pitch (◦) 4.85 (constant) Rated Rotor Speed (rpm) 72 (constant) Hub Height (m) 12.192

of 8 m/s. When turbulent inﬂow is considered, a TI of 5% is used with the IEC

Kaimal turbulence spectrum, as in Section 2.3.

This discussion analyzes wake structure, rotor power, and blade root ﬂapwise and edgewise bending moments. Rotor power results include average absolute power for single and two-turbine cases, as well as relative power for two-turbine cases. Blade root loads are studied in terms of RMS ﬂapwise and edgewise blade root bending moments. Finally, standard deviations and fatigue loads of these quantities are compared and discussed.

3.1 Wake Structure

The wake structures from FAST-FVW simulations for a range of offset distances are depicted in Figure 3.2. These results are from a past study [126] using the UAE

Phase-VI turbine, as described in Table 3.2. A 0 m offset distance corresponds to a fully waked T2 where the entirety of T2 is impacted by the wake of T1, resulting in complex wake structures. A 5 m offset distance corresponds to a partially waked

T2, where the rotor tip of T2 is aligned with the hub of T1. In this conﬁguration,

92 a) 0m Offset

10 i) Both wakes 10 ii) Upstream wakes 10 i) Downstream wakes

5 5 5

x 0 x 0 x 0

•5 •5 •5

•10 •10 •10 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 z z z b) 5m Offset

10 10 10

5 5 5 x x x

0 0 0

•5 •5 •5 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 z z z c) 10m Offset

15 15 15

10 10 10

x 5 x 5 x 5

0 0 0

Figure 3.2: Top view of wake tip vortex structure for offset distances of (a) 0 m, (b) 5 m, and (c) 10 m. Wakes are shown together (i) and separately (ii and iii). All cases are for uniform freestream velocity at 10 m/s with turbines separated by 2D. Solid black and red lines represent T1 wake structure. Dashed blue and green lines represent T2 wake structure.

93 only a portion of T2 is affected by the wake of T1. The non-interacting portion of the upstream wake exhibits a complex structure due to the turbine rotational direction and upstream wake entrainment. While the non-interacting portion of the downstream wake is primarily a helical structure, it is inclined away from the interaction. This is due to the induced velocity imparted on the downstream wake by the upstream wake. Hence, while the wake structure may not be impacted by wake interaction, the trajectory may be affected. This could inﬂuence which downstream turbines are impacted by wake interaction. For a 10 m offset, T1 and

T2 rotor tips are aligned and the upstream wake is moderately entrained by the downstream wake. This demonstrates that wake proximity without direct wake impingement can still appreciably inﬂuence the wake structure.

3.2 Rotor Power

This section discusses FAST-FVW and FAST.Farm rotor power results for varying freestream velocity, yaw angle, offset distance, separation distance, and TI. Av- erage, relative, and standard deviation of rotor power are considered along with percent differences between the computational methods. Unless otherwise noted, all plots show T1 and T2 response for FAST-FVW and FAST.Farm results. T0 response is also shown for FAST-FVW results but not for FAST.Farm results since there is no difference between T1 and T0 response.

94 T1 T2 1.00 3000 FAST.Farm_08ms 0.75 FAST.Farm_10ms FAST.Farm_12ms 2000 0.50 FAST.Farm_16ms FAST-FVW_16ms 10000.25 FAST-FVW_T0_16ms

Rotor Power [kW] 0.00 3000.0 400 5000.2600 7000.4800 3000.6400 500 6000.8 700 8001.0 Time [s] (a) Steady inﬂow T1 T2 1.00 FAST.Farm_08ms 1000 0.75 FAST-FVW_08ms FAST-FVW_T0_08ms 0.50750

0.25500

Rotor Power [kW] 0.00 3000.0 400 5000.2600 7000.4800 3000.6400 500 6000.8 700 8001.0 Time [s] (b) Turbulent inﬂow

Figure 3.3: FAST-FVW and FAST.Farm rotor power time series for T1 (left) and T2 (right) for uniform inﬂow conditions for varying steady and turbulent freestream velocities. For each color group, darkest color represents FAST.Farm results and the lighter colors represent FAST-FVW results. When three lines are present in a color group, the lightest color is for T0 FAST-FVW results.

3.2.1 Varying Freestream Velocity

Rotor power time series results are provided in Figure 3.3 for uniform and turbulent inﬂow conditions. For steady inﬂow results, negligible differences are seen between the T1 FAST.Farm results and T0 FAST-FVW results, both in terms of average value and signal unsteadiness. However, increased unsteadiness is seen in

T1 FAST-FVW results, indicating the interaction of T2 on T1 in FAST-FVW simu-

95 Table 3.3: Average percent error of rotor power for different inﬂow averaging techniques. Average Percent Error PFAST-FVW 10.84% PFAST.Farm 20.08% lations. Differences in unsteadiness and average quantity between models are less clear for T2 results. For U∞ = 10−12 m/s, the FAST-FVW T2 results have increased unsteadiness and average response. Similar observations are seen for the turbulent inﬂow time series but with more unsteadiness in all signals, as expected. In this case, there are more distinct differences between T1 FAST.Farm results and T0

FAST-FVW results, indicating that model differences become more apparent with turbulent inflow. As with steady inflow, FAST-FVW T1 results exhibit increased unsteadiness and average value compared to the other results. These differences are strengthened with increasing freestream velocity, until rated inflow speeds are reached.

It is commonly observed that wake interaction reduces the power of wake- influenced turbines, with lessened reduction as freestream velocity increases. Both of these trends are captured by the FAST-FVW and FAST.Farm methods, though to differing degrees, as shown in Figure 3.4(a). Percent errors for turbulent inflow results relative to measurements are shown in Figure 3.4(b), with average percent errors summarized in Table 3.3. For turbulent inflow cases, the average percent error is just over 10% and 20% for FAST-FVW and FAST.Farm results, respectively.

It is clear that FAST-FVW and FAST.Farm results produce a different relative ro-

96 1.1 35 Experimental FAST-FVW_TI=5% FAST-FVW_TI=0% FAST.Farm_TI=5% 1.0 30 FAST-FVW_TI=5% 0.9 FAST.Farm_TI=0% FAST.Farm_TI=5% 25 0.8 20 0.7 15 0.6

Percent Error [%] 10 0.5 Relative Rotor Power [-]

0.4 5

0.3 0 6 8 10 12 14 6 8 10 12 14 Uinf [m/s] Uinf [m/s]

(a) Relative rotor power. (b) Percent error.

Figure 3.4: (a) Relative rotor power and (b) percent error for FAST-FVW, FAST.Farm, and experimental [35] results. Cases include varying steady and turbulent freestream velocities.

tor power trend for varying freestream velocity. In particular, FAST.Farm rela-

tive power results do not change with freestream velocity until rated power is

reached. To further investigate this point, average T1 and T2 rotor power is shown

in Figure 3.5 for each model, along with the percent difference between the mod-

els. Here, acceptable agreement is seen in P T1 results, with an average percent difference of 7.82%. There are larger differences in P T2, with a percent difference

of 53.8% at U∞ = 12 m/s. Thus, the large differences in relative rotor power primarily result from the differences in T2 rotor power. This is due to a stronger wake in the FAST.Farm simulations than the FAST-FVW simulations, as shown in

Figure 3.6. For FAST-FVW results at U∞ < 14 m/s, the non-dimensionalized average wake velocity increases with increasing freestream velocity. This results in a

97 3000 3000 50 50 2500 2500 40 40 2000 2000 FAST-FVW 30 FAST-FVW 30 1500 FAST.Farm 1500 FAST.Farm % Diff. 20 % Diff. 20 1000 1000 Rotor Power [kW] Rotor Power [kW] 10 500 10 500 Percent Difference [%] Percent Difference [%] 0 0 0 6 8 10 12 14 16 18 20 6 8 10 12 14 16 18 20 Uinf [m/s] Uinf [m/s]

(a) Upstream Turbine (b) Downstream Turbine

Figure 3.5: Comparison of average turbine rotor power of (a) T1 and (b) T2 for FAST-FVW and FAST.Farm model results, including percent difference between model results. Cases include varying steady freestream velocities. weak wake and an increasing T2 relative rotor power with increasing freestream velocity. FAST.Farm results, however, show the same non-dimensionalized average wake velocity for U∞ ≤ 14 m/s and downstream distance. This results in constant relative rotor power in this freestream velocity range.

The difference in changing wake velocity with downstream distance is also a point of interest. Both FAST-FVW and FAST.Farm results show slightly increasing average wake velocity with increasing downstream distance up to 6D behind T1.

However, at 7D downstream, i.e., just upstream of T2, FAST-FVW results show a dramatic decrease in average wake velocity. This inﬂuence of the turbine on the incoming ﬂow is due to the full coupling of the FAST-FVW method and is a known phenomenon in full-scale wind farms. Alternatively, there is no upstream propagation of information in FAST.Farm. As such, this slowing of the incoming

ﬂow is not observed.

Standard deviations of rotor power for varying freestream velocities using the

98 1.4 1.4 06ms 08ms 1.2 1.2 10ms 12ms 14ms 1.0 1.0 16ms 18ms 0.8 0.8 20ms

0.6 0.6 Horizontal Location (R) Horizontal Location (D) 0.4 0.4

0.2 0.2

0.0 0.0 0.5 0.7 0.9 1.1 0.5 0.7 0.9 1.1 Non-Dimensionalized Non-Dimensionalized Wake Velocity (-) Wake Velocity (-)

(a) FAST-FVW 5D downstream. (b) FAST.Farm 5D downstream.

1.4 1.4

1.2 1.2

1.0 1.0

0.8 0.8

0.6 0.6 Horizontal Location (R) Horizontal Location (D) 0.4 0.4

0.2 0.2

0.0 0.0 0.5 0.7 0.9 1.1 0.5 0.7 0.9 1.1 Non-Dimensionalized Non-Dimensionalized Wake Velocity (-) Wake Velocity (-)

1.4 1.4

1.2 1.2

1.0 1.0

0.8 0.8

0.6 0.6 Horizontal Location (R) Horizontal Location (D) 0.4 0.4

0.2 0.2

0.0 0.0 0.5 0.7 0.9 1.1 0.5 0.7 0.9 1.1 Non-Dimensionalized Non-Dimensionalized Wake Velocity (-) Wake Velocity (-)

(e) FAST-FVW 7D downstream. (f) FAST.Farm 7D downstream.

Figure 3.6: Comparison of non-dimensionalized average wake velocity proﬁles for FAST-FVW and FAST.Farm model results. Cases include varying steady freestream velocities and downstream distances. 99 T0 15 15 T1 T2

10 10

of RotPwr [%] 5 of RotPwr [%] 5 Std. Dev. Percentage Std. Dev. Percentage

0 0 6 8 10 12 14 16 18 20 6 8 10 12 14 16 18 20 Uinf [m/s] Uinf [m/s]

(a) FAST-FVW with Constant Inﬂow (b) FAST-FVW with Turbulent Inﬂow

15 15

10 10

of RotPwr [%] 5 of RotPwr [%] 5 Std. Dev. Percentage Std. Dev. Percentage

0 0 6 8 10 12 14 16 18 20 6 8 10 12 14 16 18 20 Uinf [m/s] Uinf [m/s]

Figure 3.7: Standard deviation of rotor power for FAST-FVW and FAST.Farm results with uniform and turbulent inﬂow conditions.

FAST-FVW and FAST.Farm models with uniform and turbulent inﬂow conditions

are shown in Figure 3.7. Average standard deviations (σ) are listed in Table 3.4.

For uniform inﬂow conditions, the increase of σFAST-FVW(PT1) over σFAST-FVW(PT0)

demonstrates the effects of T2 inﬂuencing T1. This is not observed in the FAST.Farm

standard deviations because the FAST.Farm model neglects upstream inﬂuence. It

is known that wake interaction leads to increased turbulence in T2 inﬂow condi-

tions. This suggests that the standard deviations of T2 response are higher than

those for T1. The FAST-FVW model predicts this, with σFAST-FVW(PT2) equaling an order of magnitude more than σFAST-FVW(PT1) for steady inﬂow. On the other hand,

100 Table 3.4: Average percent standard deviation (σ) of average rotor power for varying freestream velocities for FAST-FVW and FAST.Farm results with uniform and turbulent inflow conditions. Uniform Inflow Turbulent Inflow σFAST-FVW (kW) σFAST.Farm (kW) σFAST-FVW (kW) σFAST.Farm (kW) PT0 0.61 N/A 6.29 N/A PT1 0.71 1.96 6.80 9.05 PT2 7.35 3.36 10.48 11.71

σFAST.Farm(PT2) varies little from σFAST.Farm(PT1). This difference arises because of the superposition method used for wake interaction in the FAST.Farm model, which smoothes the velocity deﬁcit and TI proﬁle of the upstream wake before they are used as input to T2. [16] This removes the turbulent nature of T2 operating conditions, regardless of realistic conditions. This could impact predictions of structural response and blade fatigue and requires further study.

Different responses are seen when comparing turbulent inﬂow standard deviations from Figures 3.7(b) and (d). As with steady inﬂow, FAST-FVW results show that σFAST-FVW(PT1) is increased nearly 1% above σFAST-FVW(PT0), demonstrating that downstream turbines affect the response of upstream turbines. Both methods predict a sharp drop in σ(P ) for each turbine when the freestream velocity reaches rated power. The corresponding freestream velocity differs between the methods for T1, with FAST.Farm predicting rated power at a higher freestream velocity.

Since T2 experiences a reduced inﬂow velocity due to the wake of T1, it reaches a rated power at a higher freestream velocity and thus σ(PT2) drops at a higher freestream velocity than σ(PT1).

101 a) RelativeFAST•FVW Rotor Power b) FAST.Farm

11 1

0.9 0.9 .9 0 0.8 0.8 0.8 0.7 0.7 .7 0 6 m/s

Rotor Power (kW) Power Rotor 0.6 8 m/s 0.6 .6 FVW10 m/s Relative Rotor Power Rotor Relative 0 Power Rotor Relative FAST.Farm12 m/s Experimental 0.5 0.5 .5 0 •15 •10 •5 0 5 10 15 •10 0 10 Yaw Angle (degrees) Yaw Angle (degrees)

Figure 3.8: Relative rotor power for computational results from the (a) FAST-FVW and (b) FAST.Farm methods. Cases include varying yaw angle with varying turbulent freestream velocities.

3.2.2 Varying Yaw Angle

Average rotor power from FAST-FVW and FAST.Farm for varying yaw angle and

freestream velocity are shown in Figure 3.8. As expected, both sets of results show

increasing relative rotor power as yaw angle deviates from 0◦. As depicted in Fig-

ure 2.19, the wakes are strongly interacting for yaw angles ranging from 0◦ to 5◦, which correspond to the angles with low relative power values. As yaw angle increases, the wake interaction lessens and rotor powers for both turbines approach the same value. While both methods predict this behavior, the degree of T2 rotor power reduction can differ by over 40%. Such large differences are due to the

FAST.Farm prediction of near-constant T2 relative rotor power with varying inﬂow velocity. This results in similar relative T2 rotor power for each considered inﬂow

102 velocity. Conversely, large differences are seen in FAST-FVW results for varying

inﬂow velocity. The reason for the different response between methods was dis-

cussed in Section 3.2.1. In addition to differing responses of minimum relative

rotor power, the recovery amount with increasing yaw angle also differs. For all

◦ inﬂow velocities, the FAST-FVW results recover to P T2/T1 = 1 at 10 . FAST.Farm

results, however, recover differently based on inﬂow velocity. In particular, P T2/T1

exceeds 1 for 6 m/s inﬂow velocity, while remaining below 1 for 8 and 12 m/s.

Though the differences in value are small, it is of interest that T2 FAST-FVW are

more affected in areas of strong wake interaction, while FAST.Farm is more af-

fected in areas of weaker wake interaction.

3.2.3 Varying Offset Distance

Relative average and standard deviation of rotor power results in turbulent inﬂow

conditions for varying offset distances, as well as percent difference between com-

putational methods, are shown in Figure 3.9. All relative rotor power results are

normalized by P T0 and P T1 for FAST-FVW and FAST.Farm results, respectively.

The differences between P T0 and P T1 for FAST-FVW results are shown through

T1 relative rotor power (P T1/P T0), which shows deviations of up to 2.1%. Strong

agreement is seen between the codes for T2 relative rotor, with percent differences

averaging 4.18%. For both computational methods, the T2 experiences a substan-

tial power deﬁcit in the full wake of T1. This deﬁcit lessens as T2 moves away from

103 1.0 15

0.9

10 0.8

0.7 5 Rotor Power [%]

0.6 Percent Difference of Relative Rotor Power [-] 0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 1.5 1.0 0.5 0.0 0.5 1.0 1.5 Offset Distance [D] Offset Distance [D]

(a) Relative rotor power (b) Percent difference

FAST-FVW_T0 FAST-FVW_T1 FAST-FVW_T2 FAST.Farm_T1 FAST.Farm_T2 T1 T2 160 60

140 40 120

100 20 80 Rotor Power [%] Percent Difference of

St. Dev. Percentage of 60 St. Dev. of Rotor Power [%] 0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 1.5 1.0 0.5 0.0 0.5 1.0 1.5 Offset Distance [D] Offset Distance [D]

Figure 3.9: Average relative (top) and standard deviation (bottom) of rotor power for varying offset distances for FAST-FVW and FAST.Farm results. Percent difference between computational results are included (left).

104 Table 3.5: Average percent standard deviation (σ) of average rotor power for varying offset distances for FAST-FVW and FAST.Farm results with turbulent inﬂow conditions. σFAST-FVW (kW) σFAST.Farm (kW) PT0 9.84 N/A PT1 11.08 9.99 PT2 15.82 10.81

the full wake position until ﬁnally recovering to the T1 value when the direct wake

interaction ends at ∼ ±1D. Note that there is a slight difference between T1 and

T0 power when there is a strong wake interaction, but they converge to the same

value as the interaction weakens.

Average standard deviations (σ) are listed in Table 3.5. These results are re-

ﬂected by the rotor power standard deviations, which converge to similar values with increasing offset distance. However, when there is a strong interaction both turbines are subject to increased dynamic response over T0 results with σFAST-FVW(PT2)

equaling 50 − 100% more than σFAST-FVW(PT1) for offsets between 0 and ±1D. For

this same range, σFAST-FVW(PT1) is 10% more than σFAST-FVW(PT0). Consequently,

σFAST-FVW(PT1) is on average 10% more than σFAST.Farm(PT1), because FAST.Farm

does not include the inﬂuence of downstream turbines on upstream turbines. This

demonstrates that the FAST-FVW and FAST.Farm models predict increased un-

steadiness of rotor power due to the wake interaction, though to different degrees.

Additionally, it highlights the importance of complete wake interaction to capture

the inﬂuence of the wake interaction on upstream turbines.

In addition to average differences between the computational methods, there

105 are clear differences in the response surfaces of the methods with changing offset

distance. That is, σFAST.Farm(PT2) follows a predictable pattern as the offset distance between the turbines is varied, with minimum unsteadiness when the turbines offset distance is 0D, followed by a steady increase in unsteadiness until offset

distance is ±0.5D and then decrease in unsteadiness until offset distance is ±1D.

Alternatively, although the overall trends are the same, σFAST-FVW(PT2) shows sig- niﬁcantly more erratic behavior with varying offset distance. This is likely due to the axisymmetric wake smoothing that is inherent in the FAST.Farm method. As these results were computed with a turbulent inﬂow, the erratic behavior is likely more consistent with actual turbine response. However, the smooth FAST.Farm response may be more accurate if an average response is desired. Therefore, ap- plication of the computational method will likely dictate which method is more appropriate when examining the response to a range of parameters.

3.2.4 Varying Separation Distance

Average relative and standard deviation of rotor power for varying separation distances are provided in Figure 3.10. As expected, there is a substantial power deﬁcit when T2 is located in the near wake of T1. For both methods, as T2 moves into the far wake of T1, the inﬂuence from T1 lessens and P T2 approaches P T1. While both

methods follow the same trend, FAST.Farm results exhibit a smoother response

with an average P T2/T1 percent difference of 11.9%. This increased smoothness of

106 1.0 20

15 0.8

10 0.6

Rotor Power [%] 5 Percent Difference of Relative Rotor Power [-] 0.4 0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 2.5 5.0 7.5 10.0 12.5 15.0 17.5 Separation Distance [D] Separation Distance [D]

(a) Relative rotor power (b) Percent difference

FAST-FVW_T0 FAST-FVW_T1 FAST-FVW_T2 FAST.Farm_T1 FAST.Farm_T2 T1 30 T2 140

120 20

100

80 10

60 Percent Difference of of Rotor Power [%] St. Dev. Percentage

40 St. Dev. of Rotor Power [%] 0 5 10 15 2.5 5.0 7.5 10.0 12.5 15.0 17.5 Separation Distance [D] Separation Distance [D]

Figure 3.10: Average relative and standard deviation of rotor power for varying separation distances for FAST-FVW and FAST.Farm results. Percent difference between computational results are included.

107 FAST.Farm results has been discussed in Section 3.2.3.

For all separation distances, FAST-FVW results predict P T1 to be on average

1.6% lower than P T0. This difference in T1 standard deviations reinforces the need for fully-coupled wake interactions, which are not captured by FAST.Farm. The standard deviation percentages in Figure 3.10 demonstrate the effects of wake interactions when rotor power is recovering. For all separation distances, σ(PT2) is

on average 5.7% greater than σ(PT1) for FAST-FVW results and 2.9% greater for

FAST.Farm results, with no sign of diminished effect for the largest separation dis-

tance. This highlights the obersvation that even if T2 power begins to recover in

an average sense, there can be an appreciable transient response. This is not un-

expected, since the increased wake unsteadiness has been shown to recover more

slowly than mean wake velocity. [19]

3.2.5 Varying Turbulence Intensity

The effects of varying TI on P T2/T1 and σ(P ) are investigated for each method

and compared to experimental results. Separate simulations were performed for

6−12 m/s inﬂows at each TI. Relative rotor power results are shown in Figure 3.11

for varying freestream turbulence intensities. The bars for each TI represent the

minimum and maximum values over the range of freestream velocities, with the

symbol showing the average value. Range bars are not available for experimental

results, and so only the averages are shown. As expected, P T2/T1 increases with

108 0.9

0.8

0.7

0.6

0.5 FVW FFarm Relative Rotor Power (•) Power Rotor Relative Experimental 0.4 2 4 6 8 10 12 14 Turbulence Intensity (TI)

Figure 3.11: Relative rotor power for computational and experimental results versus varying freestream turbulence intensities. Simulations were performed for U∞ = 6 − 12 m/s at each TI. Bars represent the minimum and maximum values (unavailable for experimental results), and the symbols represent the mean value.

109 increasing TI. This is due to the increased wake mixing which facilitates wake ve-

locity recovery and thus an increased P T2. Differences, however, are seen between

the resulting FAST-FVW and FAST.Farm P T2/T1 ranges based on U∞. This is not

surprising given the different P T2/T1 trends between FAST-FVW and FAST.Farm

results with increasing U∞ discussed in Section 3.2.1. The range predicted by

FAST.Farm does increase with increasing TI, indicating that in this method relative

wake velocity deﬁcit might be more sensitive to TI as opposed to U∞. In addition to differences in sensitivity to U∞, FAST.Farm average P T2/T1 increase at a slower

rate with increasing TI, compared to both FAST-FVW and experimental results.

Percent difference of rotor power standard deviation values, σ(P ), are shown

in Figure 3.12 for FAST-FVW and FAST.Farm results for varying turbulence inten-

sities. Here, σ(P ) is computed following Equation 2.18 and the percent difference is relative to the TI= 3% case. For both methods, σ(P ) increases with increas-

ing TI. Additionally, σ(PT2) is greater than σ(PT1). When using the FAST-FVW

method, this is captured without any additional models or empirical tuning. All

result show a nearly linear increase of σ(P ) with increasing TI. However, some

deviations and unexpected relationships are observed. For all turbulence intensi-

ties, σFAST-FVW(PT1) > σFAST.Farm(PT1). The opposite tends to be true for σ(PT2), with

σFAST.Farm(PT2) > σFAST-FVW(PT2). It is unclear why this would be the case and fur-

ther investigation is needed to determine the root cause of these differences. The

percent change depicted in Figure 3.12(b) demonstrates that T1 is more sensitive to

changing TI than T2. This is likely due to the already turbulent inﬂow experienced

110 40 FAST-FVW_T1 FAST.Farm_T1 125 FAST-FVW_T2 FAST.Farm_T2 30 100

75 20 50 St. Dev. of 10

Rotor Power [%] 25 Percent Difference [%] 0 0 3 5 7 9 11 13 3 5 7 9 11 13 Turbulence Intensity [%] Turbulence Intensity [%]

(a) 6 m/s inﬂow (b) Percent difference

40 125

30 100

75 20 50 St. Dev. of 10

Rotor Power [%] 25 Percent Difference [%] 0 0 3 5 7 9 11 13 3 5 7 9 11 13 Turbulence Intensity [%] Turbulence Intensity [%]

Figure 3.12: Rotor power standard deviations of average rotor power for computational results (left) and percent difference of computational results relative to computational 3% TI results (right) for FAST-FVW and FAST.Farm computational results. Cases include varying turbulence intensities for 6 and 8 m/s inﬂow velocities.

111 by T2.

3.3 Structural Response

This section discusses FAST-FVW and FAST.Farm blade-root ﬂapwise and edgewise bending moment results for varying freestream velocity, yaw angle, and offset distance. Fatigue loads of these quantities are also discussed and compared. Av- erage, relative, and standard deviation of blade-root bending moments are considered along with percent differences between the computational methods. Unless otherwise noted, all plots show T1 and T2 response for FAST-FVW and FAST.Farm results. Single-turbine responses are shown for FAST-FVW results. No such dis- tinction is made for FAST.Farm results because T0 results are identical to T1 results.

3.3.1 Varying Freestream Velocity

Flapwise blade-root bending moment time series results are provided in Figure 3.13 for steady and turbulent inﬂow conditions, respectively. Similarly to the results of Figure 3.3, minimal differences are observed between T0 FAST-FVW and T1

FAST.Farm results, with increased unsteadiness in T1 FAST-FVW results. Addi- tionally, further increased unsteadiness is evident in T2 response with overall increased unsteadiness in turbulent inﬂow results.

To further highlight differences in FAST-FVW and FAST.Farm turbine response, probability density function (PDF) results of ﬂapwise blade-root bending moment

112 T1 T2 1.00 FAST.Farm_08ms 30000.75 FAST.Farm_10ms FAST.Farm_12ms 20000.50 FAST.Farm_16ms FAST-FVW_16ms 0.25 1000 FAST-FVW_T0_16ms 0.00

Flapwise Blade-Root 3000.0 400 5000.2600 7000.4800 3000.6400 500 6000.8 700 8001.0

Bending Moment [kN-m] Time [s] (a) Steady inﬂow

T1 T2 1.00 FAST.Farm_08ms 15000.75 FAST-FVW_08ms FAST-FVW_T0_08ms 0.50 1000 0.25

0.00500

Flapwise Blade-Root 3000.0 400 5000.2600 7000.4800 3000.6400 500 6000.8 700 8001.0

Bending Moment [kN-m] Time [s] (b) Turbulent inﬂow

Figure 3.13: FAST-FVW and FAST.Farm blade-root ﬂapwise bending moment time series for T1 (left) and T2 (right) for varying steady and turbulent freestream velocities. For each color group, darkest color represents FAST.Farm results and the lighter colors represent FAST-FVW results. When three lines are present in a color group, the lightest color is for T0 FAST-FVW results.

113 T1 T2 1.00

40000.75 4000

20000.50 2000 0.25 0 0 0.00 Flapwise Blade-Root 0.00.000 0.002 0.20.004 0.0060.4 0.0000.6 0.002 0.0040.8 0.006 1.0

Bending Moment [kN-m] PDF [1/kN-m] (a) Steady inﬂow

T1 T2 60001.00 6000

40000.75 4000 0.50 2000 2000 0.25 0 0 0.00

Flapwise Blade-Root 0.00.000 0.001 0.0020.2 0.003 0.0040.4 0.0000.6 0.001 0.0020.80.003 0.0041.0

Bending Moment [kN-m] PDF [1/kN-m] (b) Turbulent inﬂow

Figure 3.14: FAST-FVW and FAST.Farm blade-root ﬂapwise bending moment PDF for T1 (left) and T2 (right) for varying steady and turbulent freestream velocities. are provided in Figure 3.14 for steady and turbulent inﬂow conditions. In Fig- ure 3.14(b), fewer U∞ results are included for clarity. Here, the differences in mean and standard deviation values are clearer. For all cases, FAST-FVW T1 results have a lower mean and increased unsteadiness than the corresponding FAST.Farm results, in particular for higher freestream velocities. Comparable differences are seen for T2 results.

Flapwise blade-root bending moment power spectral density (PSD) results are provided in Figure 3.15 for 6 m/s steady and turbulent freestream velocities. Dom-

inant low-frequency peaks correspond to the blade passing frequency, deﬁned by

114 (a) Steady inﬂow

(b) Turbulent inﬂow

Figure 3.15: FAST-FVW (left) and FAST.Farm (right) blade-root ﬂapwise bending moment PSD for T1 and T2 for steady and turbulent 6 m/s freestream velocities.

115 Equation 3.1 with [Ω] = rpm. Ω P = n (3.1) 60

For FAST-FVW results, no difference is seen between T0 and T1 FAST-FVW results for steady inflow and negligible differences are seen for turbulent inflow. For both cases, T2 FAST-FVW results show significantly increased spectral content, indicating higher unsteadiness in the incoming flow. FAST.Farm results, on the other hand, show increased spectral content of T2 results for low-frequency response only. In fact, T1 and T0 results from both FAST-FVW and FAST.Farm have higher spectral content in the higher-frequency range. This is likely due to the axisymmetric wake smoothing used in the FAST.Farm model.

Flapwise blade-root bending moment PSD results are provided in Figure 3.16 for a range of steady and turbulent freestream velocities. For all cases, negligible differences are seen between T0 and T1 results FAST-FVW results. T2 FAST-FVW results show signiﬁcantly increased spectral content for all cases, indicating higher unsteadiness in the incoming ﬂow. Additionally, passing frequency peak occur at a lower frequency due to the decreased rotor speed. The reduction of FAST.Farm

T2 spectral content reduced below T1 results holds for inﬂow velocities < 14 m/s.

Above this point, spectral content is comparable. This holds true for both steady and turbulent inﬂow.

Edgewise blade-root bending moment PDF results are provided in Figure 3.17 for turbulent inﬂow conditions. These results feature two peaks due to gravita-

116 (a) Steady inﬂow

(b) Turbulent inﬂow

Figure 3.16: FAST-FVW and FAST.Farm blade-root ﬂapwise bending moment PSD for T1 (left) and T2 (right) for steady and turbulent freestream velocities.

T1 T2 40001.00 4000

20000.75 06ms_FAST.Farm 2000 10ms_FAST.Farm 0.500 14ms_FAST.Farm 0 14ms_FAST-FVW 20000.25 14ms_FAST-FVW_T0 2000 0.00 0.0000 0.0001 0.0002 0.0003 0.0000 0.0001 0.0002 0.0003 Edgewise Blade-Root 0.0 0.2 0.4 0.6 0.8 1.0

Bending Moment [kN-m] PDF [1/kN-m]

Figure 3.17: FAST-FVW and FAST.Farm blade-root edgewise bending moment PDF for T1 (left) and T2 (right) for varying turbulent freestream velocities.

117 Table 3.6: Average percent error of minimum, mean, maximum, and range of relative ﬂapwise bending moment between computational results and experimental measurements for non-waked and waked turbines. Upstream Turbine Min. Mean Max. Range MFlap,FAST-FVW 22.10 18.44 18.97 26.68 MFlap,FAST.Farm 28.28 25.36 23.86 33.73 Downstream Turbine Min. Mean Max. Range MFlap,FAST-FVW 46.81 24.13 24.79 16.82 MFlap,FAST.Farm 19.91 27.30 29.14 31.77 tional forces acting in opposing directions on the rotating blades. Blade response differs less for edgewise loads than ﬂapwise loads for varying freestream velocity because edgewise loads are dominated by this constant gravitational loads. There is, however, some variation in T2 results, especially for higher freestream velocities because of the increasing differences in average M Edge,T2.

Average ﬂapwise bending moments, relative to the mean at 10 m/s of the non- waked turbine, are shown in Figure 3.18(a) and (c) for experimental results as well as FAST-FVW and FAST.Farm simulations. The cumulative percent error of the average minimum, mean, maximum, and range (average maximum minus average minimum) of values for the simulation results relative to experimental results are shown in Figures 3.18(b) and (d). Non-waked T1 results are shown in Fig- ures 3.18(a) and (b), while waked T2 results are shown in Figures 3.18(c) and (d).

Average percent error results over all freestream velocities are summarized in Ta- ble 3.6. Excellent agreement is observed between FAST-FVW and FAST.Farm T1 results for U∞ ≤ 10 m/s. Above this freestream velocity, FAST.Farm predicts higher

118 ] s / 400

m min_FVW min_FF

0 1.5 mean_FVW mean_FF 1 300 max_FVW max_FF @ range_FVW range_FF x

a 1.0 m 200 M / 0.5 M

[ FVW

t 100

n FFarm Percent Error [%] e 0.0 Exp m

o 0 Relative Flapwise Bending

M 6 8 10 12 14 16 18 20 6 8 10 12 14 16 18 20 Uinf [m/s] Uinf [m/s]

(a) Upstream turbine relative ﬂapwise bending (b) Percent error of T1 relative ﬂapwise bending moment moment ] s / 400

m min_FVW FVW

0 1.5 mean_FVW

1 FFarm 300 max_FVW

@ Exp range_FVW x

a 1.0 min_FF m 200 M mean_FF / 0.5 max_FF M

[ range_FF

t 100 n Percent Error [%] e 0.0 m

o 0 Relative Flapwise Bending

M 6 8 10 12 14 16 18 20 6 8 10 12 14 16 18 20 Uinf [m/s] Uinf [m/s]

Figure 3.18: Relative ﬂapwise bending moment comparisons between computational and experimental results [35] for (a) T1 and (c) T2 at varying freestream velocities. Cumulative percent error between computational and experimental results are shown for the (b) T1 and (d) T2.

119 FAST-FVW_TI=0% FAST.Farm_TI=0% FAST-FVW_TI=5% FAST.Farm_TI=5% PDiff_TI=0% PDiff_TI=5% 25 3000 3000 40 20 2500 2500 30 15 2000 2000 20 10 1500 1500 1000 10 5 Flapwise Blade-Root Flapwise Blade-Root 1000 Percent Difference [%] Percent Difference [%] Bending Moment [kN-m] Bending Moment [kN-m] 500 500 0 0 6 8 10 12 14 16 18 20 6 8 10 12 14 16 18 20 Uinf [m/s] Uinf [m/s]

(a) Upstream Turbine Flapwise (b) Downstream Turbine Flapwise

1400 4 1.25 1380 1380 3 1.00 1360 1360 0.75 1340 2 1340 0.50 1320 1 1320 Edgewise Blade-Root 0.25 Edgewise Blade-Root Percent Difference [%] Percent Difference [%] Bending Moment [kN-m] Bending Moment [kN-m] 1300 0 1300 0.00 6 8 10 12 14 16 18 20 6 8 10 12 14 16 18 20 Uinf [m/s] Uinf [m/s]

Figure 3.19: RMS of ﬂapwise (top) and edgewise (bottom) blade root bending moments for FAST-FVW and FAST.Farm results at varying uniform and turbulent freestream velocities. minimum, mean, and maximum values as compared to FAST-FVW results. As a result, both methods produce comparable percent error values for lower freestream velocities with considerably larger FAST.Farm percent error values above rated rotor power.

Average RMS results for both models with uniform and turbulent inﬂow conditions are shown in Figure 3.19. Similar trends and values for the bending moments are seen for both models. Both ﬂapwise and edgewise bending moments increase with freestream velocity until rated wind speed is reached, at which point values begin to decrease. For both bending moments, there are negligible differences be-

120 Table 3.7: Average percent difference of flapwise and edgewise blade root bending moments between FAST-FVW and FAST.Farm results for varying freestream velocities. Uniform Inflow (%) Turbulent Inflow (%) MFlap,T1 14.50 16.31 MFlap,T2 23.74 20.75 MEdge,T1 0.72 0.69 MEdge,T2 0.88 1.39 tween T0 and T1 FAST-FVW results. The FAST-FVW model yields higher flapwise bending moments than the FAST.Farm model for both turbines, with percent differences ranging from 0.58 − 41.90%, averaging 18.83% for all inflow conditions, as detailed in Table 3.7. Alternatively, edgewise bending moments for FAST.Farm are typically higher than FAST-FVW results, particularly at higher inflow velocities, with percent differences ranging from 0.03 − 3.90%, averaging 0.92% for all inflow conditions. Thus, for all inflow conditions, there are comparable trends and reasonable agreement between FAST-FVW and FAST.Farm trends for RMS blade root bending moments.

Standard deviation percentages of structural results for FAST-FVW and FAST.Farm cases are depicted in Figure 3.20. For T1 response, strong agreement is seen between the methods at U∞ ≤ 12 m/s. Edgewise moments show negligible differences for all inflow velocities. At higher inflow velocities, σFAST.Farm(MFlap) results are 85.0 − 139.5% higher than σFAST-FVW(MFlap) results, with percent difference values averaging 66.4% (38.5% for the entire inflow range). This is starkly different from the σ(P ) results in Figure 3.7, which shows comparable results between meth-

121 50 50 FAST-FVW_T0 80 80 FAST-FVW 40 40 FAST.Farm 60 60 30 30 % Diff. 40 40 20 20

20 20 10 10 Std. Dev. Percentage Std. Dev. Percentage Percent Difference [%] Percent Difference [%] of Flapwise Blade-Root of Flapwise Blade-Root Bending Moment [kN-m] Bending Moment [kN-m] 0 0 0 0 6 8 10 12 14 16 18 20 6 8 10 12 14 16 18 20 Uinf [m/s] Uinf [m/s]

(a) Upstream Turbine Flapwise (b) Downstream Turbine Flapwise

101 2.5 101 2.5

100 100 2.0 2.0 99 99

98 1.5 98 1.5

97 1.0 97 1.0 96 96 0.5 0.5 Std. Dev. Percentage Std. Dev. Percentage Percent Difference [%] Percent Difference [%]

of Edgewise Blade-Root 95 of Edgewise Blade-Root 95 Bending Moment [kN-m] Bending Moment [kN-m] 94 0.0 94 0.0 6 8 10 12 14 16 18 20 6 8 10 12 14 16 18 20 Uinf [m/s] Uinf [m/s]

Figure 3.20: Standard deviations of ﬂapwise and edgewise root-blade bending moments for T1 (left) and T2 (right) results at varying freestream velocities

122 Table 3.8: Average standard deviations of ﬂapwise and edgewise blade root bending moments as a percentage of the mean for FAST-FVW and FAST.Farm results for varying freestream velocities with turbulent inﬂow.

σFAST-FVW (kN ∗ m) σFAST.Farm (kN ∗ m) MFlap,T0 12.76 N/A MFlap,T1 12.85 21.28 MFlap,T2 18.11 21.25 MEdge,T0 100.20 N/A MEdge,T1 98.66 98.60 MEdge,T2 98.56 98.63

ods for all turbulent inﬂow conditions. Further disagreement between methods is

seen in T2 results. Though both methods show similar response trends, average

percent difference between results average 38.7% for the entire inﬂow range, with larger differences seen in the lower U∞ range than for T1 results. Though large differences are also seen for σ(MEdge) results, percent differences remain below 2.2% for all results, indicating negligible difference in σ(MEdge) response between methods. This is because the moment component is dominated by gravity, which is constant for all cases.

Average standard deviation values are detailed in Table 3.8. σ(MFlap,T2) FAST-

FVW results are on average 6% higher than all other cases. As with the power results, this is due to the increased turbulence in the wake of T1. An increase in turbulence is not seen in the FAST.Farm results because of the smoothing used in the model. Thus, while comparable trends were shown between FAST-FVW and

FAST.Farm results for time-averaged rotor power and bending moment response, a comparison of the standard deviations of these quantities indicates that the tran-

123 1.0 1.0 225

1400 1400 200 0.9 0.9 175 1200 1200 0.8 0.8 150

1000 1000 St. Dev. of 0.7 0.7 125 Flapwise Blade-Root Bending Moment [-] Flapwise Blade-Root Flapwise Blade-Root 800 Bending Moment [-] 800 100 Bending Moment [kN-m] Bending Moment [kN-m] 0.6 Bending Moment [kN-m] 0.6

20 15 10 5 0 5 10 15 20 Relative Flapwise Blade-Root 20 15 10 5 0 5 10 15 20 Relative Flapwise Blade-Root 20 15 10 5 0 5 10 15 20 Yaw Angle [deg] Yaw Angle [deg] Yaw Angle [deg] (a) FAST-FVW (b) FAST.Farm (c) Standard deviation

FAST-FVW_T0 FAST-FVW_T2 FAST.Farm_T1 FAST.Farm_T2 FAST-FVW_T1 FAST-FVW_T2/T1 FAST.Farm_T2/T1 1200 1.02 1350 1.02

1175 1325 1.01 1.01 1320 1150 1300 1.00 1.00 1125 1275 1300 0.99 0.99

1100 1250 St. Dev. of

1075 0.98 1225 0.98 Edgewise Blade-Root Bending Moment [-] Bending Moment [-] Edgewise Blade-Root Edgewise Blade-Root 1280 Bending Moment [kN-m] Bending Moment [kN-m] 1050 0.97 Bending Moment [kN-m] 1200 0.97 20 15 10 5 0 5 10 15 20 20 15 10 5 0 5 10 15 20 20 15 10 5 0 5 10 15 20 Relative Edgewise Blade-Root Relative Edgewise Blade-Root Yaw Angle [deg] Yaw Angle [deg] Yaw Angle [deg] (d) FAST-FVW (e) FAST.Farm (f) Standard deviation

Figure 3.21: RMS of ﬂapwise (top) and edgewise (bottom) blade root bending moments and standard deviations for FAST-FVW and FAST.Farm results with varying yaw angles. sient responses are quite different with the FAST-FVW model indicating a more dynamic response in wake-inﬂuenced turbines. This increased unsteadiness is not predicted by the FAST.Farm model and could have implications for prediction of turbine life.

3.3.2 Varying Yaw Angle

Figure 3.21 shows the RMS and standard deviations of ﬂapwise and edgewise blade root bending moments for varying yaw angles under turbulent 8 m/s in-

ﬂow. For the remainder of this work, σ(M) is the unaltered standard deviation, not a percentage of the mean. Though values vary, overall comparable trends are seen

124 between FAST-FVW and FAST.Farm results for both bending components. M Edge

results are asymmetric for both methods, with M Edge,T2 reducing below M Edge,T1 for

negative yaw angles and increasing for positive yaw angles. There is asymmetry in

the M Edge results because of turbine rotation. Turbine blades experience a constant

edgewise downward force due to gravity, as well a variable edgewise aerodynamic

force, as shown in Figure 3.22. The aerodynamic force switches sign based on the

blade trajectory and is reduced by wake interactions. Therefore, depending on

which side of the turbine the wake interacts with on average, loading variation can

be increased or reduced. This effect is captured by both methods, and has been ob-

served by others using different wake and aeroelastic methods. [127] Though there

is variation in M Edge with yaw angle, this variation is minimal due to the constant

gravitational force, with relative M Edge ranging from 0.978−1.019. The same asym-

metric trend is seen in σ(MEdge) results as well, with FAST.Farm predicting slightly

larger changes to unsteadiness in wake-inﬂuenced turbines (+1.9%/ − 2.2% versus

+1.2%/ − 1.5%).

When T2 is in the wake of T1, M Flap,T2 for both methods is reduced below

M Flap,T1, though to a different extent−relative FAST-FVW M Flap,T2/T1 = 0.7 whereas

relative FAST.Farm M Flap = 0.6. As expected, both methods predict M Flap,T2/T1

approaching 1 as wake interaction with T2 decreases (θ ≥ 10◦). Similar trends

are also seen for σ(MFlap) between the methods, with increased σ(MFlap,T2) when wake interaction is present. Interestingly, σ(MFlap,T1) decreases as yaw angle in-

creases from −20◦ to 20◦. This asymmetry is due to turbine rotation. However,

125 Figure 3.22: Exaggerated edgewise load differences due to different waked turbine scenarios, resulting in different combinations of aerodynamic and gravitational loads. Freestream conditions are shown as well as two partially-waked turbine conditions. Blade positions 1 and 2 are labeled on the turbine ﬁgures, which correspond to the numbered points in the loading ﬁgures on the right. Reproduced from [127] with permission.

126 FAST-FVW results for σ(MFlap,T2) increases up to 66.7% over σ(MFlap,T1), whereas

FAST.Farm results for σ(MFlap,T2) only increases up to 23.6% above σ(MFlap,T1). In

◦ fact, σ(MFlap,T2) reduces 16.2% when θ = 0 . These results suggest that, though

both FAST-FVW and FAST.Farm predict increased unsteadiness in M Flap,T2, during wake interaction, FAST-FVW predicts considerably higher unsteadiness. As before, this is likely due to the smoothing inherent in the DWM method. Addition- ally, FAST.Farm does not currently include a wake-added turbulence model. If this model were added to FAST.Farm, it is likely that T2 unsteadiness would increase more, especially in low TI situations. As no smoothing is used in the FAST-FVW methodology, no such model is needed to fully capture wake turbulence.

3.3.3 Varying Offset Distance

Blade root bending moments for varying offset distances are shown in Figure 3.23.

For both methods, M Edge and σ(MEdge,T2) increase as offset distance approaches

−0.5D, where there is a high level of wake interaction. At this point, σ(MEdge,T2)

quantities are 1.76% and 2.02% above T1 results for FAST-FVW and FAST.Farm re-

sults, respectively. As the offset distance approaches 0.5D, M Edge and σ(MEdge,T2)

decrease and σ(MEdge,T2) quantities are 1.83% and 2.43% below T1 results for FAST-

FVW and FAST.Farm results, respectively. This non-monotonic response in M Edge

is due to interaction with the constant gravitational load, as was discussed in Sec-

tion 3.3.2. Contrary to σ(MFlap,T2)) results, FAST.Farm tends to predict more un-

127 1.0 1.0

1400 1400 250 0.9 0.9

1200 1200 200 0.8 0.8

1000 1000 St. Dev. of 150 0.7 0.7 Bending Moment [-] Flapwise Blade-Root Flapwise Blade-Root Flapwise Blade-Root Bending Moment [-] Bending Moment [kN-m] Bending Moment [kN-m] Bending Moment [kN-m] 100 800 800 0.6 Relative Flapwise Blade-Root 0.6 Relative Flapwise Blade-Root 1 0 1 1 0 1 1.5 1.0 0.5 0.0 0.5 1.0 1.5 Offset Distance [D] Offset Distance [D] Offset Distance [D] (a) FAST-FVW (b) FAST.Farm (c) Standard deviation

FAST-FVW_T0 FAST-FVW_T2 FAST.Farm_T1 FAST.Farm_T2 FAST-FVW_T1 FAST-FVW_T2/T1 FAST.Farm_T2/T1

1.02 1.02 1340 1340 1330 1.01 1.01 1330 1330 1320 1320 1.00 1320 1.00 1310 1310 1310 1300 0.99 0.99

1300 St. Dev. of 1300 1290 1290

0.98 Bending Moment [-] 0.98 Bending Moment [-] Edgewise Blade-Root Edgewise Blade-Root Edgewise Blade-Root 1290

Bending Moment [kN-m] 1280 Bending Moment [kN-m] 1280 Bending Moment [kN-m] 0.97 Relative Edgewise Blade-Root 1280 0.97 Relative Edgewise Blade-Root 1 0 1 1 0 1 1.5 1.0 0.5 0.0 0.5 1.0 1.5 Offset Distance [D] Offset Distance [D] Offset Distance [D] (d) FAST-FVW (e) FAST.Farm (f) Standard deviation

Figure 3.23: RMS of ﬂapwise (top) and edgewise (bottom) blade root bending moments and standard deviations for FAST-FVW and FAST.Farm results with varying offset distance.

steadiness for σ(MEdge,T2) results than FAST-FVW. Additionally, FAST.Farm results show smoother trends than FAST-FVW results. This smoothness trend is again due to the smoothing used in FAST.Farm, resulting in more “well-behaved” trends.

For both methods, M Flap,T2 is lower than those for T1 for offset distances less

than 1D. After this point, the value settles to that of M Flap,T1. As with the rotor

power, this point corresponds to the distance where T2 is no longer in the path

of the wake from T1. σ(MFlap,T2) increases as offset distances approaches ±0.5D, where there is a high level of wake interaction. At this point, FAST-FVW and

FAST.Farm results reach maximum values of 92.1% and 23.6% above T1 results, respectively. As the offset distance is increased past this point, σ(MFlap,T2) values trend back towards σ(MFlap,T1). Though this overall trend is the same, FAST-FVW

128 results predict that σ(MFlap,T2) is elevated above σ(MFlap,T1) for all offset distances and to a much greater extent than FAST.Farm results. Alternatively, FAST.Farm predicts lower T2 unsteadiness and reduced σ(MFlap,T2) for very small offset distances (0.0 to ±0.2D). This signiﬁcant difference in T2 load unsteadiness can lead

to reduced DELs, as will be discussed below. Additionally, given the slow decay of

σ(MFlap) with increasing offset distance, it is suggested that the prolonged effects of increased turbulence should be taken into consideration for offset turbines. This could have implications for the range of inﬂuence of turbines.

3.3.4 Fatigue Loads

DELs for ﬂapwise and edgewise blade-root bending moments are presented and discussed in this section. Details on fatigue load analysis are given in Appendix D.

Results are shown for varying offset distances in Figure 3.24. This ﬁgure contains DELT2 ﬂapwise and edgewise DELs for T1 and T2, as well as relative DEL DELT2/T1 = DELT1

for FAST-FVW and FAST.Farm results. Recall that there is no upstream propa-

gation of information in FAST.Farm and so DELT1 results are unchanged for the

varying turbine conﬁgurations. Both FAST-FVW and FAST.Farm results show a

reduction of T2 ﬂapwise DEL, resulting in a DELT2/T1 < 1 until an offset distance

of ∼ ±1D. This reduction of ﬂapwise DEL is unexpected, as it is generally accepted

that increased standard deviation will lead to an increased DEL, and there are some

offset distances for which σFAST.Farm(MFlap,T1) was lower than σFAST.Farm(MFlap,T1).

129 900 2.0 900 2.0 900 2.0 Rel Rel Rel 800 1.8 800 1.8 800 1.8 1.6 1.6 1.6 700 700 700 1.4 1.4 1.4 600 600 600 1.2 1.2 1.2

500 1.0 500 1.0 500 1.0 DEL [kN*m] DEL [kN*m] DEL [kN*m] 400 0.8 400 0.8 400 0.8 OoP Bending Moment OoP Bending Moment OoP Bending Moment 0.6 0.6 0.6 300 T2 300 T2 300 T2 T1 0.4 T1 0.4 T1 0.4 200 200 200 1.5 1.0 0.5 0.0 0.5 1.0 1.5 1.5 1.0 0.5 0.0 0.5 1.0 1.5 1.5 1.0 0.5 0.0 0.5 1.0 1.5 Off [D] Off [D] Off [D] (a) FAST-FVW Flapwise (b) FAST.Farm Flapwise (c) FAST.Farm Const Flapwise

1200 2.0 1200 2.0 1200 2.0 Rel Rel Rel 1100 1.8 1100 1.8 1100 1.8

1000 1.6 1000 1.6 1000 1.6

900 1.4 900 1.4 900 1.4 800 800 800 1.2 1.2 1.2 700 700 700 1.0 1.0 1.0

DEL [kN*m] 600 DEL [kN*m] 600 DEL [kN*m] 600 0.8 0.8 0.8 IP Bending Moment 500 IP Bending Moment 500 IP Bending Moment 500 0.6 0.6 0.6 400 T2 400 T2 400 T2 T1 0.4 T1 0.4 T1 0.4 300 300 300 1.5 1.0 0.5 0.0 0.5 1.0 1.5 1.5 1.0 0.5 0.0 0.5 1.0 1.5 1.5 1.0 0.5 0.0 0.5 1.0 1.5 Off [D] Off [D] Off [D] (d) FAST-FVW Edgewise (e) FAST.Farm Edgewise (f) FAST.Farm Const Edgewise

Figure 3.24: DELs of FAST-FVW and FAST.Farm simulations for varying offset distance.

However, this reduction in DEL is observed even when σ(MFlap,T2) is nearly twice

that of σ(MFlap,T1). The ﬂapwise DELT2/T1 trend does, however, primarily follow that of M Flap,T2/T1. This is true for FAST-FVW and FAST.Farm results, as well as

FAST.Farm results with steady inﬂow. This ﬁnding indicates that, while σ(MFlap,T2)

might be increasing over that of σ(MFlap,T1), it is crucial to consider the mean value

of the bending moment as well. This is important to note, as standard deviation is

often used as a surrogate for DELs when it is too expensive to compute DELs.

However, the change of σ(M) relative to the change of M is also required, as

shown by a comparison of edgewise moment DELs. For this moment component,

DELT2/T1 > 1 for nearly half of the offset distance range. Though both M Edge,T2

and σ(MEdge,T2) are changing with the same trends with varying offset distance,

130 σ(MEdge) as a percentage of M Edge is larger for T2 than T1 for the range of offset

distances, with average values increasing from 98.7 to 99.0 (from 98.7 to 99.2% in

the ±0.5D range). Though this increase seems small, it is possible that when the

standard deviation is such a high percentage of the average load, even a small

increase can lead to an increase in DEL.

Though these ﬁndings require more investigation to better understand when

DELs should be expected to increase or decrease, this initial work is already an

important ﬁnding, particularly for optimization studies. This work shows that the

common practice of using standard deviation as a surrogate for DELs can be in-

accurate and M must also be considered. This could have significant implications for optimization studies that include DELs, which could put turbines in close proximity to upstream turbines to decrease flapwise DELs. However, this response is opposite of edgewise bending moments. Therefore, multiple loading components should be considered in optimization studies to ensure that significant DEL increase in one component is not unintentionally neglected in favor of another.

131 Chapter 4

Concluding Remarks

4.1 Principal Conclusions Obtained in This Study

In this dissertation, a single-turbine free vortex wake method was extended to include turbine-wake interactions for any number of turbine in any conﬁguration.

This involved parallelizing the code and enabling the use of variable rotor-speed turbines. This method was shown to result in less than 10% error on average when comparing two-turbine simulations to wind farm measurements. Primary conclusions are:

Multi-Turbine FAST-FVW Model Validation

1. Experimental comparisons demonstrate acceptable agreement with wind farm

rotor power and blade-root ﬂapwise bending measurements and establish

the potential of a FVW approach for wind farm wake interaction studies.

For varying freestream velocity with turbine-aligned inﬂow, computational

results averaged 10.8% for all averaging schemes. For varying inﬂow yaw an-

132 gle, comparisons yielded an overall average percent error of 9.06%. Overall

trends were predicted with less than 9.95% error on average.

Comparison to FAST.Farm Method

2. Experimental comparisons were made for FAST.Farm results. For varying

freestream velocity with turbine-aligned inﬂow, computational results aver-

aged 14.54% for all averaging schemes. For varying inﬂow yaw angle, com-

parisons yielded an overall average percent error of 7.79%. Overall trends

were predicted with less than 11.16% error on average.

3. FAST-FVW method captures wake interaction effect lessening with increas-

ing freestream velocity. FAST.Farm does not capture this trend.

4. For constant inﬂow, FAST-FVW wake interaction leads to increased unsteadi-

ness in downstream turbine rotor power response while FAST.Farm does not.

5. FAST-FVW showed signiﬁcant increase in downstream turbine blade-root

ﬂapwise moment spectral content, whereas FAST.Farm showed reduced spec-

tral content in higher-frequency range.

6. FAST-FVW results predict much higher standard deviations of downstream

turbine blade-root ﬂapwise bending moments versus FAST.Farm results.

7. The necessity of considering both transient and time-averaged turbine re-

sponse was demonstrated by studying standard deviations of wind turbine

133 power and ﬂapwise blade root bending moment for both FAST-FVW and

FAST.Farm simulations.

Inﬂuence of Turbine-Wake Interactions and Inﬂow Wind Conditions

6. FAST-FVW model predicts that non-waked turbine rotor power is more sen-

sitive to freestream TI than wake-inﬂuenced turbines, likely due to the in-

creased turbulent inﬂow conditions of the wake-inﬂuenced turbine.

7. FAST-FVW model exhibits sensitivity of relative rotor power to minor wind

inﬂow conditions, with as much as 17.2% variance in percent error. Given

wind farm measurement uncertainty, this should be considered when deter-

mining computational wake model validity.

8. Varying TI of the inﬂow wind is demonstrated to impart a marked difference

on downstream turbine response, with PT2/T1 values ranging between 0.095

and 0.16 for U∞ = 6 and 8 m/s, respectively.

9. Upstream wake structure and DELs were changed due to presence and chang-

ing location of downstream turbine, indicating the importance of upstream

propagation of information.

10. Some, but not all, blade load DELs are reduced in the wake of an upstream

turbine. Therefore, the common practice of using standard deviation as a

surrogate for DELs can be inaccurate and average blade load must also be

134 considered. This could have signiﬁcant implications for optimization stud-

ies that include DELs, which could put turbines in close proximity to up-

stream turbines to decrease ﬂapwise DELs. Additionally, due to inconsistent

response, multiple loading components should be considered in optimiza-

tion studies to ensure that signiﬁcant DEL increase in one component is not

unintentionally neglected in favor of another.

4.2 Recommendations for Future Research

While the ﬁndings of this dissertation are promising, there are additional steps that could be taken towards model veriﬁcation and validation; code improvement; and potential code applications.

4.2.1 Further Veriﬁcation and Validation

First and foremost, further comparison to low- and high-ﬁdelity models is necessary to fully benchmark FVW model performance relative to other models. Specif- ically, comparisons using large-eddy simulations would provide valuable information and insight. The LES code SOWFA is also coupled with FAST and could provide high-ﬁdelity comparisons for both wind power and structural response.

A number of assumptions and variations were made when comparing computational and experimental results. Primarily, it was necessary in this work to generate a representative Vestas V90 turbine for use with comparison to experi-

135 mental results. However, it would be a more direct comparison if an actual Vestas

V90 turbine model could be obtained. In addition to more accurately predicting the power curve, this would allow for comparison of structural results which is currently not possible because the structural model primarily used in this work is based off of the NREL 5-MW turbine and not the Vestas V90 turbine. Another assumption was comparison of two-turbine simulations to turbine measurements in a wind farm setting. It would be beneﬁcial to perform three- or four-turbine

FVW simulations to further investigate turbine power and structural response in multi-turbine situation. This would provide insight into whether the FVW method will result in more accurate wind farm measurement comparisons when additional turbines are added to the simulations. It is likely that the presence of additional turbines influenced the response of the actual turbines considered for this work, especially given that downstream turbines can influence upstream turbines. Fi- nally, due to the complexity of actual wind inflow conditions, further analyzing

FAST-FVW results of turbine response to changes in simulated inﬂow conditions could lead to more realistic comparisons with experimental measurements. The use of physics-based inﬂow conditions could aid in this comparison.

4.2.2 Code Improvements and Applications

One of the biggest areas of improvement is code speedup. Little effort has been made to optimize the code and there is huge potential for speedup. In particu-

136 lar, the biggest bottleneck for code speed is the Biot-Savart computations, which currently happens in serial for every Lagrangian marker on a single processor.

OpenMP could be implemented at this step and potentially provide signiﬁcant speed-up.

It is currently necessary for the user of FAST-FVW to specify the desired wake cutoff length. This requires a convergence study to be performed for a new turbine, and potentially any new configuration of turbines. A more sophisticated implementation would involve introducing a criteria that would stop tracking La- grangian markers when, e.g., the marker circulation reaches a specified value. This would require testing to ensure that markers are not excluded when potential to markedly influence surrounding turbines exists.

A potentially exciting avenue for code development involves providing the option to model the wake using a vortex point approach. This would allow for dual- rotor turbine and vertical axis turbine simulations. Dual-rotor turbines are an area of active research which the DWM method is unable to handle due to signiﬁcant assumptions in near-wake modeling. [128, 129] While vertical axis wind turbines

(VAWT) were disregarded in the early development of wind turbines in favor of horizontal axis turbines, technology advancements have recently renewed interest in this technology. [130–133] Again, due to the limitations of the DWM method formulation, it would not be applicable for such research. Both dual-rotor turbine and VAWTs would be difficult to model with a filament-based FVW method due to the significant blade-wake interaction in the near wake. Instead, a point-based

137 FVW method would be able to handle such interactions in a simpler manner. Hav- ing the option to model the wake using a point- or ﬁlament-based approach would indeed be very powerful and allow for diverse research.

In addition to improvements to the code itself, it would be of great benefit to the community to integrate the FAST-FVW method into FAST.Farm, once it is made publicly available. This would allow for the new features of FAST.Farm to be uti- lized in FVW-simulation studies. In particular, enhanced visualization capabilities would provide visualization of the entire flow field, as opposed to the tracked La- grangian markers. Additionally, FAST.Farm includes an advanced unsteady aerodynamics module that better captures blade aerodynamics. Finally, FAST.Farm allows for full LES ABL wind simulations to be read in as the inflow wind component of FAST. This would allow for FAST-FVW simulations using physics-based inflow, thus providing more realistic inflow conditions and comparisons to experimental measurements.

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150 Appendix A

Multi-Variate Taylor Series Expansion

The general form for a multi-variate Taylor series expansion (TSE) is given in Equa- tion A.1.

f(x, y) = 1 f(a, b) + f (a, b)(x − a) + f (a, b)(y − b) + [f (a, b)(x − a)2 (A.1) x y 2! xx

2 + 2fxy(a, b)(x − a)(y − b) + fyy(a, b)(y − b) ] + H .O.T .

For the purpose of deriving the difference operators to use in the FVW predictor- corrector method, these variables are deﬁned for all TSEs as:

∆ψi • a = ψ + 2

∆ζ • b = ζ + 2

• f(a, b) = ~r(a, b)

As there are eight vertices in the stencil, eight TSEs are required. These are taken

at the following points:

151 1. x = ψ, y = ζ

2. x = ψ + ∆ψi, y = ζ

3. x = ψ + ∆ψi, y = ζ + ∆ζ

4. x = ψ, y = ζ + ∆ζ

5. x = ψ − ∆ψi, y = ζ

6. x = ψ − ∆ψi−1, y = ζ + ∆ζ

7. x = ψ − ∆ψi−1 − ∆ψi−2, y = ζ

8. x = ψ − ∆ψi−1 − ∆ψi−2, y = ζ + ∆ζ

152 Appendix B

Representative V90 Wind Turbine

As the Vestas V90 turbine is proprietary, exact parameters were not available for

constructing a model. However, a representative turbine was developed using in-

formation from NREL [134] and basing other parameters as needed on the NREL

5MW turbine. [135] In particular, controller parameters were developed using a

trial-and-error approach until the rotor power of the representative Vestas V90 tur-

bine reasonably matched that of the Vestas V90 for constant inﬂow conditions,

shown in Figure B.1. Percent error does not exceed 23% and averages 7.00% in

the range of velocities considered in this study. The most notable discrepancy be-

tween the actual and representative V90 turbine is the U∞ at which rated power

is reached. For the representative turbine, Prated is reached when U∞ = 12 m/s, whereas for the actual turbine it is reached when U∞ = 15 m/s . While this does

not lead to unreasonable rotor power error, it is likely to affect the power deﬁcit ex-

perienced by the downstream turbines. Though it is well documented that PT2/T1

approaches 1 with increasing U∞, it should not reach unity until after rated power

153 25 00 30 20

00 20 15

0 00 Percent Error (%) Error Percent Rotor Power (kW) Power Rotor 1 Ideal V90 FVW 5 % Error

0 05 10 15 20 25 5 10 15 20 25 Freestream Velocity (m/s) Freestream Velocity (m/s)

Figure B.1: Single-turbine power comparison of expected representative V90 turbine and computational FVW results. is reached. Since this point occurs 3 m/s sooner for the representative V90 turbine, this is likely a source in discrepancy PT2/T1 when comparing FVW to experimental results. While improvements can be made to this turbine model to better match the actual V90 turbine, it is considered adequate for the purposes of this study.

154 Appendix C

Turbulent Inﬂow Modeling

Wind turbines exclusively operate with turbulent inflow. The characteristics of this inflow are continually evolving and heavily impact the performance and structural response of the wind turbines. As such, significant research has gone into developing tools that accurately represent the atmospheric turbulent inflow experienced by wind turbines. The NREL code TurbSim is a stochastic pre-processing tool that generates turbulent, full-field wind inflow that can be used in FAST simulations. [121] It uses a statistical model to numerically simulate the time series of the three-dimensional velocity field onto a two-dimensional rectangular grid. [121]

First, the velocity component and spatial coherence spectra are defined in the frequency domain and then converted into the time domain using an inverse Fourier transform. [121] These rectangular grids are then propagated through the domain using Taylor’s frozen turbulence hypothesis to obtain local wind speed. [121] In this way, a turbulent wind field is generated in both time and space. Fundamen- tally, this process assumes that the turbulent inflow is a stationary process. It sim-

155 Table C.1: Summary of input parameters used by TurbSim for turbulent inflow generation. Parameter Value Grid Spacing 10 m Time Step 0.05 sec Turbulence Model IEC Kaimal Turbulence Intensity Specified percentage Turbulence Type Normal Coherence Parameters Default ulates non-stationary turbulent components, TurbSim can simulate additional coherent turbulent structures. [121] The stochastic nature of turbulence is included by use of random seed parameters which generate random phases for the velocity time series. [121] By changing the value of this random seed, statistically similar turbulent files can be generated while changing the transient components. When the same random seed parameters are used, the same random phases are produced, allowing for comparisons of other input parameters.

Within TurbSim, there are many options for how the turbulent ﬁeld can be de-

fined. Options are provided for spatial and temporal resolution, mean velocity and flow angle, turbulence model, average turbulence intensity, wind profile, and spatial coherence parameters. In addition to generic options, there are options for user-defined wind profiles, time series, and spectral content. The TurbSim parameters used for this research are summarized in Table C.1.

156 Appendix D

Fatigue Load Modeling

When a structure is subject to uniform loading, as shown in Figure D.1, it is a straight-forward process to determine the amplitude of the cycle as well as the cycle count. However, when a structure is subject to non-periodic loading, a method is required to reduce the complex, irregular loading history into a series of constant- amplitude loads. Rainﬂow counting is a method that was developed by Matuishi and Endo [136] for exactly this purpose. In this method, a time-history of load, stress, or strain, is oriented such that time is on the y-axis. The peaks and valleys of the signal are imaged to be roofs of a pagoda with water dripping down them, hence the name “rainﬂow” method. As the “water” drips down the “roof”, it either encounters another roof or falls off, as depicted in Figure D.2. Following this method, the amplitude of each peak and valley is compared to the surrounding peaks and valleys to determine if the cycle terminates or continue to the next

“roof”. This identiﬁes the amplitude and count of each half-cycle. These half- cycles are then grouped by equivalent amplitudes to determine the total number

157 Uniform Non•Uniform 60

Load (kN*m) •30

•60 0 50 100 Time (seconds)

Figure D.1: Example time-series plots of uniform and non-uniform loading.

158 85 kN*m ½ cycle 45 kN*m ½ cycle 50 kN*m ½ cycle

Figure D.2: Sample qualitative rainﬂow analysis for non-uniform loading.

159 Table D.1: Sample chart of amplitude and count of each cycle in a non-uniform signal. Amplitude Whole Cycles Half Cycles 10 2 0 13 0 1 19 1 0 24 0 1 27 0 1 of cycles for each amplitude. By this process, a complex, non-uniform load signal can be reduced into a chart of cycle amplitudes and the number of occurrences of each cycle, such as that shown in Table D.1

Fatigue load analysis is a convoluted process that necessitates assumptions.

There are, however, established standard for computing wind turbine fatigue loads.

Following Annex G of IEC 61400-1, [137] rainﬂow counting is used to quantify the

ﬂuctuating loads. Damage is then assumed to accumulate linearly and indepen- dently for each cycle, and Miner’s rule is used to compute the total damage imparted on the structural component. The fatigue analysis code MLife accumulates fatigue damage due to ﬂuctuating loads over the design life of the wind turbine.

Following Annex G of IEC 61400-1, these ﬂuctuating loads are broken down into individual cycles by matching local minima with local maxima in the time-series by way of rainﬂow counting. Each cycle is characterized by a mean value and range. [138] Following Miner’s rule, [139] damage is assumed to accumulate linearly with each cycle. Total damage from all cycles, D, is expressed as the sum of

160 damage from each individual cycle.

X ni D = (D.1) N i i

Here, i indicates each load cycle, Ni is the number of cycles to failure for a given

RF load amplitude (Li ) and ni is the cycle count. Based on S-N curves, the relationship between load range and cycles to failure is given by: [138]

Lult − |LMF|m N = i 1 RF (D.2) 2 Li

where Lult is the ultimate design load of the component, LMF is the ﬁxed load mean,

and m is the Who¨hler exponent of the considered component.

In this formulation, the fatigue cycle is assumed to occur over a constant mean.

However, the actual cycles occur over a spectrum of load means. To account for

this, the Goodman correction is used to allow for the data to be analyzed as if it

were occurring over a ﬁxed mean. [138] Using this correction results in a mean

load expressed as: [138] Lult − |LMF| LRF = LR i i ult M (D.3) L − |Li |

RF R where Li is the Goodman-corrected load range about a ﬁxed mean; Li is the load range for cycle i; Lult is the component ultimate load; and LMF is the ﬁxed mean of the load.

These components can be combined to compute the short-term DELs for a

161 given time series. A DEL is a constant-amplitude fatigue load that occurs at a

ﬁxed mean and frequency, with the end of producing equivalent damage as the

variable spectrum load. [138]

P RF m 1/m ( i ni(L ) ) DEL = Steq (D.4) nj

where DEL is the DEL for the time series about a ﬁxed mean; ni is the damage

STeq count for cycle i; and ni is the total equivalent fatigue counts for the time series given by: [138]

STeq eq nj = f T (D.5)

where f eq is the DEL frequency and T is the elapsed time of the time series.

162