Precision and Microsystems Engineering

Towards High-Fidelity AeroelasticAnalysis of Wind TurbinesCoupling and verication

S.F. van den Broek

Mas

tero

fScie

nce

Thes

is

Report No: EM 2013.021Coaches: Dr. T. Ashuri Ir. E.A. FeredeProfessor: Prof.dr.ir. F. van KeulenSpecialization: Engineering MechanicsType of report: MSc thesisDate: 2013-08-04

TowardsHIGH FIDELITY AEROELASTIC

analysis of

WIND TURBINES

m

Coupling and Verication

Sander van den Broek

August 2013

Submitted in partial fulllment of the requirementsfor the degree of Master of Science in Mechanical Engineering

to the

Department of Precision and Microsystems EngineeringFaculty of Mechanical, Maritime and Materials Engineering

Del University of Technology

is thesis was typeset using the LATEX typesettingsystem originally developed by Leslie Lamport,based on TEX created by Donald Knuth.

e formatting of this thesis is based onthe master thesis of Elvind Uggedal.e cover is based on the cover of the DCSCmaster thesis template.e photo on the coveris taken by the author, it features one of one ofthe testing turbines at NREL’s NWTC site atLouisville, CO, USA.

e body text is set 12/14.5pt on a 26pc measure withMinion Pro designed by Robert Slimbach.is neohumanistic font was rst issued by AdobeSystems in 1989 and have since been revised.Other fonts include Sans and Typewriter fromDonald Knuth’s Computer Modern family.

a

ABSTRACT

e design of wind turbines is a multidisciplinary process. Most of thecodes currently used in industry and academia use low delity mod-els, which are not suitable for detailed design.is thesis describes thedevelopment of an aerostructural model for wind turbines with a highdelity structural solver.

is is done by coupling an existing high delity structural solverwith an aerodynamic solver. Coupling is done by interfacing the com-ponents in Python. e aeroelastic solution is computed using thenon-linear block Gauss-Seidel method.e code makes it possible todo steady-state analysis of a wind turbine blade. A high-delity modelis created based on a Sandia National Labs model detailing the designof a 5 MW turbine.

Results from a comparative study between a low delity model andthe developed model of this work show that the structural torsion has asignicant eect on the performance of large wind turbines. Torsionaldegrees of freedom are not included in most low delity models.is isa clear indication that current aeroelastic codes have severe limitationswhen used to design exible blades. e developed model makes itpossible to model bend-twist coupling due to composite layups as wellas geometry. Future work will focus on time-domain simulation of theaeroelastic response and high-delity optimization.

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CONTENTS

Abstract i

Contents iii

List of Figures v

List of Tables vii

Nomenclature ix

Preface xiii

1 Introduction 11.1 Background 11.2 Motivation 21.3 Research questions 31.4 Research method 31.5 esis outline 4

eoretical background

2 Wind turbine aerodynamics 92.1 Wind turbine aerodynamic design 92.2 Blade Element Momentum theory 13

3 Wind turbine structures 293.1 Structural design 293.2 Mechanical analysis of composites 33

Implementation of aeroelastic framework

4 Aeroelastic coupling and modeling 454.1 Overview of the aerostructural framework 454.2 pyAeroStruct 454.3 pyTACS, the TACS-pyAeroStruct interface 474.4 AeroDyn interface 484.5 pyAeroDyn, the AeroDyn-pyAeroStruct interface 52

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4.6 pyAeroDyn functions 554.7 Modeling an aeroelastic problem 65

5 Verication of developed code 695.1 Verication of composite plate 695.2 Turbine model used for verication 705.3 Results of turbine blade model 72

Conclusions

6 Conclusions and future work 816.1 Conclusions 816.2 Future work 82

Bibliography 85

Appendices

iv

LIST OF FIGURES

1.1 Cumulative global wind energy capacity. 1

2.1 Torque extraction of blade. 102.2 Apparent wind speed along the blade. 112.3 Planform shape of turbine blade. 122.4 Steam turbine model of wind turbine. 142.5 Pressure and velocity throughout the steam turbine. 152.6 Notation of rotating annular stream as well as trajectory of

air particle. 172.7 Change in angular momentum. 182.8 BEM elements. 202.9 Flow onto turbine blade. 212.10 Forces acting on the turbine blades. 212.11 NACA 0012 li and drag coecients. 222.12 Denition of the coordinates used in the skewed wake cor-

rection model. 25

3.1 Cross section of turbine blade. 303.2 Displacements and their relationship to the midplane. 383.3 Coordinates of the composite’s plies. 403.4 Kinematic assumptions in rst order shear theory. 42

4.1 Relationship between codes 464.2 Location of blade node. 514.3 Plot of the intermediate mesh. 564.4 e intermediate mesh and the structural mesh plotted at

the same time. 564.5 Intermittent mesh representation of an aerodynamic ele-

ment. 564.6 Graphical representation of the dimensions used for mesh

generation. 574.7 Nodes of example element 594.8 Process used to determine the traction forces applied to the

nodes. 624.9 Mean traction force distribution 624.10 Moment traction force distribution 634.11 Moment correction traction force distribution 644.12 Total traction force distribution 64

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4.13 Process used to create the structural mesh. 65

5.1 Deformed plate. 695.2 Relative error between TACS and ANSYS for test case. 705.3 Deformation results for a variety of mesh sizes. 725.4 Mass distribution along the span of the blade with interpo-

lation. 735.5 Power curve with FAST control settings. 735.6 Pitch angle eect on power coecient. 735.7 Pitch angle setting of "tuned" TACS vs FAST controlled

model. 745.8 Power output of "tuned" TACS compared to other models.

745.9 x-direction deection [m]. 765.10 y-direction deection [m]. 765.11 z-direction deection [m]. 765.12 x-rotation [deg]. 775.13 y-rotation [deg]. 775.14 z-rotation [deg]. 77

vi

LIST OF TABLES

5.1 Orthotropic material properties 695.2 Deformation results for the composite plate test case [mm]. 695.3 Mass calculation results, ANSYS and FAST models are Nu-

MADmodel results. 725.4 Mass distribution along elements 725.5 Deection and twist of the blade at 11 m/s wind conditions

FAST vs "tuned" TACS. 76

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NOMENCLATURE

Mathematical notationa Scalara VectorA Matrix

Latin symbolsA Area m2

a Axial ow induction factor -a′ Angular induction factor -B Number of blades -C Stiness matrix Pac Chord length mCD Drag coecient -CL Li coecient -Cp Power coecient -CT rust coecient -D Orientation matrix -d Distance from aerodynamic center to LE/TE mE Young’s modulus PaE Tolerance -Fθ Tangential force NFL Li force NG Shear modulus Pag Gravitational acceleration 9.80665 m/s2

h Height mh Laminate thickness (Chapter 3) mI Mass moment of inertia kg⋅m2L Angular momentum N⋅m⋅sLelm Length of element mM Mass (Chapter 2) kgM Moment per unit length (Chapter 4) N⋅m/m

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m Mass ow kg/sN Force per unit length N/mni Position vector of node i mnsha Sha direction vector mp Pressure Pap Momentum (chapter 2) kg⋅m/sp Element position vector (chapter 4) mpcent Spanwise location of the center of the element mQ Reduced transformed stiness matrix -Q Reduced stiness matrix for unidirectional lamina -Qhub Hub correction factor -Qtip Tip correction factor -R Rotation matrix -R Rotor radius mr Local radius mRhub Hub radius mS Planform area m2

S Compliance matrix Pa−1

t Torque exerted by blade N⋅mT Torque N⋅mtaf Airfoil thickness N⋅mu Deection in x-direction mU Wind velocity m/sv Deection in y-direction mW Eective wind velocity m/sw Deection in z-direction myac y-direction relative position of the aerodynamic center -yo y-direction oset of blade -

Greek symbolsα Angle of attack radβ Angle of inow radγ Twist angle (Chapter 2) radγ Shear strain (Chapter 3) -γ In plane deection angle (Chapter 4) radε Strain -θ Fiber orientation (Chapter 3) radθ Blade azimuth angle (Chapter 4) radκ Angle of deection (Chapter 3) rad

x

κ Sha tilt angle (Chapter 4) radλr Tip speed ratio -ν Poisson’s ratio -ξ Pitch angle (Chapter 4) radρ Density kg/m3

σ ′ Local solidity (Chapter 2) Paσ Stress (Chapter 3) Paτ Shear stress Paυ Out of plane deection angle (Chapter 4) radΦ Angle to mid-surface radϕ Airfoil inow angle (Chapter 2) radϕ Element twist (Chapter 4) radχ Skew wake angle radψ Ellipsoidal coordinate (Chapter 2) radψ Precone angle (Chapter 4) radΩ Blade rotational velocity rad/sω Wake rotational velocity rad/s

Subscripts0 Undeformed∞ Upstream1 In the 1-direction (Chapter 3)2 In the 2-direction3 In the 3-directionBA Blade attached coordinate systemd At the diskEL Element coordinate systemGL Global coordinate systemmc Moment correctionmtr Mean traction forceR Root-side of the elementT Tip-side of the elementw In the wake (down-steam)x In the x-direction (Chapter 3)y In the y-direction (Chapter 3)z In the z-direction (Chapter 3)

Superscripts+ Front of the actuator disk

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− Back of the actuator disk

AbbreviationsBEM Blade element momentum methodCAD Computer aided designGFRP Carbon-ber reinforced polymerCK Coupled Krylov methodCLT Classical lamination theoryCNK Coupled Newton-Krylov methodECN Energieonderzoek Centrum NederlandFEA Finite element analysisFF Full-eld turbulent owFRP Fiber reinforced polymerFSDT First order shear deformation theoryGFRP Glass-ber reinforced polymerHAWT Horizontal-axis wind turbineHH Hub heightIEC International Electrotechnical CommissionLBGS Linear block Gauss-SeidelNASA National Aeronautics and Space AdministrationNLBGS Nonlinear block Gauss-SeidelNREL National Renewable Energy LabNuMAD Numerical Manufacturing And Design ToolNWTC National Wind Technology CenterSWIG Simplied Wrapper and Interface GeneratorTACS Toolkit for the Analysis of Composite Structures

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PREFACE

is thesis was written between fall 2012 and summer 2013.e majorityof the work was done at the University of Michigan. I was also fortunateenough to be spend two weeks of my research at National Wind Tech-nology Center (NWTC) of the National Renewable Energy Lab (NREL)in Colorado.

I would like to thank Prof. Joaquim Martins for hosting me duringmy stay at the University of Michigan. e funding for my research wasmade possible by the Justus & Louise van Een research grant. I am verythankful to Prof. Fred van Keulen for supporting my application to thisgrant.

In addition to this I would like to thankmy supervisors, Turaj Ashuriand Etana Ferede.ey were very supportive of my research and werealways available for questions. My oce mates, Gaetan Kenway andMarco Ceze, who were always up for discussing things on my mind. Athank you to Prof. van Bussel who made my stay at NREL possible. Andof course Jason Jonkman, who hosted me during my stay at NREL. Hewas very helpful in making me feel welcome, discussing options andinviting me to developmental meetings.

And let me not forget my friends and family, both in the US andin the Netherlands. In particular my parents, who supported me goingabroad for over 7 months. Douglas Leavy and family, who hosted me inColorado, and proofreadmy thesis.e lunch group inMichigan alwaysprovided a welcome diversion during the day and featured stimulatingconversation.

Sander van den BroekAnn Arbor, MI. USA

August 2013

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1

INTRODUCTION

1.1 backgrounde trend the last few years has been to reduce dependency on fossil fuels.is is done by creating an energy mix which consists of multiple fuelsources. Wind energy is promoted by many countries through subsidies(Saidur et al., 2010).e usage of wind energy has rapidly increased overthe last few years. is is illustrated in Figure 1.1. Renewable sourcesare expected to produce more energy than natural gas sources by 2016(Goossens, 2013).

Rapid increase in population, expected by 2040, as well as emerg-ing economies in Asia and Africa, will lead to a signicant increase inglobal energy usage. Other green alternatives, such as nuclear fusionand wave energy are at the proof of concept stage (Tokimatsu et al., 2003;Ashuri et al., 2012).e nite supply of fossil fuels combined with theenvironmental impact of their usage warrant an increase in the usageof renewable energy (ExxonMobil, 2013).is is reected in the globalannual spending on renewable energy (REN21, 2013, p. 15).

Wind energy production will increase greatly over the next fewdecades.ere are plans in theUS to increase wind energy’s contribution

Figure 1.1: Cumulative wind energy usage. Figure fromWikiMedia Commons,source of data is from Global Wind Energy Council (2012).

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to the national energy production to 20% by 2030 (DoE, 2008). econtribution in 2008 was a mere 2%. e rapid increase in installedcapacity means that even small increases in eciency will have a verylarge eect.is has further peaked the interest in industry and academiato design wind turbines with more advanced tools.

1.2 motivationModern wind energy is a relatively young sector compared to aerospaceand it did not become a serious research eld until the oil crisis inthe 1970’s (Burton et al., 2011). When it comes to design, aeroelasticanalysis and optimization in the eld of aerospace engineering is farmore advanced.ere are several reasons for this:

• Aviation has been an active industry for over 100 years. Researchin wind energy began roughly 40 years ago.

• Both World Wars as well as the Cold War allocated a signicantamount of defense money into the research and development ofmilitary aircra.

• Development of wind turbines is spearheaded by the cost of energy.Wind turbine manufacturers are not fully aware of the potentialdesign improvements possible with higher delity aeroelasticmod-els.

Considering these facts, many advances in aerospace can be adaptedand used for wind turbine design.e majority of these design codescurrently used for wind turbines in academia and industry are of a rela-tively low delity (Ashuri and Zaaijer, 2007).ese design codes tendto use linear beam or modal models for the structure and blade elementmomentum to do an aeroelastic analysis. Higher delity models exist,but they are generally limited to mono-disciplinary analysis (Hansenet al., 2006).

e majority of the research in this thesis was performed at the Uni-versity of Michigan’s MDOlab.e MDOlab has extensive experiencein high delity optimization of aircra congurations (Haghighat et al.,2012; Mader and Martins, 2012). It also has a number of powerful tools,one of which is TACS (Kennedy, 2012) used as a structural solver. An-other is pyAeroStruct, an aerostructural managing code (Kenway et al.,2012).ese tools are designed with aircra congurations in mind, butcould potentially be adapted for usage in wind turbine design.

e goal of this thesis is to modify and adapt theMDOlab tools to doaerostructural analysis of wind turbines.is work is similar to previouswork done by Yan (2012).e work of Yan not been veried or validated.

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is means its results are somewhat questionable. His aerodynamic codeis also oversimplied.

is thesis uses AeroDyn, a wind turbine aerodynamic solver. Itis coupled with TACS, a high delity structural solver. e resultingaeroelastic code is compared and veried with a lower delity code.

e resulting code will be used for future research in e.g.

• Time domain analysis.

• (Multi-delity) optimization.

1.3 research questionsere are a number of research questions which will be addressed in thecourse of this thesis.e main questions are:

1. What required changes andmodications to the exisitingMDOlabcodes must be made to support wind turbine blade design?

2. How can AeroDyn be coupled with the existing MDOlab codes?

3. How do the results of the coupled code compare to a low-delitymodel?

1.4 research methodis thesis is part of a broader eort being undertaken at theUniversity ofMichigan. It’s important to limit the scope of this project to a reasonablesize for amaster thesis. Other researchers will investigate other aspects ofthis eort not included in this thesis. Research is limited to the followingitems:

• Changes to existing codes should be limited to minimize the nec-essary verication and validation.

• Limit the aeroelastic analysis to steady state solutions.

• Do not do any optimization, but when possible accommodate thisfor future research.

e rst step taken during this research is an extensive literaturestudy on wind turbine aerodynamic and structural theories widely usedto make aeroelastic codes.is includes the blade element momentum(BEM) for aerodynamic and classical lamination theory (CLT) for struc-tural disciplines.ese items are written down for use in the thesis.

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Aer becoming familiar with the basic theories, the AeroDyn in-terface is investigated. is is done by learning Fortran, studying themanual (Laino and Hansen, 2002; Jonkman and Jonkman, 2013) andlearning to write Fortran test programs.

Once familiar with AeroDyn and its interface at the Fortran level,research is done into wrapping the code to the Python level. is isachieved by learning Python and learning to wrap Fortran codes usingF2py. Once familiar with wrapping simple codes AeroDyn is wrapped.

With a functional Python-wrapped AeroDyn interface, research isdone into the existing codes and how they can be interfaced. is isdone by talking to people from theMDOlab and studying code examples.e existing AeroDyn wrapper is then incorporated into these codesallowing for aeroelastic analysis.

Once generated, the code must be veried against an existing code.Applicable models were discussed while visiting NREL’s NWTC inGolden, CO.e model is then created using the tools developed by theMDOlab (when possible).

e verication model is then compared to the results. Any dif-ferences found between the codes are to be analyzed and physicallyexplained.

1.5 thesis outlineChapter 2 starts the literature research with an overview of the aerody-namic eects of wind turbines. It also gives a derivation of BEM, withan emphasis on the implementation of BEM in AeroDyn.

e structural considerations of wind turbines are reviewed in Chap-ter 3. is chapter concerns the structure of wind turbine blades andhow they are aected by certain variables.e section also gives a reviewof the classical lamination theory which is used in composite analysis. Italso briey discusses other composite models.

Chapter 4 discusses the interface of AeroDyn and the method inwhich it’s coupled to TACS. It also includes a brief overview of the aeroe-lastic solver and its convergence criteria.

Chapter 5 compares the composite model of TACS to ANSYS and theaeroelastic response with that of FAST. In order to analyze multi-laminacomposites, certain additions weremade to TACS and pyTACS to get theconstitutive class working.e rest of the chapter discusses the choice

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of a verication model and the results of the model compared to thereference model.

Results and future work are discussed in Chapter 6. Particular em-phasis is placed on the role of the code in wind turbine analysis and howit compares to the landscape of current analysis codes.

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PART I

THEORETICAL BACKGROUND

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WIND TURBINEAERODYNAMICS

is chapter discusses the aerodynamic design of wind turbines. erst part of the chapter discusses the aerodynamic considerations ofwind turbines. e nal part of the chapter discusses Blade ElementMomentum (BEM). BEM is the method used in this thesis to calculatethe aerodynamic loads acting onto the structure.

2.1 wind turbine aerodynamicdesign

2.1.1 e sitee amount of kinetic energy contained in wind depends on the velocitysquared. It is therefore imperative to the wind turbine design to have agood idea of the wind speed distribution on the site.

An ideal site would have a constant wind speed that allows for aconstant energy production. In practice, however, there are times inwhich there’s hardly any wind and other times in which there are stronggusts. Designing a wind turbine to be able to extract the energy of thesestrong gusts is usually not economically feasible.

Wind speed varies by time of day as well as due to the terrain.eviscous friction causes a phenomenon known as wind shear. Wind sheardescribes the wind velocity distribution in the earth’s boundary layer.e friction on the earth’s surface causes the wind speed to be muchlower near the surface. e eect depends on the surface roughness.More information on this phenomenon can be found in Elliott andCadogan (1990).

2.1.2 Number of bladese more blades a design incorporates, the less each blade has to extract.is means that the loads on the blade are smaller and the rotation speedis lower.is reduces structural loads and allowing for lighter blades.

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A consequence of this is that the blade has to become narrower inorder to maintain aerodynamic eciency.e term for the ratio of bladearea to annular area is called the solidity.e optimum ratio is for theblade area to be just a few percent of the total annular area. In practice itis too expensive to produce the very narrow blades needed for 4+ bladeturbines.

Other considerations in the selection of blades are the dynamics.e turbine experiences a shock twice per rotation, this occurs when theblades are aligned with the tower. At this instance the forces are at thehighest on the blade pointing up, and at its lowest on the blade pointingdown.is shock has been reduced by some manufacturers by using ateetering pin.is pin gives the blade some freedom to move, reducingthe shock on the structure.ree-bladed turbines are also reported tobe less visually disturbing than two-bladed turbines (Gurit, 2013) and(Burton et al., 2011, sec. 6.5.6). More details on the performance eectsof the number of blades can be found in (Burton et al., 2011, sec. 6.5).

2.1.3 Blade designAirfoil

Horizontal axiswind turbines (HAWT) extract energy fromwind throughtorque generated from airfoils. HAWT extract this torque by utilizingthe li force of an airfoil, this allows the turbine to extract more energythan drag-based devices.ese airfoils have a shape which is similar tothat of airplanes. e li generated by the airfoil generates a componentwhich is translated to a torque.is is shown in gure Figure 2.2. Moreinformation can be found in (Johnson, 2001, ch. 4)

WE Handbook- 2- Aerodynamics and Loads

There is, unfortunately, also a retarding force on the blade: the drag. This is the force parallel to the wind flow, and also increases with angle of attack. If the aerofoil shape is good, the lift force is much bigger than the drag, but at very high angles of attack, especially when the blade stalls, the drag increases dramatically. So at an angle slightly less than the maximum lift angle, the blade reaches its maximum lift/drag ratio. The best operating point will be between these two angles.

Since the drag is in the downwind direction, it may seem that it wouldn’t matter for a wind turbine as the drag would be parallel to the turbine axis, so wouldn’t slow the rotor down. It would just create “thrust”, the force that acts parallel to the turbine axis hence has no tendency to speed up or slow down the rotor. When the rotor is stationary (e.g. just before start-up), this is indeed the case. However the blade’s own movement through the air means that, as far as the blade is concerned, the wind is blowing from a different angle. This is called apparent wind. The apparent wind is stronger than the true wind but its angle is less favourable: it rotates the angles of the lift and drag to reduce the effect of lift force pulling the blade round and increase the effect of drag slowing it down. It also means that the lift force contributes to the thrust on the rotor.

The result of this is that, to maintain a good angle of attack, the blade must be turned further from the true wind angle.

Apparent wind angles

Blade at low, medium & high angles of attack

Figure 2.1: Torque extraction through li, from Gurit (2013).

e ideal airfoil shape is very thin, about 10-15% of the chord length(Gurit, 2013). In practical applications the airfoil shape might have to

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be adapted for structural reasons or in case of airplanes for fuel andincorporation of control surfaces. In wind turbines the bending mo-ment is the highest near the root of the blade. is means that moststructural adaptations have to take place at the root of the blade. Inwind turbines with active pitch control1 1. Active pitch control means that

the blade can rotate along the root-tip direction.is allows the bladeto actively change its angle of attack,reducing power ouput and otherforces.

the root of the blade oen has acircular connection to the hub of the turbine.

Twist

e angle of attack for the airfoil is relative to the apparent wind speedwhich is a combination of the incoming wind and the motion of thewind turbine blade.

WE Handbook- 2- Aerodynamics and Loads 4

Twist

The closer to the tip of the blade you get, the faster the blade is moving through the air and so the greater the apparent wind angle is. Thus the blade needs to be turned further at the tips than at the root, in other words it must be built with a twist along its length. Typically the twist is around 0-20° from root to tip. The requirement to twist the blade has implications on the ease of manufacture.

Blade section shape

Apart from the twist, wind turbine blades have similar requirements to aeroplane wings, so their cross-sections are usually based on a similar family of shapes. In general the best lift/drag characteristics are obtained by an aerofoil that is fairly thin: it’s thickness might be only 0-5% of its “chord” length (the length across the blade, in the direction of the wind flow).

Blade twist

Typical aerofoil shapes offering good lift/drag ratio

Figure 2.2:e apparent wind speed at sections of the blade, from Gurit (2013).

e combination of the motions means that the apparent wind direc-tion diers along the length of the blade.is is usually in the range of10-20°. It should be noted that the twist of the blade signicantly aectsthe manufacturing costs of a blade.

Planform shape

To get the most out of the incoming air ow it is necessary to slow downthe air as uniformly as possible across the annular area. Having air slowdown too much will cause turbulence which leads to wasted energy. Notslowing it down enough reduces the converted energy.

e tip of the blade passes through more volume of air compared tothe root of the blade.e apparent air speed at the tip is also a lot higher.In order to get a uniform slowing down of the air it is necessary for the

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tip of the blade to extract more energy than the root of the blade.

e li force is proportional to the air velocity squared22.e li force can be describedas FL = 1

2 ρ∞V 2∞SCL in which S isthe planform area (Anderson, 2001,

sec. 1.5).

. It is alsoproportional to the chord length.is translates to an optimal taperingedge which is approximately inversely proportional to the radius.isis shown in gure Figure 2.3. Most designs use a linear approximationof the tapering edge.

WE Handbook- 2- Aerodynamics and Loads

Because the tip of the blade is moving faster than the root, it passes through more volume of air, hence must generate a greater lift force to slow that air down enough. Fortunately, lift increases with the square of speed so its greater speed more than allows for that. In reality the blade can be narrower close to the tip than near the root and still generate enough lift. The optimum tapering of the blade planform as it goes outboard can be calculated; roughly speaking the chord should be inverse to the radius. So if the chord was 2m at 0m radius, it should be 0m at m radius. This relationship breaks down close to the root and tip, where the optimum shape changes to account for tip losses.

In reality a fairly linear taper is sufficiently close to the optimum for most designs, structurally superior and easier to build than the optimum shape.

Rotational speed

The speed at which the turbine rotates is a fundamental choice in the design, and is defined in terms of the speed of the blade tips relative to the “free” wind speed (i.e. before the wind is slowed down by the turbine). This is called the tip speed ratio.

High tip speed ratio means the aerodynamic force on the blades (due to lift and drag) is almost parallel to the rotor axis, so relies on a good lift/drag ratio. The lift/drag ratio can be affected severely by dirt or roughness on the blades.

Optimum blade planformFigure 2.3: Typical planform shape of turbine blade, from Gurit (2013).

2.1.4 Rotational speede rotational speed chosen for a design is characterized by the wing tipratio. is is the ratio of the velocity of the wing tip to that of the freewind speed.

A high tip speed ratio implies that the li and drag are nearly parallelto the rotor axis.is makes the turbine’s eciency highly dependenton its li/drag ratio. It also makes the blade’s eciency sensitive to dirtor other roughness on the blade.Nevertheless, a high tip speed ratio is preferred over a low tip speed

ratio.is is due to a number eects33. Çetin et al. (2005), Gurit (2013)and Ragheb (2011)

:

• e torque generated onto the turbine has an equal and oppositeeect on the wind.e air swirls around in the direction oppositethat of the blades. e swirl represents lost potential energy. Alow tip speed ratio means higher torque, and therefore larger swirllosses44.is is also called the whirlpool

eect..

• Another eect is called tip losses.ese are caused by incoming airescaping around the blade tip. As described in the previous sectionthe optimal chord length at the tip depends on the apparent windspeed. Which itself depends on the rotation speed. A low tip

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speed ratio requires a large torque which in turn requires a longerchord at the tip.e result is that the tip represents a larger ratioto the total blade area resulting in larger losses.

• A smaller tip speed ratio results in an increase of torque. isrequires a supporting structure which can support the higherloads.

ere are of course limits to the tip speed. A notable design consid-eration is noise, which scales with approximately the h power to thetip rotation speed (Burton et al., 2011, sec. 6.4.4). Erosion and damagefrom rain, sand, bird strikes also becomes a problem at high tip speeds.A typical turbine design uses a wind tip ratio of 7-10.is means that fora design wind speed of 15 m/s the tip speed is about 120 m/s (432 km/h).

2.1.5 Power controlWind turbines are designed to have a rated power.is is determinedby weighing the cost of the structure to the increase in converted energy.At higher wind speeds the turbine has to be controlled to limit the forcesacting on the turbine. To ensure that the wind turbine does not suerdamage at high wind speeds it is necessary to regulate extraction of windenergy.

e regulation is usually achieved by changing the angle of attackthrough active pitch control. is also allows the blade to be be adjustedat low speeds at which the apparent wind velocity is in the rotor axisdirection. e term for adjusting the angle of attack at wind speedsabove the rated wind speed is called feathering.

During a storm a blade may be feathered to stop the hub’s rotationentirely.ough some designs instead increase the angle of attack untilthe blade stalls. e latter method has the advantage that stalling can belimited to gusts thus making it useful in gusty winds.

A lot of research is being done in aeroelastic tailoring5 5. Aeroelastic tailoring reers todesigning components with theaeroelastic eects in mind.

. Aeroelastictailoring allows the blade to twist while being deected through bend-twist coupling.is could in the future allow for large scale passive-pitchcontrolled wind turbines (Veers et al., 1998).

2.2 blade element momentumtheory

e most commonly used method of calculating aerodynamic loadson wind turbine blades is the Blade Element Momentum Method. It isderived by combining the blade element method and combining it with

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forces this would require. Pressure energy can be extracted in a step-like manner,however, and all wind turbines, whatever their design, operate in this way.The presence of the turbine causes the approaching air, upstream, gradually to

slow down such that when the air arrives at the rotor disc its velocity is alreadylower than the free-stream wind speed. The stream-tube expands as a result of theslowing down and, because no work has yet been done on, or by, the air its staticpressure rises to absorb the decrease in kinetic energy.As the air passes through the rotor disc, by design, there is a drop in static

pressure such that, on leaving, the air is below the atmospheric pressure level. Theair then proceeds downstream with reduced speed and static pressure – this regionof the flow is called the wake. Eventually, far downstream, the static pressure in thewake must return to the atmospheric level for equilibrium to be achieved. The risein static pressure is at the expense of the kinetic energy and so causes a furtherslowing down of the wind. Thus, between the far upstream and far wake condi-tions, no change in static pressure exists but there is a reduction in kinetic energy.

3.2 The Actuator Disc Concept

The mechanism described above accounts for the extraction of kinetic energy but inno way explains what happens to that energy; it may well be put to useful work butsome may be spilled back into the wind as turbulence and eventually be dissipatedas heat. Nevertheless, we can begin an analysis of the aerodynamic behaviour ofwind turbines without any specific turbine design just by considering the energyextraction process. The general device that carries out this task is called an actuatordisc (Figure 3.2).Upstream of the disc the stream-tube has a cross-sectional area smaller than that

of the disc and an area larger than the disc downstream. The expansion of thestream-tube is because the mass flow rate must be the same everywhere. The massof air which passes through a given cross section of the stream-tube in a unit lengthof time is rAU, where r is the air density, A is the cross-sectional area and U is the

Figure 3.1 The Energy Extracting Stream-tube of a Wind Turbine

42 AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES

Figure 2.4: Steam turbine model of wind turbine, used with permission from Burtonet al. (2011)

the momentum method.e BEMmethod is also utilized by Aerodyn,which is the soware package used to calculate the aerodynamic loadsin the coupled soware. is section gives an overview of the origin,assumptions and methodology used in the BEMmethod.is sectionis based on the explanation of the theory in Burton et al. (2011); Ingram(2011); Hansen (2012); Moriarty and Hansen (2005).

2.2.1 e concept of an actuator discSteam turbine model

A wind turbine is designed to convert some of the kinetic energy ofair (wind) into useful mechanical energy.is causes the wind to losesome of its energy, resulting in a reduced airspeed. Assuming that theair passing through the rotor disc stays separate from the air that does, itis possible to draw the boundary layer of these two air ows as a steamturbine as shown in Figure 2.4.is is to say that the air ow diameterincreases downstream of the rotor. e boundary layer expands aergoing through the rotor disc because the air speed has decreased.edecrease is not instantaneous, as shown in Figure 2.5. As air gets closerto the actuator disc the pressure increases and the air slows down. Aerpassing through the actuator disc there’s a sudden drop in pressure, sincethe atmospheric pressure aer the rotor disc is the same as in front of it,the boundary layer is further expanded and the air speed is decreasedeven more.

Since the velocity decreases it is possible to state that from conserva-tion of mass and Figure 2.566.

m Mass ow [kg/s]

ρ Air density [kg/m3]

A Flow area [m2]

U Flow velocity [m/s]

:

m = ρA∞U∞ = ρAdUd = ρAwUw

In which∞ denotes the conditions upstream, d at the disc and w farinto the wake (downstream).

14

flow velocity. The mass flow rate must be the same everywhere along the stream-tube and so

rA1U1 ¼ rAdUd ¼ rAwUw (3:1)

The symbol 1 refers to conditions far upstream, d refers to conditions at the discand w refers to conditions in the far wake.

It is usual to consider that the actuator disc induces a velocity variation whichmust be superimposed on the free-stream velocity. The stream-wise component ofthis induced flow at the disc is given by aU1, where a is called the axial flowinduction factor, or the inflow factor. At the disc, therefore, the net stream-wisevelocity is

Ud ¼ U1(1 a) (3:2)

3.2.1 Momentum theory

The air that passes through the disc undergoes an overall change in velocity,U1 Uw and a rate of change of momentum equal to the overall change of velocitytimes the mass flow rate:

Rate of change of momentum ¼ (U1 Uw)rAdUd (3:3)

The force causing this change of momentum comes entirely from the pressuredifference across the actuator disc because the stream-tube is otherwise completelysurrounded by air at atmospheric pressure, which gives zero net force. Therefore,

( pþd pd )Ad ¼ (U1 Uw)rAdU1(1 a) (3:4)

To obtain the pressure difference ( pþd pd ) Bernoulli’s equation is applied sepa-rately to the upstream and downstream sections of the stream-tube; separate equa-

Stream-tube

Velocity

Pressure

Actuator disc

p∞p∞

U∞Uw

p+d

Ud

p–d

Velocity

Pressure

Figure 3.2 An Energy Extracting Actuator Disc and Stream-tube

THE ACTUATOR DISC CONCEPT 43

Figure 2.5: Pressure and velocity throughout the steam turbine. From Burton et al.(2011), used with permission.

Introducing the axial ow induction factor or inow factor a whichis dened as:

a = U∞ −Ud

U∞

(2.1)

It can easily be shown that velocities Ud can be expressed as:

Ud = U∞(1 − a) (2.2)

2.2.2 Momentum theoryLinear momentum

As mentioned in the previous section the air speed decreases aer pass-ing through the actuator disc. UsingNewton’s second law7 7. f = dp

dt , in which p is the mo-mentum p = Mv. M is an absolutequantity of mass [kg].

, the associatedforce can be calculated by calculating the change in momentum per unitof time (M = m).

Change in momentum = (U∞ −Uw)´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

∆v

ρAdUd´¹¹¹¹¹¹¸¹¹¹¹¹¹¶

m

(2.3)

e force associated with this change in momentum can be attributedto the pressure dierence between the front and back of the actuatordisc. is is shown in Figure 2.5 as p+d and p−d . Combining this withEquation (2.2) it is possible to write:

(p+d − p−d)Ad = (U∞ −Uw)ρAdU∞(1 − a) (2.4)

Calculating p+d − p−d is possible by using the Bernoulli equation whichstates: 1

2ρU2 + p + ρgh = constant

e Bernoulli equation represents an energy balance within the uid,the assumption is made that under steady conditions provided no workis done by or on the uid the energy does not change. e balanceupstream can be written as:

12

ρ∞U2∞ + p∞ + ρ∞gh∞ = 1

2ρdU2

d + p+d + ρd ghd (2.5)

15

By further assuming that the ow is incompressible88. Pressure and temperature changesfor speeds below Mach 0.3 can beconsidered negligible (Shaughnessyet al., 2005, p. 80). Tip velocitiesmay a bit higher than this, this

could introduce an error for highrotational speeds (Vermeer et al.,

2003).

and completelyhorizontal it is possible to equate ρ∞ = ρd and h∞ = hd . is allowsEquation (2.5) to be rewritten to:

12

ρU2∞ + p∞ = 1

2ρU2

d + p+d (2.6)

For downstream: 12

ρU2w + p∞ = 1

2ρU2

d + p−d (2.7)

Subtracting Equations (2.6) and (2.7) gives:

(p+d − p−d) =12

ρ(U2∞ −U2

w) (2.8)

Which when combined with Equation (2.4) gives:

12

ρ(U2∞ −U2

w)Ad = (U∞ −Uw)ρAdU∞(1 − a)

Which can be solved to show that:

Uw = U∞(1 − 2a) (2.9)

is shows that half of the axial loss of speed takes place upstream ofthe actuator disc, and half downstream.

Combining Equations (2.4) and (2.9) allows the following expressionfor the force on the actuator disc:

Fx = (p+d − p−d)Ad = 2ρAdU2∞a(1 − a) (2.10)

is expression can be used to calculate the thrust caused by the changeof momentum in the air.e unknowns have been reduced to the axialinduction factor a.

Angular momentum

e torque exerted by the wind turbine generates an equal and oppositetorque on the passing air. is causes the air in the wake to rotate inthe direction opposite of that of the rotor.is means that the air gainsangular momentum, and therefore has a velocity component which istangential to the rotation along with its axial component. is is shownin Figure 2.6.e increase in kinetic energy that is associated with thiswake is compensated by a lower pressure downstream (Burton et al.,2011, p. 47).

e tangential force component on an annular (ring-shaped) elementcan be described by rst recounting that the following equations are

16

3.3.1 Wake rotation

The exertion of a torque on the rotor disc by the air passing through it requires anequal and opposite torque to be imposed upon the air. The consequence of thereaction torque is to cause the air to rotate in a direction opposite to that of the rotor;the air gains angular momentum and so in the wake of the rotor disc the airparticles have a velocity component in a direction which is tangential to the rotationas well as an axial component (Figure 3.4).

The acquisition of the tangential component of velocity by the air means anincrease in its kinetic energy which is compensated for by a fall in the staticpressure of the air in the wake in addition to that which is described in the previoussection.

The flow entering the actuator disc has no rotational motion at all. The flowexiting the disc does have rotation and that rotation remains constant as the fluidprogresses down the wake. The transfer of rotational motion to the air takes placeentirely across the thickness of the disc (see Figure 3.5). The change in tangentialvelocity is expressed in terms of a tangential flow induction factor a9. Upstream ofthe disc the tangential velocity is zero. Immediately downstream of the disc thetangential velocity is 2ra9. At the middle of the disc thickness, a radial distance rfrom the axis of rotation, the induced tangential velocity is ra9. Because it isproduced in reaction to the torque the tangential velocity is opposed to the motionof the rotor.

An abrupt acquisition of tangential velocity cannot occur in practice. Figure 3.5shows the flow accelerating in the tangential direction as it is ‘squeezed’ betweenthe blades; the separation of the blades has been reduced for effect but it is theincreasing solid blockage that the blades present to the flow as the root is ap-proached that causes the high values of tangential velocity close to the root.

3.3.2 Angular momentum theory

The tangential velocity will not be the same for all radial positions and it may wellalso be that the axial induced velocity is not the same. To allow for variation of bothinduced velocity components consider only an annular ring of the rotor disc whichis of radius r and of radial width r.

The increment of rotor torque acting on the annular ring will be responsible for

r

Ω

Figure 3.4 The Trajectory of an Air Particle Passing Through the Rotor Disc

ROTOR DISC THEORY 47

Figure 2.6: Notation of rotating annular stream as well as trajectory of air particle.Reused with permission from Burton et al. (2011)

valid for an annular shape9 9.

I Moment of inertia [kg⋅m2]M Mass [kg]

ω Wake rotational velocity [rad/s]

T Torque [N⋅m]L Angular momentum [N⋅m⋅s]

:

I = Mr2

L = Iω

T = dLdt

T = dIωdt

= d(Mr2ω)dt

= dMdt

r2ω

So for a small annular element the torque can be described as (Manwellet al., 2002, eqn. 3.28):

dT = dMωr2 = dmωr2 (2.11)m = ρAUd (2.12)m = ρ2πrdrUd (2.13)

→ dT = ρ2πr drUd ωr2 = ρUd ωr22πr dr (2.14)

Introducing the angular induction factor a′ as:

a′ = ω2Ω

(2.15)

And using Equation (2.2) it is possible to write the tangential forcecomponent as:

dT = 4a′(1 − a)ρVΩr3π dr (2.16)

is gives an expression for the moment acting on the actuator disk.is moment is due to the change in angular momentum (assumed non-existent upwind). e increased angular momentum is caused by theblade moving through the air, this is shown in Figure 2.7.

2.2.3 e power coecientSince power equals force times velocity it is possible to use Equations (2.10)and (2.10) to get to the following expression:

Power = FUd = 2ρAdU 3∞a(1 − a)2 (2.17)

17

P1: OTA/XYZ P2: ABCJWST051-03 JWST051-Burton March 14, 2011 13:39 Printer Name: Yet to Come

ROTOR DISC THEORY 45

r

W

Figure 3.4 The trajectory of an air particle passing through the rotor disc

the disc (see Figure 3.5). The change in tangential velocity is expressed in terms of a tangentialflow induction factor a′. Upstream of the disc the tangential velocity is zero. Immediatelydownstream of the disc the tangential velocity is 2rΩa′. At the middle of the disc thickness,a radial distance r from the axis of rotation, the induced tangential velocity is rΩa′. Becauseit is produced in reaction to the torque the tangential velocity is opposed to the motion ofthe rotor.

An abrupt acquisition of tangential velocity cannot occur in practice and must be gradual.Figure 3.5 shows the flow accelerating in the tangential direction as it is ‘squeezed’ betweenthe blades: the separation of the blades has been reduced for effect, but it is the increasing

2a'Wr

pd-_ ρ(2a'Wr)2_

21

U•(1–a)

U•(1–a)U•(1–a)

a'Wr

Wr

pd+

Rotor motion

ρ

Figure 3.5 Tangential velocity grows across the disc thicknessFigure 2.7: Change in angular momentum. Reused with permission from Burtonet al. (2011)

e Power coecient is dened as:

Cp =Power12ρU 3

∞Ad(2.18)

e denominator represents the power available in the air in the absenceof the actuator disc. Combining Equations (2.17) and (2.18) Cp can berepresented as:

Cp = 4a(1 − a)2 (2.19)

e Betz limit

It is not possible to transform all of the kinetic energy contained inthe air into mechanical energy. e air has to ow to do work on theturbine, this is why a wall does not generate energy. To nd out what themaximum energy, is a maximum has to be found for Cp.is is done bytaking the rst derivative of Equation (2.19) and checking whether it’s amaximum by taking the second derivative.

dCp

da= 4(1 − a)(1 − 3a) = 0 (2.20)

d2Cp

da2= 8(3a − 2) < 0 (2.21)

Equation (2.20) has the solutions a = 1 and a = 13 . Checking these solu-

tions in Equation (2.21) it is clear that only a = 13 is a local maximum.

18

e corresponding value for Cp,max = 1627 ≈ 0.593.is is the theoretical

maximum achievable power coecient. It is named aer the Germanaerodynamicist Albert Betz who rst derived this limit in 1919 Betz(1966).

It would perhaps be a bit more fair to compare actual performance tothis limit in the form of Cp = Power extracted

Power available = Power extracted1627 (

12 ρU 3∞Ad)

.is is however,not the generally accepted denition of Cp.

ere are other factors oen used to quote energy eciency ofwind turbines such as the annual capacity factor dened in Burtonet al. (2011):

e annual capacity factor is dened as the energy generated during theyear (MWh) divided by wind farm rated power (MW) multiplied by thenumber of hours in the year. e determination of the capacity factor will,of course, be based on the best available information of wind speeds andturbine performance.

It should be noted that it is theoretically possible to break through theBetz limit by increasing the mass ow through the use of a diuser.ishas never been used in full-scale turbines andmight not be economicallyfeasible due to the required structural additions (Hansen, 2012, p. 41-44)and (Gilbert and Foreman, 1983).

2.2.4 Blade Element Methode blade element method makes the following key assumptions:

• ere are no aerodynamic interactions between blade elements.

• e forces on the blades are determined only by the li and dragcoecients.

is means that BEM (without corrections) is only valid for steady cases.Eects such as tower shadow10 10. Local wind speed drops due to

ow around the tower., dynamic stall and dynamic inow are

not taken into account. Its assumptions also become invalid as axialinduction factor a becoming greater than 0.4. In addition to this, tiploss eects due to the creation of vortices at the blade tip aren’t correctedin standard BEM.

e blade element method divides the blades into N rings, depend-ing on the size of the turbine somewhere between 10 and 20 rings areusually used.is is illustrated in Figure 2.8 Each segment has dierentforces acting on it due to having a dierent tangential velocities (Ωr),chord length (c) and twist angle (γ).e blade element method calcu-lates these per segment and assumes that the surrounding segments do

19

Figure 2.8: Blade element model, from Ingram (2011).

not aect the result.e overall performance characteristics are deter-mined by summing up all of the individual rings.

To nd the correct li and drag force it is necessary to compare theow to that of experimental data. Most airfoils are tested stationary inwind tunnels. To use this data on the wind turbine’s blades it is necessaryto calculate the relative air ow. To estimate the ow at the turbine theaverage speeds of the upstream and downstream are used.

Prior to the air entering the turbine it is not rotating, aer passingthrough the air is rotating at the rotational speed ω.e average rotatingspeed is therefore ω

2 .e blade itself is rotating at a rotational speed Ω.e average tangential velocity the blade experiences is therefore equal toΩr + ωr

2 = Ωr(1 = a′).is is shown in Figure 2.9. From Equations (2.2)and (2.15) and g. 2.9 it is clear that:

Ωr + ωr2

= Ωr(1 + a′) (2.22)

tan β = Ωr(1 + a′)U∞(1 − a) (2.23)

W = U∞(1 − a)cos β

(2.24)

e value for the angle of inow (β) will vary per element. By introducingthe local tip speed ratio λr dened as:

λr =ΩrU∞

(2.25)

Equation (2.23) can be further simplied to:

tan β = λr(1 + a′)1 − a

(2.26)

20

U!(1-a)W

r

rr

2

x

r

blade rotation

wake rotation

Figure 2.9: Flow onto turbine blade, slightly modied from Ingram (2011).

L

x

F

Fx

D

φ

Figure 2.10: Forces acting on the turbine blades, slightly modied from Ingram(2011).

21

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

0 10 20 30 40

CL o

r CD

incidence / [degrees]

CL

CD

Figure 2.11: NACA 0012 li and drag coecients, from Ingram (2011).

Forces on blades

e forces on a blade element can be calculated as:

dFθ = dL cos β − dD sin β (2.27)dFx = dL sin β + dD cos β (2.28)

In which the li and drag coecients are calculated as:

dL = CL12

ρW2c dr (2.29)

dD = CD12

ρW2c dr (2.30)

e li and drag coecients are based on experimental data, an exampleof lit and drag coecients of a NACA 0012 are shown in Figure 2.11egure shows that at an angle of attack of around 14 stall occurs. isindicates that the ow across the airfoil becomes increasingly separated,reducing li and increasing drag.

Combining Equations (2.27) to (2.30) it is possible to write:

dFx = B 12

ρW2(CL sin β + CD cos β)c dr (2.31)

dFθ = B 12

ρW2(CL cos β − CD sin β)c dr (2.32)

(2.33)

In which B represents the number of blades on the turbine.e torquedT being the tangential force Fθ multiplied by the radius:

dT = B 12

ρW2(CL cos β − CD sin β)cr dr (2.34)

22

Dening the local solidity σ ′ as:

σ ′ = Bc2πr

(2.35)

It is possible to write Equations (2.31) and (2.34) using Equations (2.24)and (2.35) to:

dFx = σ ′πρ U2∞(1 − a)2cos2 β

(CL sin β + CD cos β)r dr (2.36)

dT = σ ′πρ U2∞(1 − a)2cos2 β

(CL cos β − CD sin β)r2 dr (2.37)

ese expressions can be used to calculate the force and moment actingon an element.e only unknown in this equation is the axial inductionfactor.

2.2.5 Corrections made to the blade element methode blade element method is an engineering method, that is to say that itmakes a lot of assumptions. Some of the assumptions mademight not bevalid or induce an error in the nal results.is section introduces someof the corrections used in BEM theory and soware such as aerodyn.More information on various corrections can also be found in (Sant,2007, sec. 3.5)

Some models make an estimate of all errors using empirical data.An example from

Tip loss model

One of the eects not taken into account in the blade element methodis tip loss. e tip loss phenomenon describes the creation of helicalstructures in the wake. e strength of which increase exponentiallytowards the tip of the rotor blades.is eect plays a major role in theinduced velocity eld. An overview of various methods used to correctfor the tip loss eect can be found in Shen et al. (2005). Aerodyn usesthe method developed by Prandtl. It uses a correction factor Q on theinduced velocity eld. is factor is then applied for an angular ringsection using Equations (2.10) and (2.16) as:

Qtip =2πcos−1 e−(

B(R−r)2r sin ϕ ) (2.38)

dFx = 4QπrρU2∞a(1 − a) dr (2.39)

dT = 4Qa′(1 − a)ρVΩr3π dr (2.40)

Prandl’s method is used in a lot of engineering applications due to itseasy formulaic formulation (Moriarty and Hansen, 2005). It does haveits limitations, the most signicant is that the wake does not expand.

23

is has a signicant inuence on lightly loaded rotors. e accuracywhen compared to more comprehensive (and computationally moredemanding) methods is also adversely aected when dealing with lessthan three blades or high tip speed ratios.

An alternative tip loss correction method oered by Aerodyn is anempirical model based on the Navier-Stokes solutions described in Xuand Sankar (2002).ese solutions are based on a specic turbine designat a certain wind speed. It therefore has limited use.

Hub Loss

Similar to the tip loss blade also sheds a vortex at the hub of the turbine.Aerodyn corrects this in similar way to the tip loss correction using acorrection factor:

QHub =2πcos−1 eη , η = B(r − Rhub)

2Rhub sinϕ(2.41)

Glauert correction

When the induction factor a is greater than 0.4 the BEM theory is nolonger valid. is occurs when the turbine operates at high tip speedratios (lowwind speed, constant speed turbine). At this point the turbineenters the turbulent wake state (a > 0.5). When this happens the wakeow starts to propagate upstream, this is not physically realistic andviolates BEM assumptions. What actually happens is that air enters fromoutside the wake and generates more turbulence.

A correction for this eect was developed by Glauert (1935), it isbased on experimental measurements done on helicopter rotors underhigh induced velocities. It was originally adapted to correct the thrustof an entire rotor, but has been adapted for use on the local coecientsof the individual blade elements used in BEM.

Since the occurrence of a turbulent wake is oen accompanied witha large tip loss due to the induced velocities being high.ere are a num-ber of empirical correctionmethods available such as Burton et al. (2011);Manwell et al. (2002).ese methods have a problem with numericalinstability when used in conjunction with tip loss correction. is iswhy a new model has been developed by Buhl (2005). is model takesinto account the deviation from the momentum theory when a > 0.4.

As shown in gures in Buhl (2005); Moriarty and Hansen (2005)the model with this Glauert correction corresponds a lot better to theempirical data. It should be noted that this correction was not originallyintended to be used for this purpose, it is used BEM because no bettermodel exists.

24

Figure 2.12: Denition of the coordinates used in the skewed wake correction model.Taken fromMoriarty and Hansen (2005).

Correction for skewed wake

e BEM is designed to be used for axi-symmetric ows through theturbine. It is not uncommon however for wind turbines to operate atrelative (yaw) angles in relation to the incoming wind. is creates avelocity component in the plane of rotation.is leads to unsteady aero-dynamic forces and a gradient on the inow velocity along the rotordisk (Leishman, 2006) and (Sant, 2007, sec. 7.2). Most BEM correctionsfor skewed wake correct the axial induction factor to model the gradient.

Aerodyn uses a method developed by Pitt and Peters (1981) to correctthe induced velocity. It is based on a model developed by Glauert foruse with autogyro aircra (Glauert, 1928, p. 571-573).

e formulation used is:

askew = a (1 + 15πr32R

tanχ2cosψ) (2.42)

e denitions of χ and ψ can be seen in Figure 2.11. e wake skewangle χ can be calculated as:

tan χ =U∞(sin γ − a tan χ

2 )U∞(cos γ − a) (2.43)

Which is approximately (Burton et al., 2011):

χ = (0.6a + 1)γ (2.44)

is method scales the local axial induction factor with that of the aver-age induction factor of the rotor disk. is is an engineering method,and is based on experimental results.

e limitations of this approach is the assumption that the wake iscylindrical in shape.is is only a valid assumption for rotors under a

25

light load.ere us also no theoretical basis to use this with BEM theory.

Research papers Xudong et al. (1988); Snel et al. (1995) have showna better correlation with measurements using this approach, thoughsome suggest that the eect is overly compensated Eggers (2000). It issafe to say that despite its limitations it can be empirically stated that itimproves results.

Other factors

ere are a number of factors not taken into account in the currentversion of Aerodyn, these include (Moriarty and Hansen, 2005):

• e blade thickness in the local angle of attack.

• Cascade eects for high solidity wind turbines.

• Span wise gaps used in partial span pitch control.

e rst two can have a signicant eect on in-plane yaw forces near thehub. Hints are made in the Aerodyn documentation that corrections forthese eects may be added in a future version (Moriarty and Hansen,2005, p. 9). Spanwise gaps are not included because wind turbines don’ttend to use partial span pitch control.

2.2.6 Blade Element Momentum MethodIn the previous sections two methods have been introduced.

Blade element method A local method which can calculate the li anddrag forces on annular blade section using an induction factor.

Momentummethod A global method which calculates the inductionfactor as a function of thrust.

e blade element momentum theory combines these two theories. esolution for axial induction factor a is derived through iterations, thealgorithm used by Aerodyn (Moriarty and Hansen, 2005) is describedbelow.

BEM algorithm. To get a rst estimate for a an estimate is taken byassuming:

• Inow angle ϕ is small, sinϕ ≈ ϕ)

• e tangential induction factor a′ = 0

• e tip and hub loss corrections Q = 1

• Drag coecient CD = 0

• Li coecient CL = 2πa

26

• Angle of attack α = ϕ − β

Which gives an estimate (Moriarty and Hansen, 2005, eq. 20):

a = 14(2 + πλrσ ′ −

√4 − 4πλrσ ′ + πλ2r σ ′(8β + πσ ′))

Aer which the estimates are made for:

Inow angle

tanϕ =U∞(1 − a) + ve−op

Ωr(1 + a′) + ve−ip(2.45)

rust coecient

CT = σ ′(1 − a)2(CL cosϕ + CD sinϕ)sin2 ϕ

(2.46)

Tip lossesQtip =

2πcos−1 e−(

B(R−r)2r sin ϕ ) (2.47)

Hub lossesQhub =

2πcos−1 e−(

B(r−Rhub)2Rhub sin ϕ

) (2.48)

Total lossesQ = QtipQhub (2.49)

e next step depends on the load of the element. If the coecient ofthrust CT > 0.96 the element the axial induction factor is calculatedusing the modied Glauert correction described in Section 2.2.5. If thisis not the case standard BEM theory is used.

a =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

18Q − 20 − 3√

CT(50 − 36Q) + 12F(3Q − 4)36Q − 50 , CT > 0.96Q

(1 + 4Q sin2 ϕσ ′(CL cosϕ + CD sinϕ

)−1, CT < 0.96Q

(2.50)

e tangential induction factor a′ is calculated using:

a′ = (−1 + 4Q sinϕ cosϕσ ′(CL sinϕ − CD cosϕ)

)−1

(2.51)

With the nal correction being for the skewed wake of Equation (2.42):

askew = a (1 + 15πr32R

tanχ2cosψ)

27

is process is repeated from Equation (2.45) until the results con-verge to a nal value for a and a′ (and through blade element theory,the forces and moments). e options in Aerodyn to e.g. not take tipand hub losses into account are small modications to the algorithmabove. In the aforementioned case Q = 1 would be used.

Aerodyn does not directly couple the resulting forces to dynamicstall1111. Dynamic stall describes the

latency of the li coecient to re-spond to a changing angle of attack.

is means that the coecient be-comes a function of α in addition to

α.

routines. It uses static airfoil data which it then processes with adynamic stall routine before returning the output to the aeroelastic code.ere are some situations in which the used assumptions are not alwaysvalid (Moriarty and Hansen, 2005).

28

3

WIND TURBINE STRUCTURES

Wind turbines have shown a steady trend to increase in diameter overtime.is causes the centrifugal and gravitational forces associated withthe greater diameter to also increase greatly (Ashuri and Zaaijer, 2008).One way to reduce these forces is to reduce the mass of the blade.is isdone in modern turbines through the use of composite materials.issection describes the structural design of wind turbine blades. It alsodiscusses some of the theoretical background of composites.

3.1 structural designAs described in Chapter 2, the forces on the blade are the highest at thetip of the blade and are reduced proportionally with the radius.

e bending moment at the tip is zero, it increases to its maximumwhich is located at the root of the blade. It is therefore necessary for theroot of the blade to be the strongest part of the blade while the bladetip can be relatively weak.is is fortunate from an aerodynamic pointof view as this allows for a thin blade at the tip, resulting in a relativelysmall amount of drag at the part of the blade responsible for the mostconverted energy.e wind speed at the hub is a lot lower than that atthe tip. is means that the blade near the root has to be designed togenerate the li at a relatively low apparent air speed. is is usuallydone by increasing the thickness of the blade (which increases the li),as increasing the chord length is usually considered too costly.

In general the thickness needed for structural reasons is greaterthan that needed for aerodynamic reasons.is means that the use ofmaterials with a high tensile strength can directly inuence the potentialeciency of a turbine blade.

3.1.1 Box structuree structure of a blade consists of a thin shell airfoil.e airfoil itselfis compressed downwind and elongated in the upwind direction. Togive the blade some stiness, shear webs are included in the center ofthe airfoil.ese are connected to spar caps which are bonded to the top

29

and bottom of the airfoil.is is similar to the design of a steel I-beam.

ere are two common ways in which this is constructed:

1. e spar caps are built as part of the shell and a separate shearweb is bonded between the caps.

2. e spar caps and shear web are built together and bonded ontothe shells.

A turbine blade created using the latter method is shown in Figure 3.1.

Figure 3.1: Cross section of turbine blade.

Laminate orientation

Turbine blades are usually made using ber reinforced plastics (FRP).Wind turbine blades are very suitable for the use of ber reinforcedmaterials because the direction of the stresses are fairly predictable.Stresses in the shells tend to be in the root-tip direction. Stresses in theshear web are at a 45° angle to the chord direction. ese directions aregeneralized. Ideal ber orientation depends on exact blade geometryand load distribution.

3.1.2 FatigueRepeated variations in loads can lead to fatigue damage in structures(Sutherland, 1999).ough this is usually associated with metals, thisphenomenon also occurs with other materials such as FRP.

Load variations originate from wind shear, gusts, yaw error11. Term used for forces whichoccur when the turbine is notpointed directly into the wind.

and tur-bulence.e fatigue life depends on the amplitude of the load variations.Turbine blades are considered to be a component which should last the

30

entire life of a turbine (replacement would be too costly).

e actual allowable stress variation depends on the material used.Carbon ber in prepreg resin can withstand cycles of 50% ultimate loadfor 10 million cycles2 2. Approximately 20 years of use.while lower quality glass-reinforced resins can onlywithstand 20% of ultimate load variations.

ere are two main methods in designing for fatigue. e rstmethod is to limit the allowable load to be within the fatigue limit.is method is more restrictive than necessary since damage occursduring the variation of load instead of the application thereof.e sec-ond, more advanced method, is rainbow counting which assesses theload variation (damage caused) per cycle and takes the distribution ofthe load into account through what is called the Palmgren-Miner law.Additional unknowns and assumptions mean that an additional safetyfactor is required when assessing the fatigue load variation amplitude.

3.1.3 Blade shellemain function of the shell is to give the blade an airfoil shape.oughthe shell also gives the blade a lot of its torsional stiness.e blade doeshave a small amount of contribution in the ap-wise stiness, though themain contribution of the shell’s ber reinforcement is the the edgewisedirection.

e center of gravity of the blades is near the root of the blade.ismeans that the edgewise bending stress is also the highest near the root.Extra reinforcement is oen added to the trailing edge to strengthen thestructure to be able to withstand the extra bending stress.

FRP sheets only needs to be a few mm thick to get the the necessarystrength for its structural requirements. e large surface areas makethe stiness of the shells an issue. To increase the in-plane shell-stiness,two layers of FRP are usually glued together with a honeycomb structureor low density foam.

3.1.4 StinessMost HAWT turbines have the blade upwind.is is done because thesupport structure (usually a tube-shaped tower) generates a turbulentairow behind the tower which reduces the aerodynamic eciency. It’salso less likely to suer from fatigue loads caused by the osculations inloads that are associated with this phenomenon.

One of the consequences of an upwind placement of the wind turbineblades is that the blades deform towards the tower. A minimum towerclearance of 30% of the undeformed conguration is usually considered

31

the minimum when designing a wind turbine.is is particularly chal-lenging when upscaling the blade’s dimensions. e clearance as well asthe stresses have to be kept similar to smaller scales (Capponi et al., 2010).

One trick oen used when designing wind turbines is to give the un-loaded some prebending or coning.e blade then straightens out whenunder operating loads.is "trick" does increase the manufacturing costof a blade. It can also reduce the aerodynamic eciency at low windspeeds.

Care should be taken to avoid engine orders33. An integer multiple of the rota-tional frequency.

being the eigenfrequen-cies of the blade.is can be done by a smart distribution of mass andstiness along the blade.

3.1.5 Fiber

e two most common types of bers used in turbine design are glassand carbon bers.e latter is much stronger, but also much more ex-pensive.is means that carbon ber usage tends to be limited to largeturbine design and localized to high stress areas in which its propertiesare of most use.

It can be approximated that the mass increases with the third powerof the length while the aerodynamic performance only increases with thesecond power. From this fact it is easy to see that it is not economicallysound to create the largest possible wind turbine.e exact optimumsize evolves with the development of new technologies and the use ofmaterials such as carbon ber reinforcement.

3.1.6 Structural design process

e dominant force experienced by the blade is the liing force. isforce can be estimated fairly accurately using methods such as BEMexplained in Chapter 2. e structural design starts by the design ofthe spar caps and shear web. e aerodynamic eciency lowers at largethicknesses which may be required structurally.e design process istherefore a combination of material and manufacturing costs and aero-dynamic eciency.

e actual design is usually done iteratively or through the use ofmono-disciplinary optimization44. Mono-disciplinary optimiza-

tion optimizes the aerodynamicsand structure separately.e re-sulting optimum is usually not

found. More advenced techniquesoptimize the aeroelastic perfor-

mance (Martins and Lambe, 2013).

studies. Low delity aeroelastic toolsare also widely used. A design is checked more thoroughly with FE tomake sure that fatigue requirements are met, and that design assump-tions on loads are accurate for the nal design.

32

3.1.7 e choice for composite materials

Comparing various materials it is clear that glass and ber composites(GFRP & CFRP) have the best properties (Burton et al., 2011, p. 386-390).ese do however have anisotropic properties on a macroscopic scale.e low young’s modulus also makes it more likely for the material tofail due to buckling in thin structures than for it to fail due to yielding.is is exemplied by the panel stability factor5 5. E

(U CS)2 in which UCS is theuniaxial compressive strength.

. is value is around 0.1for glass and ber reinforced composites and around 0.8 for steel andaluminum. e fatigue life of composites is much better as shown in(Burton et al., 2011, table 7.1).is becomes ever more important as theblade diameter increases and gravity associated loads become larger.

A laminate material consists of several lamina which are layersaround 125 µm in thickness. A lamina consists of a bers which canbe oriented randomly or in a specic direction.is report focuses onoriented bers due to to their use in aerostructures. ese bers aresuspended in amatrix material, usually an epoxy resin.

3.2 mechanical analysis ofcomposites

ere are several methods that can simplify the anisotropic propertiesof composites and reduce it to a set of comparable isotropic properties(Ashuri et al., 2010a). It is still necessary to use a theory which can beused to analyze the stiness of the composite materials and its deforma-tion to a random load in an accurate way.

is section introduces classical lamination theory.is is a widelyused theory used to analyze thin composite structures. It’s based on theCauchy-Poisson-Kircho-Love plate theory. It also briey discusses therst order shear deformation (FSDT) theory, the method used in TACS.

3.2.1 Hooke’s law

e stress-strain relationship for a material can be described by thestiness matrixC and its inverse, the compliance matrix S. Both of thesematrices are of the dimensions 6 × 6.

[σ1 σ2 σ3 τ1 τ2 τ3 ]T = C [ε1 ε2 ε3 γ23 γ31 γ12]T (3.1)

[ε1 ε2 ε3 γ23 γ31 γ12]T = S [σ1 σ2 σ3 τ1 τ2 τ3 ]T (3.2)

33

Isotropic material

It can be easily shown that for an isotropic material these 36 constantsreduce 3 according to:

S11 =1E= S22 = S33 (3.3)

S12 = −νE= S13 = S21 = S23 = S31 = S32 (3.4)

S44 =1G

= S55 = S66 =2(1 + ν)2E

(3.5)

And all others being zero, there is no coupling between axial strain andshear strain.

Anisotropic materials

Anisotropic materials require more coecients to accurately describethe material. It can easily be seen that due to symmetry of the C and Smatrices (Jones, 1999, p. 57-63) the number of independent coecientsreduces to 21.

Monoclinic material

Amonoclinic material is a material that has a plane of symmetry.esymmetry allows a number of terms to be canceled out. If for instance.a material is symmetric on a plane on the center of the material thestiness matrix can be written as:

C =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

C11 C12 C13 0 0 C16C12 C22 C23 0 0 C26C13 C23 C33 0 0 C360 0 0 C44 C45 00 0 0 C45 C55 0

C16 C26 C36 0 0 C66

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Orthotropic material

A material is orthotropic when it has three axis of symmetry. In thiscase a stiness matrix can be reduced to:

C =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

C11 C12 C13 0 0 0C12 C22 C23 0 0 0C13 C23 C33 0 0 00 0 0 C44 0 00 0 0 0 C55 00 0 0 0 0 C66

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(3.6)

34

Transversely isotropic material

If amaterial is isotropic on a plane it is possible to reduce to ve constants.If the material is isotropic in direction 1, normal to plane 2-3 the stinessmatrix reduces to:

C =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

C11 C12 C12 0 0 0C12 C22 C23 0 0 0C12 C23 C22 0 0 00 0 0 C22−C23

2 0 00 0 0 0 C55 00 0 0 0 0 C55

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

3.2.2 Hook’s law and laminaUnidirectional lamina

Working under the assumption that the the thickness is very small whencompared to the other dimensions and no external forces work on thesurface it can be assumed that the lamina is under plane stress. Planestress is the state in which stress in the direction normal to the planeequal zero. is allows for the three dimensional stress analysis to bereduced to two dimensional stress-strain equations.

A unidirectional lamina can be considered an orthotropic material,using Equation (3.6) it is easy to derive:

ε3 = S13σ1 + S23σ2, γ23 = γ31 = 0

And thus the reduction to:⎡⎢⎢⎢⎢⎢⎣

ε1ε2γ12

⎤⎥⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎢⎣

S11 S12 0S12 S22 00 0 S66

⎤⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎣

σ1σ2τ12

⎤⎥⎥⎥⎥⎥⎦(3.7)

And stiness: ⎡⎢⎢⎢⎢⎢⎣

σ1σ2τ12

⎤⎥⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎢⎣

Q11 Q12 0Q12 Q22 00 0 Q66

⎤⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎣

ε1ε2γ12

⎤⎥⎥⎥⎥⎥⎦(3.8)

Which has the reduced stiness coecients:

Q11 =S22

S11S22 − S212(3.9)

Q12 = −S12

S11S22 − S212(3.10)

Q22 = −S11

S11S22 − S212(3.11)

Q66 =1

S66(3.12)

35

Two-dimensional lamina

Laminae tend to have its ber reinforcement at angles due to low sti-ness and strength.is is done by using a local coordinate system withcoordinates 1 and 2, at an angle θ from the global coordinate system x-y.

As is shown in (Kaw, 2005, Appendix. B) the relationship is:⎡⎢⎢⎢⎢⎢⎣

ε1ε2γ122

⎤⎥⎥⎥⎥⎥⎦= T

⎡⎢⎢⎢⎢⎢⎣

εxεyγx y2

⎤⎥⎥⎥⎥⎥⎦(3.13)

⎡⎢⎢⎢⎢⎢⎣

σxσyτx y

⎤⎥⎥⎥⎥⎥⎦= T−1

⎡⎢⎢⎢⎢⎢⎣

σ1σ2τ12

⎤⎥⎥⎥⎥⎥⎦(3.14)

In which the transformation matrix T equals:

T =⎡⎢⎢⎢⎢⎢⎣

cos2 θ sin2 θ 2 sin θ cos θsin2 θ cos2 θ −2 sin θ cos θ

− sin θ cos θ sin θ cos θ cos2 θ − sin2 θ

⎤⎥⎥⎥⎥⎥⎦,

T−1 =⎡⎢⎢⎢⎢⎢⎣

cos2 θ sin2 θ −2 sin θ cos θsin2 θ cos2 θ 2 sin θ cos θ

sin θ cos θ − sin θ cos θ cos2 θ − sin2 θ

⎤⎥⎥⎥⎥⎥⎦Combining this with Equation (3.8) it is possible to write:

⎡⎢⎢⎢⎢⎢⎣

σxσyτx y

⎤⎥⎥⎥⎥⎥⎦= T−1Q

⎡⎢⎢⎢⎢⎢⎣

ε1ε2γ12

⎤⎥⎥⎥⎥⎥⎦(3.15)

Using measurements it is possible to determine the engineering con-stants of the material using the compliance and stiness matrices (Kaw,2005, sec. 2.6).e actual relationship between stress and strain in thematerial direction requires a combination of the reduced stiness ma-trix for unidirectional lamina Q, transformation matrix T as well asthe Reuter matrix R6

6.e Reuter matrix equals⎡⎢⎢⎢⎢⎢⎣

1 0 00 1 00 0 2

⎤⎥⎥⎥⎥⎥⎦

e relationship of the strains in local and globalcoordinate systems can be expressed as:

[ε1ε2γ12

] = RTR−1⎡⎢⎢⎢⎢⎢⎣

εxεt

γx y

⎤⎥⎥⎥⎥⎥⎦(3.16)

Combing this expression for the local strains with Equation (3.15) allowsfor the relationship between the global stresses and strains:

⎡⎢⎢⎢⎢⎢⎣

σxσyτx y

⎤⎥⎥⎥⎥⎥⎦= T−1QRTR−1´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

Q

⎡⎢⎢⎢⎢⎢⎣

εxεt

γx y

⎤⎥⎥⎥⎥⎥⎦(3.17)

In whichQ is the reduced transformed stiness matrix.e expressionsfor the terms are not listed here, but can be found in many referencebooks such as (Kaw, 2005, p. 112) or (Jones, 1999, eqn. 2.85).

36

3.2.3 Laminate materialsA real structure will not consist of a single lamina, as previously men-tioned, they have a thickness of just 125 µm.e failing stress along thebers of a glass-epoxy lamina is around 130 MPa. For transverse loadsthis is much less. To maximize the strength to weight ratio it is necessaryto optimize the ply orientation of the laminae.

When discussing laminate materials it is normal to use laminate codeto describe the layers. For example a laminate with the following orderof orientations:

0−459090600

Has the laminate code of [0/-45/902/60/0].ere are further shorthandnotations for symmetry and alternating ply directions:

0−309090−300

Has the laminate code of [0/-30/90]s. Similarly for an uneven amountof layers:

0−3090−300

Has the laminate code of [0/-30/90]s.

MA 0MB −30MB 30MB 30MB −30MA 0

Has the laminate code of [0MA/∓ 30MB]s. Notice how ∓/± describes theorder and superscript the material type.

37

Macromechanical Analysis of Laminates

321

• The laminate is thin and is loaded only in its plane (plane stress) (

σ

z

=

τ

xz

=

τ

yz

= 0).• Displacements are continuous and small throughout the laminate

(

u

,

v

,

w

<<

h

), where

h

is the laminate thickness.• Each lamina is elastic.• No slip occurs between the lamina interfaces.

Consider a side view of a plate in the Cartesian

x–y–z

coordinate systemas shown in Figure 4.4. The origin of the plate is at the midplane of the plate,that is,

z

= 0. Assume

u

0

,

v

0

, and

w

0

to be displacements in the

x

,

y

, and

z

directions, respectively, at the midplane and

u

,

v

, and

w

are the displacementsat any point in the

x

,

y

, and

z

directions, respectively. At any point other thanthe midplane, the two displacements in the

x–y

plane will depend on theaxial location of the point and the slope of the laminate midplane with the

x

and

y

directions. For example, as shown in Figure 4.4,

FIGURE 4.3

Resultant forces and moments on a laminate.

FIGURE 4.4

Figure showing the relationship between displacements through the thickness of a plate tomidplane displacements and curvatures.

xx

y

(a) (b)

Nx

MxMxy

Myx

MyNxyNyxNy

y

z z

Mid-plane

Cross-sectionbefore loading

Cross-sectionafter loading

h/2

h/2z

z

x

κ

zα

AAʹ

uo

wo

1343_book.fm Page 321 Tuesday, September 27, 2005 11:53 AM

© 2006 by Taylor & Francis Group, LLC

κ

Figure 3.2: Displacements and their relationship to the midplane. From Kaw (2005),reproduced with permission.

3.2.4 Classical lamination theory

is section describes the behavior of laminates to axial forces and bend-ing/twisting moments. Classical lamination theorymakes the followingassumptions:

• A line perpendicular to the center remains so during deformation(γxz = γyz = 0).

• e laminate is thin and under plane stress (σz = τxz = τyz = 0)

• All displacements are small and continuous.

• e lamina are elastic.

• No slip occurs between lamina interfaces.

A diagram of displacements in shown in Figure 3.2. From this it caneasily be seen that:

α = ∂w0∂x

(3.18)

u = u0 − zα → u = u0 − z ∂w0∂x

(3.19)

→ v = v0 − z ∂w0∂y

(3.20)

38

From the denition of strain e.g. εx = ∂u∂x it is possible to combine

Equations (3.19) and (3.20) to redene the strain as:

εx =u0∂x

− z ∂2w0∂x2

(3.21)

εy =v0∂y

− z ∂2w0∂y2

(3.22)

γx y =∂u∂y

+ ∂v∂x

(3.23)

= ∂u0∂y

+ ∂v0∂x

− 2z ∂2w0

∂x∂y(3.24)

It is possible to separate the midplane strains (ε0x ε0y γ0x y]T) and mid-plane curvatures ([κx κy κx y]T) as:

⎡⎢⎢⎢⎢⎢⎣

εxεyγx y

⎤⎥⎥⎥⎥⎥⎦=

⎡⎢⎢⎢⎢⎢⎢⎣

∂u0∂x∂v0∂y

∂u0∂y + ∂v0

∂x

⎤⎥⎥⎥⎥⎥⎥⎦´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶[ε0x ε0y γ0x y]T

+z

⎡⎢⎢⎢⎢⎢⎢⎣

− ∂2w0∂x2

− ∂2w0∂y2

−2 ∂2w0∂x∂y

⎤⎥⎥⎥⎥⎥⎥⎦´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶[κx κy κx y]T

(3.25)

Combining this with Equation (3.17) it is possible to separate the indi-vidual contributions to the stress:

⎡⎢⎢⎢⎢⎢⎣

σxσyτx y

⎤⎥⎥⎥⎥⎥⎦= Q

⎡⎢⎢⎢⎢⎢⎣

ε0xε0yγ0x y

⎤⎥⎥⎥⎥⎥⎦+ zQ

⎡⎢⎢⎢⎢⎢⎣

κxκyκx y

⎤⎥⎥⎥⎥⎥⎦(3.26)

is shows that the stress distribution within a lamina is linear. estress distribution across the entire laminate depends on the orientationand composition of the laminae which comprise the laminate.

Consider a k layer laminate. Its lamina thickness is tk, the totalthickness:

h =n∑k=1

tk (3.27)

Measuring from the center of the plane, with down being positive thetop of the laminate has the coordinates h0 = − h

2 , similarly the bottomhas of the top lamina has the distance h1 = − h

2 + t1. For a graphicalrepresentation see Figure 3.3.

Integrating the stress across a cross section should give the resultingforces per unit of length in direction i (Ni):

Ni = ∫h2

− h2

σi dz (3.28)

Similarly the moment (Mi) can be calculated across the laminate thick-ness as:

Mi = ∫h2

− h2

σiz dz (3.29)

39

326 Mechanics of Composite Materials, Second Edition

loads applied to the laminate are known. This question is answered in thenext section.

4.3.4 Force and Moment Resultants Related to Midplane Strains and Curvatures

The midplane strains and plate curvatures in Equation (4.16) are theunknowns for finding the lamina strains and stresses. However, Equation(4.18) gives the stresses in each lamina in terms of these unknowns. Thestresses in each lamina can be integrated through the laminate thickness togive resultant forces and moments (or applied forces and moments). Theforces and moments applied to a laminate will be known, so the midplanestrains and plate curvatures can then be found. This relationship betweenthe applied loads and the midplane strains and curvatures is developed inthis section.

Consider a laminate made of n plies shown in Figure 4.6. Each ply has athickness of tk. Then the thickness of the laminate h is

(4.19)

Then, the location of the midplane is h/2 from the top or the bottom surfaceof the laminate. The z-coordinate of each ply k surface (top and bottom) isgiven by

Ply 1:

FIGURE 4.6Coordinate locations of plies in a laminate.

h0

123 h/2

h/2 z

Mid-plane

hn

hn−1

hk

hk−1 k−1

k+1tk

n

k

h1

h2 h3

h tk

k

n

==

∑ .1

hh

top surface0 2= − ( ),

1343_book.fm Page 326 Tuesday, September 27, 2005 11:53 AM

© 2006 by Taylor & Francis Group, LLC

Figure 3.3: Coordinates of the composite’s plies. From Kaw (2005), reproduced withpermission.

Using Equation (3.26) it is possible to rewrite this to:

⎡⎢⎢⎢⎢⎢⎣

NxNyNx y

⎤⎥⎥⎥⎥⎥⎦=

n∑k=1∫

hk

hk−1Q

⎡⎢⎢⎢⎢⎢⎣

ε0xε0yγ0x y

⎤⎥⎥⎥⎥⎥⎦dz +

n∑k=1∫

hk

hk−1Q

⎡⎢⎢⎢⎢⎢⎣

κxκyκx y

⎤⎥⎥⎥⎥⎥⎦z dz (3.30)

and similarly:

⎡⎢⎢⎢⎢⎢⎣

MxMyMx y

⎤⎥⎥⎥⎥⎥⎦=

n∑k=1∫

hk

hk−1Q

⎡⎢⎢⎢⎢⎢⎣

ε0xε0yγ0x y

⎤⎥⎥⎥⎥⎥⎦z dz +

n∑k=1∫

hk

hk−1Q

⎡⎢⎢⎢⎢⎢⎣

κxκyκx y

⎤⎥⎥⎥⎥⎥⎦z2 dz (3.31)

As shown in (Jones, 1999, p. 197-198) (Kaw, 2005, p. 330-331) this can beequated to six equations with six unknowns, which can be solved:Which can be solved.

⎡⎢⎢⎢⎢⎢⎣

NxNyNx y

⎤⎥⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎢⎣

A11 A12 A16A12 A22 A26A16 A22 A66

⎤⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎣

ε0xε0yγ0x y

⎤⎥⎥⎥⎥⎥⎦+⎡⎢⎢⎢⎢⎢⎣

B11 B12 B16B12 B22 B26B16 B22 B66

⎤⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎣

κxκyκx y

⎤⎥⎥⎥⎥⎥⎦(3.32)

⎡⎢⎢⎢⎢⎢⎣

MxMyMx y

⎤⎥⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎢⎣

B11 B12 B16B12 B22 B26B16 B22 B66

⎤⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎣

ε0xε0yγ0x y

⎤⎥⎥⎥⎥⎥⎦+⎡⎢⎢⎢⎢⎢⎣

D11 D12 D16D12 D22 D26D16 D22 D66

⎤⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎣

κxκyκx y

⎤⎥⎥⎥⎥⎥⎦(3.33)

In which the terms are:

Ai j =n∑k=1

[Q i j]k(hk − hk−1), i = 1, 2, 6; j = 1, 2, 6 (3.34)

Bi j =12

n∑k=1

[Q i j]k(h2k − h2k−1), i = 1, 2, 6; j = 1, 2, 6 (3.35)

Di j =13

n∑k=1

[Q i j]k(h3k − h3k−1), i = 1, 2, 6; j = 1, 2, 6 (3.36)

(3.37)

In general the steps to analyze a laminated composite are (Kaw, 2005,p. 332):

40

1. Find the reduced stiness matrix Q for each lamina using thematerial properties and Equations (3.9) to (3.12).

2. Determine the transformed reduced stiness matrix using the pre-vious result. is can be done through e.g. (Kaw, 2005, eqn. 2.104,2.137-2.138).

3. Determine the coordinates of the top and bottom of each ply.

4. Using the results of the previous two steps determine A B Dma-trices.

5. Use these matrices and applied forces to calculate the strains andcurvatures on the mid-plane using Equations (3.32) and (3.33).

6. Global strains can be found using Equation (3.26).

7. Find associated global stresses using Equation (3.17).

8. Find the local strains using Equation (3.13).

9. Find the local stresses using Equation (3.14).

3.2.5 First order shear theoryFirst order shear theory deformation theory7 7. Also known as Mindlin–Reissner

plate theory(FSDT) is an expanded

version of the Kircho-Love plate theory that allows for transversal stressand/or strain8 8.ese were assumed negligible in

Kircho-Love plate theory.(Carrera et al., 2011a).is means that the kinematic as-

sumptions used in Kircho-Love plate theory Equations (3.19) and (3.20)are transformed to:

u(x , y, z) = u0(x , y) + zΦx(x , y) (3.38)u(x , y, z) = v0(x , y) + zΦy(x , y) (3.39)

w(x , y, z) = w0(x , y) (3.40)

ese assumptions are illustrated in Figure 3.4. It is clear that Kircho-Love can be formulated so if it’s stated that: Φi = ∂w0

∂i .

ere are many dierent implementations of composites in FSDTe.g. (Yu, 2005; Carrera, 1996). A comparison of various implementationsof composite shells can be found in Carrera et al. (2011b).e methodused in TACS is derived in Kennedy (2012, sec. 3.1.1)

41

P1: TIX/XYZ P2: ABCJWST092-03 JWST092-Carrera July 14, 2011 13:1 Printer Name: Yet to Come

44 CLASSICAL PLATE/SHELL THEORIES

Figure 3.10 Undeformed and deformed geometries of a plate according tothe Reissner–Mindlin hypotheses.

The hypotheses of FSDT are clearly shown in Figure 3.10. The dis-placement field of FSDT has five unknowns (there were three for CLT): themidsurface displacements (u0, v0, w0) and the rotations of a transverse normalaround the x- and y-axes (y,x ). In the case of CLT, the rotations coincidewith the derivatives x = −∂w0/∂x and y = −∂w0/∂y. Only strain εzz iszero, therefore stresses σxz and σyz are different from zero. Figure 3.11 shows

Figure 3.11 FSDT: displacements u and v, and transverse shear stressesthrough the thickness direction z.

Figure 3.4: Kinematic assumptions in rst order shear theory, reproduced from(Carrera et al., 2011a) with permission.

42

PART II

IMPLEMENTATION OFAEROELASTIC FRAMEWORK

4

AEROELASTIC COUPLING ANDMODELING

is chapter describes the aeroelastic coupling and its components. Ageneral overview of the framework used for this thesis is found in Sec-tion 4.1. Section 4.2 discusses the controlling script and its conver-gence criteria. Section 4.3 briey discusses the interface with TACSto pyAeroStruct. An overview of the interface of AeroDyn is given inSection 4.4. Section 4.5 describes the creation of an interface to AeroDyn,allowing for aerostructural analysis of wind turbine blades.

4.1 overview of the aerostructuralframework

e aerostructural framework consists of a combination of various dif-ferent codes.eir relationship is illustrated in Figure 4.1. At the heartof the framework is pyAeroStruct.is code controls the processes ofthe coupled codes.rough a combination of standardized functions,the structural and aerodynamic solver are instructed in their calculation,displacements and other measures. For this thesis the coupled codes arepyTACS and pyAeroDyn. Sections 4.3 and 4.5 discuss these "modules".ey both are Python level function-standardized interfaces for pro-grams written in a lower-level programming language1 1. Lower level programming lan-

guages such as C++ and Fortranhave the advantage of being compu-tationally much faster.ey havethe disadvantage of being moredicult to program for than higherlevel languages such as Python.

.e emphasisis on pyAeroDyn since that has been written especially for this thesisby the author. is is why Section 4.6 goes into further details on theimplementations these functions.

4.2 pyaerostructpyAeroStruct is the aerostructural controlling script developed by theUniversity of Michigan’s MDOlab. It is designed to do high-delityaerostructural optimization, e.g. Kenway andMartins (2013). It supportsfour dierent aerostructural solvers:

• Non-linear block Gauss-Seidel method (NLBGS)

• Coupled Newton-Krylov method (CNK)

• Linear block Gauss-Seidel (LBGS)

45

Figure 4.1: Relationship between codes

• Coupled Krylov method (CK)

All of these methods are described in (Kenway et al., 2012).e resultsin this thesis were produced using the Non-Linear block Gauss Seidelmethod 22.e Non-linear block Gauss-

Seidel method is considered the tra-ditional and intuitive way of solvingan aerostructural problem (Martins

et al., 2005).

.is method runs the analyses sequentially, passing the loadsto the structure and passing the deformations to the aerodynamic loadsto the structure until the convergence criteria are met. pyAeroStruct isbuilt in a modular way. It uses classes for its solvers and uses functionswithin a class to, for example, get forces or displacements.is makesit relatively easy to implement new codes into the pyAeroStruct frame-work. e functions used for the coupling of AeroDyn are describedin Section 4.5. e relation of pyAeroStruct to TACS and AeroDyn isshown in gure Figure 4.1.

4.2.1 Convergence of the aeroelastic problem.e convergence criteria uses three dierent types of tolerances:

• EA - Aerodynamic tolerance

• ES - Structural tolerance

• EAS - Aerostructural solution tolerance

For convergence EA and ES are in the order of 10−1 and 10−3, respectively.EAS is in the range of 10−3 − 10−6 depending on the desired accuracy.e aerodynamic residual is calculated at the beginning of an iteration.

46

It uses the previous iteration’s structural displacements and aerodynamicloads3 3. Since AeroDyn uses BEM, which

does not use convergence the resid-ual passed to the aerostructuralsolver always equals 0 in the cur-rent implementation.

.e structural residual is calculated at the beginning of a structural so-lution using the previous iteration’s deformed shape with the currentstep’s force calculation.pyAeroStruct uses an implementation of Aitken acceleration (Irons andTuck, 1969) to vary the numerical damping dynamically.is increasesthe speed of convergence, it can have problems with large deformationsand can be manually overwritten.

4.2.2 Load and displacement transfere coupling between the structure and aerodynamics done by utilizingtwo dierent meshes:

• A structural mesh which is utilized by the FEA solver.

• An aerodynamic4 4. In the context of pyAeroDyn, thiswill be referred to as intermediatemesh to make it clear that it’s not aCFD solver.

mesh which is used by the aerodynamic solverto determine the loads on the structure.

Deformations to the structuralmesh are passed to the aerodynamicmesh.Similarly the loads are transferred using a conservative scheme. ework is based on the work of Brown (1997).e current implementationis within TACS and is described in Kennedy (2012, sec. 5.2.2).

4.3 pytacs , the tacs-pyaerostructinterface

TACS is aC++ code. To get a python interface to theC++ code, awrapperis created with the use of SWIG5 5. SWIG (Simplied Wrapper and

Interface Generator) is an opensource tool that allows users togenerate wrappers around librarieswritten in C or C++ with scriptinglanguages such as Python. Also see(Beazley, 2003)

. To use this code with pyAeroStruct itwas necessary to create an additional interfacing level between TACSand pyAeroStruct.is level is referred to as pyTACS and it consists oftwo les:

pyTACS.py is le contains everything needed to dene a linear anal-ysis.is includes load transfer, coordinate transfer etc.

pyTACS_Mesh.py is le contains the conguration of the structuralmodel.is includes the mesh, material assignments etc.

As part of this project pyTACS_Mesh.py was modied to work withcomposite materials. e original implementation only allowed forisotropic materials.

47

4.4 aerodyn interfaceis section describes the interface and usage of AeroDyn. AeroDyn isan implementation of the blade element momentum method used tocalculate the aerodynamic loads for wind turbines. It has been developedby NREL in Golden, CO. USA and continues to be developed with newversions coming out regularly.

4.4.1 Winde rst input to AeroDyn discussed is the wind. ere are two types ofwind les: the steady ow at hub height (HH) and full eld turbulentow (FF).

Hub height steady ow

Steady ow at the hub height (HH) uses data at certain time periods withintermittent data being interpolated from those points. An example le:

1 ! S t e ady wind f i l e2 ! Time Wind Wind Ver t . Hor iz . Ver t . LinV Gust3 ! Speed Dir Speed Shea r Shea r Shea r Speed4 0 .0 5 0 0 0 0 0 05 630 . 0 20 0 0 0 0 0 0Which scales the wind speed linearly from t = 0 to t = 630 from 5 to 20m/s. As described in (Laino and Hansen, 2002, table 7.2)e data pointsallow for a number of changes in the wind including shear, direction andintensity.e entries can be entered manually using measurement dataor assumptions.ey can also be generated using TurbSim, a stochasticwind ow generator developed by NREL.

Full eld dynamic ow

e other method is the full eld simulation of wind data. is methoduses a variety of simulation models to generate coherent turbulent struc-tures on a random background turbulence.e methods used by thesemodels are beyond the scope of this thesis. An interested reader can turnto B.J. Jonkman (2012); Kelley and Jonkman (2007) formore informationon the methods and their assumptions.

e FF les has wind speed data in a grid shape.e velocity of thepoints in the grid is a function of time. AeroDyn interpolates the windspeed of an element based on the wind speeds of the nearest elements.To use a FF wind le with AeroDyn the grid must be greater than thediameter of the turbine. Full eld dynamic ow les can be generatedusing TurbSim.e options and inputs used by TurbSim can be foundin B.J. Jonkman (2012).

48

4.4.2 Description of a conguration in AeroDyne conguration is described through a series of markers. ese mark-ers contain the position and orientation relative to the global coordinatesystem as well as their rst derivative in time.e global coordinate sys-tem is dened from the undeformed tower center line with the XY planebeing the ground6 6. Or water level, in oshore tur-

bines.. X points downwind, Y to the le of the positive X di-

rection andZ points up against gravity. Markers are dened in Fortran as:1 TYPE , PUBLIC : : Marker2 REAL( ReKi ) : : P o s i t i o n ( 3 )3 REAL( ReKi ) : : O r i e n t a t i o n ( 3 , 3 ) ! D i r e c t i o n c o s i n e ma t r i x4 REAL( ReKi ) : : T r a n s l a t i o nV e l ( 3 ) ! T r a n s l a t i o n a l v e l o c i t y5 REAL( ReKi ) : : Ro t a t i onVe l ( 3 ) ! R o t a t i o n a l v e l o c i t y6 END TYPE Marker

e orientations are in the form of a rotation matrix from the global co-ordinate system to the marker’s rotated coordinate system. e rotationmatrix has the inherent property ofDDT = I.

4.4.3 Initizalising AeroDyne rst part in initializing AeroDyn involves reading the AeroDyninitialization options dened as:

1 TYPE : : AD_In i tOpt ions2 CHARACTER( 1 0 2 4 ) : : ADInputF i l e ! Name o f t h e AeroDyn inpu t f i l e3 CHARACTER( 1 0 2 4 ) : : OutRootName ! Root name o f AeroDyn ou tpu t f i l e s4 LOGICAL : : WrSumFile ! Wri te an AeroDyn summary f i l e ?5 END TYPE AD_In i tOpt ions

e AeroDyn input le denes a number of items such as:

• e use of dynamic inow, stall, induction factor pitchingmoment,tip loss, hub loss models.

• Environmental constants such as air pressure and viscosity.

• Airfoil data names and assignments.

• e length, chord and twist of the airfoil.

Other options include the location to write output les and whether ornot to write a summary le.

Another factor that has to be dened is the undeformed initial con-guration. is is dened in the AeroConfig type in AeroDyn. Itsdenition is:

1 TYPE , PUBLIC : : AeroConf ig2 TYPE( Marker ) , ALLOCATABLE : : B l ade ( : )3 REAL( ReKi ) : : B l adeLeng th4 TYPE( Marker ) : : Hub5 TYPE( Marker ) : : Ro t o rFu r l6 TYPE( Marker ) : : N a c e l l e7 TYPE( Marker ) : : T a i l F i n8 TYPE( Marker ) : : Tower9 TYPE( Marker ) : : S u b s t r u c t u r e

10 TYPE( Marker ) : : Foundat ion

49

11 END TYPE AeroConf ig

e description of the type shows that the initial conguration is denedby the position and orientation of a number of points on the turbine.ese points are:

Blade e position of the intersection of the blade’s pitch axis and theblade’s root.e y-axis pointing towards the trailing edge of theblade and the z-axis pointing towards the tip of the blade.

Bladelength e total length of the undeected blade from the root tothe tip.

Hub is marker is located in the apex of rotation of the hub and ro-tates with the hub. It is oriented so that the x-axis is pointing inwhat would nominally be the downwind direction and the z-axisperpendicular to the hub’s centerline with the same azimuth asthe rst blade.

RotorFurl e rotorfurl denes the position of the teeter pin for twoblade rotors, or the apex of rotation for three-bladed rotors. Itdoes not rotate with the rotor but does move with the tower. Italso yaws with the nacelle and furls with the rotor.

Nacelle e nacelle marker is located at the intersection between thetower’s yaw axis and the yaw bearing. it is oriented with the x-axispointing downwind and the z-axis pointing up.

Tailn is is not currently used in AeroDyn.

Tower e tower marker is xed to the base of the tower. Its positionis on the intersection of the tower-base centerline and the tower-base connection to the substructure. e orientation is denedwith the x-axis being nominally downwind and the z-axis beingupwards.

Substructure is is not currently used in AeroDyn.

Foundation is is not currently used in AeroDyn.

e last input is an integer referred to as ErrStat.is integer returnsa non-zero ErrStat code if an error occurs during initializing. If theinitialization went correctly ErrStat remains zero.

AD_Init outputs the positions of the markers in the local coordinatesystem in the AllAeroMarkers type of format. AeroDyn needs themarkers to be relative to the global coordinate system.

50

AeroDyn User’s Guide, Version 12.50 December 24, 2002 13

21

PRINT or NOPRINT

-- The sixth and final entry on each line is an optional string flag to control whether output is written for that element to the element file (element.plt for YawDyn, reqelem.plt for ADAMS, [rootname].elm for FAST). If the string PRINT is found, output is written for that element. If PRINT is not found (or NOPRINT is found), element data is not written. AeroDyn responds to whichever string occurs first on each line. If no string is found on a line, no data is written for that element. The element file is not created if no element data is requested (i.e., PRINT is not found on any line). This file is useful when debugging the aerodynamics of a model. It contains data such as angle of attack, aerodynamic coefficients and forces, and induction factors for the selected elements. The output frequency to the element file will coincide with the frequency requested for other time series data in the dynamics simulation.

22

SINGLE or ReNum or

USER

-- This value is optional, and will likely be missing for most models (as it is in the example of Table 6.1). It determines whether you will use a SINGLE air-foil table (or single interpolation point between two airfoil tables), or multiple airfoil tables (or multiple interpolation points between tables) during the simu-lation. If your airfoil files have only one table in them, this line is ignored. You should use SINGLE if you want to use a single table from a file contain-ing multiple tables (or if you want to use a single interpolation point between two tables) for the duration of the simulation. Use ReNum if you want to move between multiple airfoil tables in your files based on Reynold’s Num-ber. AeroDyn is set up to handle those calculations, but be sure to provide an accurate viscosity (ID #16). Use USER only if you supply your own algo-rithm in the dynamics code to move between airfoil tables during a simulation. Most AeroDyn users will not need this option. It is useful for those who wish to implement control algorithms, or investigate roughness or Reynolds num-ber effects in AeroDyn.

Aerodynamic center

Element location(RELM)

(DR)Element

Span

Element Blade

Blade root (Hinge or hub center)

Figure 6.1 Sketch of the blade element geometry and nomenclature.

7.0 The Hub-Height Wind Data File

In order to run a simulation using AeroDyn, a wind input file is required. There are 2 types of wind input files. This section will discuss hub-height (HH) wind files. Section 8.0 addresses full-field (FF) turbu-lence files. The wind file name appears on line 8 of the aerodyn.ipt file. The name can be up to 100 char-acters long.

The HH wind file option in AeroDyn is flexible in that it allows several different parameters to be updated throughout the simulation, though re-programming is required to exercise some of these. How to repro-

Figure 4.2: Location of blade node.

Calculating loads

To calculate loads it is necessary to rst set themarkers in theAllAeroMarkerstype to the global coordinate system.e AllAeroMarkers type is de-ned as:

1 TYPE , PUBLIC : : A l lAeroMarker s2 TYPE( Marker ) , ALLOCATABLE : : B l ade ( : , : )3 TYPE( Marker ) , ALLOCATABLE : : Hub ( : )4 TYPE( Marker ) , ALLOCATABLE : : Ro t o rFu r l ( : )5 TYPE( Marker ) , ALLOCATABLE : : N a c e l l e ( : )6 TYPE( Marker ) , ALLOCATABLE : : Tower ( : )7 TYPE( Marker ) , ALLOCATABLE : : T a i l ( : )8 END TYPE Al lAeroMarker s

e denitions dier slightly from the AeroConfig type.

Blade e blade markers now describe the position of the node locatedin the center of the blade’s annular element.e location is illus-trated in Figure 4.2. e markers are oriented so that the z-axisis towards the blade’s tip and the y-axis along the chord towardsthe trailing edge. e other items are placeholders. e currentversion of AeroDyn is only able to calculate the loads on the blades.e calculation of loads on other components is planned for futurereleases of AeroDyn.

e loads

e loads are organized in an AllAeroLoads type. is is dened inFortran as:

1 TYPE , PUBLIC : : A l lAeroLoads2 TYPE( Load ) , ALLOCATABLE : : B l ade ( : , : )3 TYPE( Load ) , ALLOCATABLE : : Hub ( : )4 TYPE( Load ) , ALLOCATABLE : : Ro t o rFu r l ( : )5 TYPE( Load ) , ALLOCATABLE : : N a c e l l e ( : )6 TYPE( Load ) , ALLOCATABLE : : Tower ( : )7 TYPE( Load ) , ALLOCATABLE : : T a i l ( : )8 END TYPE Al lAeroLoads

In which the Load type is dened as:1 TYPE , PUBLIC : : Load2 REAL( ReKi ) : : Fo rce ( 3 )

51

3 REAL( ReKi ) : : Moment ( 3 )4 END TYPE Load

4.5 pyaerodyn , theaerodyn-pyaerostructinterface

is section describes the development of a python interface for Aero-Dyn. Section 4.5.1 describes some of the background behind the devel-opment of the wrapper. Section 4.5.2 discusses the relationship betweenpieces of codes and how they t together. It also gives a general overviewof the pyAeroDyn code. Section 4.6 discusses the implementation ofthe various functions used by pyAeroStruct and the assumptions madeduring their implementation.

4.5.1 Development of wrapperSince pyAeroStruct is written in Python and AeroDyn in Fortran, itwas necessary to write a wrapper which allows AeroDyn to be used at aPython level.

e easiest and most common method to wrap Fortran to Pythonis F2py77. F2py is a piece of code designed

to generate Python level interfacesfor Fortran code (Peterson, 2009).

. Compiling a Fortran module with F2py generates a Pythonmodule which can be imported to execute Fortran functions or subrou-tines. e main hurdle in the development of the AeroDyn wrapperwas the fact that the latest version of F2py does not support derivedtypes88. A derived type is a way to or-

ganize data in Fortran 90(+).It is similar to a data structurein C and shares some similari-ties with classes in C++. It canalso be compared to a class inPython without functionskk.

(Pletzer et al., 2008). As described in Section 4.4 these are usedextensively in AeroDyn to describe the initial conguration, currentconguration, options and loads.

To get around this problem, a wrapper within a wrapper was writ-ten which takes input arrays, strings, integers and scalars and assignsthem into the correct part of a derived type before calling the function.e derived structure containing the loads is then split o into separatearrays before being passed back to the python level.

e development of pyAeroDyn had a number of milestones:

1. Get results from AeroDyn (pure Fortran)

2. Make a Fortran function which calls AeroDyn without using de-rived types as an input.

3. Wrap input functions with F2py

4. Write pyAeroDyn (AeroDyn interface to pyAeroStruct)

52

e history of various iterations and methods of implementation of eachstep will not be discussed in this thesis. It instead has a focus on the nalimplementation. Needless to say, everything in the nal code has beenreprogrammed at least twice.

4.5.2 Structure of pyAeroDynAeroDyn is accessed by pyAeroStruct through functions inside pyAero-Dyn.ese functions call functions in the wrapped python level inter-face of AeroDyn. pyAeroStruct manages the dierent processes andmakes the calls to both pyAeroDyn and pyTACS which are then passedto their wrapped codes.

Overview of functions in pyAeroDyn

During the aerostructural analysis there are a number of functions thatpyAerostruct calls from the aerodynamic solver (pyAeroDyn).e onescurrently implemented in pyAeroDyn are:

Initialization During initialization:

• Options are loaded into memory.• e intermediate mesh is generated.• Rotation matrices are generated for blade orientation, shatilt and blade pre-cone.

• Arrays are allocated.• Initialization markers are calculated.• AeroDyn is initialized.

Solve During a solve step:

• e time is updated9 9. AeroDyn only recalculates aconguration if the time changes.So even though only steady-statecongurations are analyzed (for thisthesis) it is still necessary to changethe time.

.• Positions, orientations and velocities of each element arecalculated.

• Forces and moments acting on each element are calculatedby AeroDyn.

• Centrifugal forces are added10 10.e centrifugal forces are imple-mented as a pseudo-aerodynamicforce. Future work will integratethese directly into TACS.

.• e forces in the global reference frame are calculated inorder to calculate themoment generated by the aerodynamicforces.

getForces e getForces command returns the traction force per node.It does this by:

• Determining the distance from the aerodynamic center tothe leading and trailing edges.

53

• Calculating the forces distribution needed to generate thenecessary moment.

• Compensating for the added net force.

• Apply forces in the element coordinate system.

• Rotating forces to the global reference frame1111. AeroDyn’s forces are in the ele-ment coordinate system.

.

• Returning arraywith forces per node in the global coordinatesystem.

getSurfaceCoordinates is passes an array with all the node coordi-nates of the intermediate surface.

setSurfaceCoordinates is function changes the coordinates of thenodes of the intermediate surface.

getSurfaceConnectivity is passes an array of all the elements of theintermediate mesh. It consists of the nodes which constitute theelement.

writeSurfaceSolutionFile is function parses the current congura-tion of the aerodynamic mesh to a tecplot mesh le.

e details of these functions can be found in Section 4.6.

Coordinate systems

It is important to distinguish between the dierent coordinate systemsused in pyAeroDyn. When it’s important to distinguish between thereference frames, GL, BA or EL may be added in subscript to a vector ormatrix to denote its reference frame.

global coordinate system (GL) dened as being on the ground/meanwater level of the turbine with the x-direction pointing downwindand the z-direction pointing up. e y-direction pointing in sucha way to create a right-handed Cartesian coordinate system.isis the coordinate system used by AeroDyn for inputs.

Body attached coordinate system (BA) Used internally in pyAeroDyn.It rotates with the blade around the hub’s rotational axis and ispitched with its precone and/or sha tilt.

Element coordinate system (EL) is is the coordinate system inwhichthe forces andmoments of the elements are dened.e y-directionpoints towards the trailing edge.e z-direction points towardsthe blade tip.e x-direction is oriented in such a way to generatea right-handed Cartesian coordinate system.

54

4.6 pyaerodyn functionsis section goes into detail on the methods and procedures behind thefunctions in pyAeroDyn. It can be used to understand the pyAeroDyncode or to simply understand the methods pyAeroDyn uses.

4.6.1 InitializationBefore any calculation can be done it’s necessary to initialize AeroDyn.is is done automatically when an instance of the pyAeroDyn object12 12. To understand how pyAeroStruct

operates it is necessary to be famil-iar with object oriented program-ming. A good introductory bookavailable online is Budd (1997).

is created.

Loading of options

ere are a number of options dened in pyAeroDyn.ey have defaultvalues but can be overwritten when an instance of pyAeroDyn is created.Examples of options in pyAeroDyn are path to AeroDyn input le andangles for pre-cone and sha tilt.

Generation of intermediate mesh

As described in Section 4.2 the load and displacement transfer is donethrough the use of two separate meshes: the structural mesh and theintermediate mesh. e intermediate mesh has a pivotal role in pyAero-Dyn. It is used to calculate the position, orientation and velocities ofeach element as well as providing the surface to apply the traction force.Figure 4.3 features a plot of the intermediate mesh.e intermediate mesh is generated using airfoil shape data found in aninput le.e formatting of the input le is based on the formatting ofthe AeroDyn input le.e le contains the:

1. Spanwise position of the element’s center.

2. Twist of the element.

3. Spanwise length of an element.

4. Chordlength of the element.

5. Airfoil number (not used).

6. Relative chordwise oset of the blade center13 13. For a 2 m chord blade:0.5 would have the blade in themiddle of the y-axis0.25 would have the center ofthe blade oset 0.5 m to the y-direction.

.

7. Relative position of the aerodynamic center.

pyAeroDyn has been designed to use intermediate meshes which matchup to the AeroDyn elements. is means that only the last two columnshave to be added.e chordwise oset can be set to 0.5 throughout theblade if the blade doesn’t have any sweep.

55

Figure 4.3: Plot of the intermediate mesh. Figure 4.4:e intermediate mesh and the structural meshplotted at the same time.

e spatial coordinates of the intermediate mesh have been calcu-lated to give it the same position as the structural mesh.is is illustratedin Figure 4.4.

Each aerodynamic element is represented by four shell elements.e intermediate shell’s representation of an aerodynamic element canbe found in Figure 4.5.

Orientation18

01

23

4 5

67

z

y

x ~xloc = R~xglob

m~! (~! ~r)

Y

XFigure 4.5: Intermittent mesh representation of an aerodynamic element.

e undeformed position of nodes 4-5-6-7 is the same as nodes 0-1-2-3 with the addition of the element length in the z-direction. e

56

position of nodes 0-1-2-3 are calculated using the following equations14 14. In which:

yoff y-direction oset (sweep)

ϕ Twist of element

taf Airfoil thickness (assumed 0.5m)

c Chord length

pcenter Spanwise position of thecenter of the element.

Lelem Length of element.

:

n0 (LE) =⎡⎢⎢⎢⎢⎢⎣

−yoc sinϕ−yoc cosϕpcenter − Lelem

2

⎤⎥⎥⎥⎥⎥⎦(4.1)

n1 (top) =⎡⎢⎢⎢⎢⎢⎣

taf2 cosϕ − (yac − yo)c sinϕ− taf2 sinϕ + (yac − yo)c cosϕ

pcenter − Lelem2

⎤⎥⎥⎥⎥⎥⎦(4.2)

n2 (TE) =⎡⎢⎢⎢⎢⎢⎣

(1 − yo)c sinϕ(1 − yo)c cosϕ

pcenter − Lelem2

⎤⎥⎥⎥⎥⎥⎦(4.3)

n3 (bottom) =⎡⎢⎢⎢⎢⎢⎣

−taf2 cosϕ − (yac − yo)c sinϕ

taf2 sinϕ + (yac − yo)c cosϕ

pcenter − Lelem2

⎤⎥⎥⎥⎥⎥⎦(4.4)

e parameters are also illustrated graphically in Figure 4.6.e meshis stored as an n × 3 array containing the node coordinates and an m × 4array containing the connectivity of the shell elements15 15. For a mesh of n nodes and m

elements..

Figure 4.6: Graphical representation of the dimensions used for mesh generation.

Rotation matrices

As described in Section 4.5.2 pyAeroDyn uses multiple coordinate sys-tems.e coordinate system used internally by pyAeroDyn is the bladeattached coordinate system. AeroDyn’s markers have to be passed inthe global coordinate system. is means that the markers have to betransformed from the local to the global coordinate system.

e position of the origin of the coordinate system can be oset byadding the hub height of the z-component in the position vector.eother elements of the marker require a set of rotation matrices to takethe change of orientation into account.Four dierent types of rotation are taken into account in pyAeroDyn.

• Blade azimuth angle (θ) (x-rotation)

• Sha tilt angle (κ) (y-rotation)

57

• Precone angle (ψ) (y-rotation)

• Pitch angle (ξ) (z-rotation)

To get the rotation applied correctly the rotations are applied in thefollowing order:

R = RκRθRψRξ (4.5)

Fully written out this equals1616. In which si and c j are shorthandfor sin i and cos j.

:

R =⎡⎢⎢⎢⎢⎢⎣

cos κ 0 sin κ0 1 0

− sin κ 0 cos κ

⎤⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎣

1 0 00 cos θ sin θ0 − sin θ cos θ

⎤⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎣

cosψ 0 sinψ0 1 0

− sinψ 0 cosψ

⎤⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎣

cos ξ − sin ξ 0sin ξ cos ξ 00 0 1

⎤⎥⎥⎥⎥⎥⎦

=⎡⎢⎢⎢⎢⎢⎣

sξ(cκcψ − cθsκsψ)cθsκsψ −sξ(cκcψ − cθsκsψ) − cξsκsθ cκsψ − cψcθsκcθsξ cθcξ + sψsθsξ cψsθ

−cξ(cψsκ + cκcθsψ) − cκsθsξ sξ(cψsκ + cκcθsψ) − cκcξsθ −sκsψ − cκcψcθ

⎤⎥⎥⎥⎥⎥⎦(4.6)

is matrix will be used in future sections to switch reference frames.

Allocation of arrays

During initialization, a number of arrays are used during load solving.ese are assigned during initialization because they only have to beallocated once.

Initialization markers

During initialization it is necessary to pass the entries of an aerocongtype to AeroDyn (Section 4.4.3). Some of these entries are determinedby the options set during initialization, others are calculated using datafrom input les. As an example, the position of the blade’s root is calcu-lated as the mean position of four blade root nodes and rotated usingthe rotation matrix R Equation (4.6).

Initialization of AeroDyn

During the nal part of initialization the AeroDyn initialization functionis run. is loads certain positions such as the blade root and hubposition into AeroDyn’s memory. It also causes AeroDyn to read theAeroDyn input le and the corresponding wind input le. Once thishas been done pyAeroDyn is fully initialized and ready to perform loadcalculations.

4.6.2 Solving loads for a given congurationBefore node forces can be calculated it’s necessary to rst calculate theforces per aerodynamic element.

58

Time update

e rst step is to update the time17 17.is is necessary even for non-transient analyses because AeroDyndoes not recalculate loads unlessthe time has changed.

. pyAeroDyn is set up to automaticallyincrease the analysis time by 0.025 s per solving iteration.

Marker calculationOrientation

18

01

23

4 5

67

z

y

x ~xloc = R~xglob

m~! (~! ~r)

Y

XFigure 4.7: Nodes of a sample ele-ment (repeated from Figure 4.5).

In order to calculate the forces and moment on an element AeroDynneeds to know the following properties of each element:

• Position

• Orientation

• Translational velocity

• Rotational velocity

ese are calculated from the node positions of the intermediate meshat every calculation.is means that mesh deformations are taken intoaccount making this a coupled analysis.

e position is calculated using the average position of the aero-dynamic center nodes18 18. Nodes 1, 3, 5 & 7 in Figure 4.5.. is position is then premultiplied with therotation matrix R dened in Equation (4.6).

e orientation are dened in AeroDyn as a rotation matrix19 19. In which:

xel e element attached coordi-nate system.

D Element orientation matrix.

xgl e global coordinate system.

:

xel = DGL→ELxgl

is is also described in Section 4.4.2. e rotation matrix is createdby rst calculating the element to body attached rotation matrix andpremultiplying the projected vectors with the body attached to globalcoordinate system matrix R Equation (4.6). e orientation matrix iscalculated by calculating the directions of the element coordinate systemand creating a transformation matrix. Using the element illustrated inFigure 4.5 as an example:

1. Direction vector y is calculated as y = n0 − n2 + n4 − n6

2. Direction vector is normalized to unit length y = y∣y∣

3. x∗20 20. x∗ direction is used here.is isbecause this is only x if x∗ and y areperpendicular to each other.is istrue for the undeformed state, butnot necessarily for a deformed state.Also see Section 4.4.2.

is determined through x∗ = n1 − n3 + n5 − n7

4. Direction vector is normalized to unit length x∗ = x∗∣x∗∣

5. e z-direction is determined with z = x∗ × y

6. Direction vector is normalized to unit length21 21. To take into account the case ofx ≠ x∗

z = z∣z∣

7. x is calculated as x = y × z

59

8. Orientation matrix is assembled as:

DGL→EL = R [x y z] (4.7)

e body attached to element orientation matrix is also assembledas:

DBA→EL = [x y z] (4.8)

e translational velocities of the element are calculated by rstcalculating the in plane (γ) and out of plane (υ) deection angle of theblade.is is done by calculating tan−1 of the components of the elementposition:

γ = − tan−1pel,ypel,z

(4.9)

υ = tan−1 pel,xpel,z

(4.10)

ese angles are used to take the deformation into account while calculat-ing the translational velocity of the elements.e velocity componentsare rst calculated in the body-xed coordinate system:

vBF =⎡⎢⎢⎢⎢⎢⎣

0−Ω∣pt,el∣ cos(ψ + υ) cos(θ + γ)−Ω∣pt,el∣ cos(ψ + υ) sin(θ + γ)

⎤⎥⎥⎥⎥⎥⎦(4.11)

ese results are then rotated to compensate for the sha tilt angle:

vt,GL = Rκvt,BF (4.12)

e Rotational velocity is taken as:

vr,GL = R [Ω 0 0] (4.13)

Calculating forces and moments

ese markers are passed to AeroDyn, which returns the forces andmoments of each element in the element coordinate system. Optionally,an additional pseudo-centrifugal force can be added. As is derived frommany mechanics textbooks, e.g. Rixen (2011); Lanczos (1970),Landauand Lifshitz (1982, p. 39) and Taylor (2005, p. 344-348) the centrifugalforces are implemented as:

fcfgl = −mΩ × (Ω × r) (4.14)

is force is calculated as2222. In which mel is the mass perunit length of the element.

:

fcfgl = −mel cos(ψ + γel)⎡⎢⎢⎢⎢⎢⎣

Ω00

⎤⎥⎥⎥⎥⎥⎦×⎛⎜⎝

⎡⎢⎢⎢⎢⎢⎣

Ω00

⎤⎥⎥⎥⎥⎥⎦× pel,BA

⎞⎟⎠

(4.15)

e cos term is added to take into account the eect of the pre-cone angleon the amplitude of the force. ese forces are in the body-attachedreference frame, which is the same reference frame used to apply forcesto the structure.

60

Calculate blade moment

In order to calculate the moment acting on the blade it’s necessary totransform the force results to the body-attached reference frame.

fBA = DTBA→ELfel (4.16)

ese forces are then used to calculate the moment on the blade withthe equation:

t =nelem−1∑0cos(ψ + υel)pel,BA × fBA (4.17)

e moment acting on the sha of the turbine is then calculated as23 23. In which nshaft is the directionvector of the sha. For a sha with-out tilt nshaft = [1 0 0]T.

:

tsha = t nsha (4.18)

Node force retrieval

AeroDyn calculates the forces and moments acting on an aerodynamicelement. Forces are applied to TACS as a traction on the intermediatemesh and transfered to the structural mesh.is section describes themethod used to calculate the traction forces to correspond to the forceand moment24 24. Only the moment acting in the

z-direction is applied, momentsacting in other directions are neg-ligeble.

results calculated by AeroDyn. e method used is simi-lar to that of Berg et al. (2011), though that work uses forces on nodesinstead of traction forces. In the explanation of the method, the airfoil isoen illustrated in 1D.is is for the sake of simplicity. e actual loadsare applied on the 3D intermediate mesh.

AeroDyn’s force and moment results are one dimensional.e loadson the intermediate mesh are applied as a three dimensional tractionforce.e addition of twomore dimensions allows for an innite amountof traction distributions which when summed equal AeroDyn’s forceand load results. For the sake of simplicity a linear force distribution isassumed25 25.ere was discussion of using

e.g. xfoil to come up with a betterpressure distribution along thechord.is was decided against asit would take too much time andthere are plans to add CFD in thefuture to get these results.

.

When applying a force ormoment to a structure it is easy to introducean unintended moment or force. ese have to be corrected during theprocess in order to correctly transfer loads to the structure. e processused in pyAeroDyn is:

1. Determine distances of the leading and trailing edges to the aero-dynamic center.

2. Calculate the moment that the mean traction force generates andcorrect the moment term of the AeroDyn results to correct this.

3. Calculate the traction force distribution used to generate moment.

4. Calculate and correct for the generated mean traction force addedin the previous step without adding any additional moment.

61

Figure 4.8: Process used to determine the traction forces applied to the nodes.

is process is illustrated in Figure 4.8 and further elaborated in the restof this section.

e rst step taken is to calculate the distance from the aerodynamiccenter (where the moment is applied) to the leading and trailing edge.e distance is calculated using a vector equation derived in Weisstein(2013). Using this method the shortest distance d between a point x0and a line dened by two x1 and x2 can be found as:

d = ∣(x2 − x1) × (x1 − x0)∣2∣x2 − x1∣2

(4.19)

is method is used on the intermediate mesh to nd the shortestdistance from the edge nodes to the line between the aerodynamiccenter’s nodes. For the element shown in Figure 4.5 this would be2626. R and T are used to distin-

guish between the root-side andthe blade tip-side of the element.

:

dLE,R =∣(n3 − n1) × (n1 − n0)∣2

∣n3 − n1∣1(4.20)

dTE,R =∣(n3 − n1) × (n1 − n2)∣2

∣n3 − n1∣1(4.21)

dLE,T =∣(n7 − n5) × (n5 − n4)∣2

∣n7 − n5∣1(4.22)

dLE,R =∣(n7 − n5) × (n5 − n6)∣2

∣n7 − n5∣1(4.23)

For the undeformed conguration the root and trailing edge distance isthe same. ey are calculated separately to allow for variations in thetraction force application for chord-wise deformation.

AC

fmean,x

Figure 4.9: Distributionof mean traction force

along the chord direction.

62

When applying a pure and equal traction force on an airfoil in thechordwise direction (as shown in Figure 4.9) there will be an additionalmoment applied at the same time (unless the aerodynamic center is inthe middle of the airfoil).is moment can be derived as:

Mmtr = fmean,x (d2LE2

− d2TE2

) (4.24)

is moment is subtracted from the AeroDyn calculated moment inorder to neutralize it.

getForces• Amount of force offset for LE/TE nodes

22

http://mathworld.wolfram.com/Point-LineDistance3-Dimensional.html

Fm,i = 3 ~M

2di

di^2

ACfm,

fm,

Figure 4.10: Distribution of tractionforce to generate the moment alongthe chord direction.

With the distance from the aerodynamic center to the edges calcu-lated, and the element moment corrected for the mean traction force it’spossible to calculate the force oset necessary to generate the moment.e traction forces are applied on the nodes and are automatically lin-early interpolated between the nodes.is means that the contributedforce distribution on the blade will look similar to Figure 4.10.

e force distribution can be described mathematically as:

fi = −ydifm,i (4.25)

e moment of the aerodynamic center to either edge is the M2 .e net

force fm,i ,x caused by the moment force distribution can easily derivedas:

M2= ∫ y fi ,xdy (4.26)

= ∫d i

0

y2di

fm,i ,xdy = −d2i3

fm,i ,x (4.27)

→ fm,i = −32d2i

M (4.28)

e net traction force acting on a cross section of an element in they-direction is:

fM ,y =dLE fLE,x2

− dTE fTE,x2

(4.29)

is only equals zero if dLE = dTE, which is equivalent to having theaerodynamic center in the center of the blade.

e nal step in the node force calculation is to compensate for thenet-force applied Equation (4.29). For the limited force distributionthat’s possible with the limited amount of nodes in the intermediatemesh, there’s a unique solution to apply the correcting force withoutapplying a moment27 27.is method is not used for all

forces. It would mean that no forceacts on the aerodynamic center. Itwould also cause a large amount ofthe force acts on the edge.

. By stating that:

∑M = fmc,1d2LE2

− fmc,2d2TE2

= 0 (4.30)

fM ,y = fmc,1dLE2

− fmc,2dTE2

(4.31)

63

It can easily be derived that:

fm,1 =2 fM ,y

dLE + d2LEdTE

(4.32)

fm,2 =2 fM ,y

dTE + d2TEdLE

(4.33)

e approximate shape of this corrective force is shown in Figure 4.11.

getForces• Amount of force offset for LE/TE nodes

23

http://mathworld.wolfram.com/Point-LineDistance3-Dimensional.html

Fm,i = 3 ~M

2di

di^2

AC

fmc,fmc,

Figure 4.11: Distributionof traction force to correct

force introduced by moment,along the chord direction.

With the forces known they can be applied to to the nodes.e approxi-mate shape of this distribution is shown in Figure 4.12. For the nodes atthe root section of Figure 4.5 they are calculated as:

fn0,EL = fmean −⎡⎢⎢⎢⎢⎢⎣

fm,100

⎤⎥⎥⎥⎥⎥⎦+⎡⎢⎢⎢⎢⎢⎣

fmc,100

⎤⎥⎥⎥⎥⎥⎦(4.34)

fn1,EL = fmean (4.35)

fn2,EL = fmean +⎡⎢⎢⎢⎢⎢⎣

fm,200

⎤⎥⎥⎥⎥⎥⎦+⎡⎢⎢⎢⎢⎢⎣

fmc,200

⎤⎥⎥⎥⎥⎥⎦(4.36)

fn3,EL = fmean (4.37)

e nal step is to rotate these forces to the body attached reference frame(used by TACS).is is done by simply rotating it with Equation (4.8):

AC

Figure 4.12: Distributionof total traction force

along the chord direction.

fn,BA = DTBA→ELfn,EL (4.38)

At the end of the getForces funtion AeroDyn returns an n × 3 array witha list of the forces on all n nodes.

Mesh interface with pyAeroStruct

It’s vital that pyAeroStruct can read the mesh and pass any deformationsto the mesh back to pyAeroStruct. ese functions are implementedwith three dierent functions.

getSurfaceCoordinates is function passes2828. In which n is the number ofnodes in the mesh.

an n × 3 array back topyAerostruct.

getSurfaceConnectivity is function passes2929. In which m is the number of ele-ments in the mesh.

an m × 4 array backwith the connectivity of the elements. e numbers in the ar-ray correspond to the order in which the nodes are listed in thegetSurfaceCoordinates function.

setSurfaceCoordinates During an aerostructural analysis the meshwill be deformed. To get the deformations passed into pyAeroDyn(and thereby changing the inputs of AeroDyn) an n × 3 arraycan be passed in with this function which updates all the nodecoordinates in pyAeroDyn.

64

Figure 4.13: Process used to create the structural mesh.

Output intermediate mesh to le

e last function built into pyAeroDyn is a function which parses thecurrent (deformed) intermediate mesh to le. e le is parsed in atecplot mesh format (Tecplot, 2008) which can be loaded in a compatibleviewer.

4.7 modeling an aeroelasticproblem

In order to run an aeroelastic analysis it’s necessary to rst model thestructure and aerodynamics.is section describes the modeling proce-dure normally used within the MDOlab as well as the modeling proce-dure used in Chapter 5.

Modeling of structure

Most of the work done by the MDOlab is related to aircra congura-tions. e lab has developed a number of tools to ease the modelingof wings for aeroelastic analysis and optimization. From the structuralside TACS uses a NASTRAN input le for the mesh generation. It alsosupports families30 30. A family in this context is a

grouping of elements.which allow for dierent constitutive objects to be

assigned to areas of the mesh.is means that it’s possible to assign dif-ferent material properties to each family of the mesh31 31. It’s important to note that ma-

terial assignments in the .bdf leare not used.ese are assigned toTACS directly.

. e relationshipof these codes, as well as the method used in Chapter 5, is shown in 4.13

e rst step used to create a mesh is using pyGeo to get the outergeometry of the airfoil dened. pyGeo takes a number of airfoil shapesat specied points and interpolates the surface between these points.is results in an IGES le which has the outer geometry of an airfoil.e MDOlab then normally uses pyLayoutGeo to add spars, stringers

65

and ribs to the airfoil and export it to a .tin le3232. .tin les are that native ge-ometry format used by ICEM

. pyLayoutGeo is oflimited use for wind turbine blades. Its current implementation doesnot model the trailing edge of an airfoil3333.e main focus of the MDOlab’s

structural optimization lies inthe wingbox.e trailing edgeof an aircra has limited struc-tural contribution to stiness.Wind turbine blades tend tobe thicker and don’t have con-trol surfaces at the trailing edge.

. is is a problem with theprojection method used by pyLayoutGeo.e current work-around isto model the spars and surface splits in CAD (PTC Creo Parametric).Creo Parametric has built-in linking with ICEM allowing the modelerto assign families within the CAD environment.is makes it easier tomake minor changes to the geometry without having to reassign thesewithin ICEM.

Both techniques have a .tin output le that is used by ICEM CFD3434. A mesh generation tool devel-oped by ANSYS.

.e mesh has to be generated as an all-quad mesh because TACS doesnot support other types of shell elements.

e material assignment is done during the aeroelastic analysis’ ini-tialization script. More details on the MDOlab’s approach to structuralmeshing can be found in Kenway (2013, sec. 2.1.2) and Kenway et al.(2010).

Modeling of the aerodynamics

ere are a number of inputs to the aerodynamic solver.ey are:

AeroDyn input le is is the input le used by AeroDyn.

Wind input le ewind input le which is referenced by theAeroDyninput le.

ADcfg.py Contains some options used by the aerodynamic solver suchas rotational speed.

adata Species a number of geometrical points which are used whengenerating the intermediate mesh. e layout is very similar tothe AeroDyn input le.

e AeroDyn input le species a number of options and variables usedduring the analysis.is includes:

• Usage of dynamic stall, dynamic inow, tip loss, etc. models.

• Filename of wind input le.

• Air density and viscocity.

• Airfoil table names.

• Element positions, lengths and airfoil types.

66

More details on the wind input le can be found in Laino andHansen(2002, ch. 6.0).e choice of the element distribution and shape has tobe done strategically since this directly aects the performance charac-teristics of the turbine blade.e lename of the AeroDyn input le isone of the initialization options when initializing pyAeroDyn.

e wind input le is loaded by AeroDyn during its initialization. Itis used to determine the wind velocity of a given point during calcula-tion. More information on the dierent kinds of les can be found inSection 4.4.1 or Kelley and Jonkman (2007); B.J. Jonkman (2012).

e previous two les are used directly by AeroDyn. In addition tothese pyAeroDyn also uses an input le.is le species points whichare used later in the analysis to specify the markers or forces.ey are:

Numbl e number of blades used in the analysis.

Nelem e number of elements in one blade.

Hh e hub height [m]

Hubradius e radius of the hub [m]

Bladelength e length of a blade (root-to-tip) [m]

Ω e rotational speed of the blade [rad/s]

Adata e adata le is loaded into an array.

e adata le contains a parsed array with the geometry used tocreate the intermediate mesh (Section 4.6.1). e points in the adata leare in the same order as les generated using NuMAD (Berg and Resor,2012). is is because pyAeroDyn was developed with a test modelgenerated in NuMAD in mind. It also contains the mass of each elementper unit length [kg/m].

67

5

VERIFICATION OF DEVELOPEDCODE

is chapter discusses the results of verication done on the aeroelasticcode developed. Section 5.1 starts o by doing a simple verication to acomposite plate using ANSYS as a reference.is is done because priorto this project, the constitutive object had never been used from thePython level, and never with multi-lamina composites.Section 5.2 discusses the choice of themodel used to verify the aeroelasticcode and how it was created.e results of the aeroelastic analysis arepresented in Section 5.3.

5.1 verification of composite plateSince multi-lamina composites are a new feature in pyTACS, a shortverication study was done using ANSYS as a baseline.e vericationof the composite constitutive objects is done by modeling a 2x2 m platewith a varying core material thickness and mesh renement1 1. 10-20-40 mm and 16x16-32x32-

64x64 mesh sizes..e model

uses four layers of orthotropic lamina in a [45/-45]s orientation. eproperties of the lamina can be found in Table 5.4.

E1 70 GPaE2 70 GPaG12 4.5 GPaG13 4.5 GPaG23 3.2 GPaν 0.3ρ 2810 kg/m3

Table 5.1: Material properties of theorthotropic lamina.

A 1 kPa pressure is applied on the surface of the plate.e maximumdeformation is then measured and compared.

5.1.1 Resultse max. deformation results are found in Table 5.2. e values are veryclose to each other. When the relative error is calculated and plotted itis clear that the values converge when the mesh gets ner (Figure 5.2).is indicates that they tend to converge to the same exact solution.

Figure 5.1: Deformation result ofthe 10 mm plate with a 16x16 mesh.

ickness TACS ANSYS[mm] 16 × 16 32 × 32 64 × 64 16 × 16 32 × 32 64 × 6410 mm 795.9 797.2 798 798.4 798.5 798.620 mm 100.1 100.3 100.3 100.4 100.4 100.440 mm 12.7 12.7 12.7 12.7 12.7 12.7

Table 5.2: Deformation results for the composite plate test case [mm].

69

15 20 25 30 35 40 45 50 55 60 650.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

%di

ere

nce

10 mm20 mm40 mm

Figure 5.2: Relative error between TACS and ANSYS for test case.

5.2 turbine model used forverification

e verication is an essential nal step in the development of theaerostructural code. It compares the developed model with existingmatured models.

5.2.1 Choice of modelere aren’t a lot of detailed turbine blade models publicly available.eonly production blade that has publicly available steady-state data is froman old report (Hand et al., 2001) of a wind turbine test done by NRELat NASA Ames.is blade is relatively small and very sti, making it apoor candidate for aeroelastic code verication.e lack of suitable datameans that the choice was made to do puremodel-to-model verication.

e most popular reference model used in academia and other non-commercial wind energy research centers is the NREL 5 MW blade(Jonkman et al., 2009).e structural properties of which are based onthe LM-Glassber LMH64-5. e sectional blade properties are pub-lished by ECN as part of the DOWEC project (Lindenburg, 2002).eECN report only mentions sectional properties of the LMH-64-5 blade,as the composite layup is proprietary information.

Sandia National Lab has reverse engineered a shell-model based on

70

the NREL 5 MW (Resor, 2013).is model’s composite layout is basedon a reference turbine (Nijssen et al., 2007) and the 10 MW referenceturbine also designed by Grith and Ashwill (2011).

e Sandia 5 MWmodel is created by optimizing the blade to corre-spond with the mass and stiness distribution of the NREL 5 MW blade,as well as the total mass.eir blade is modeled using NuMAD (Bergand Resor, 2012) which allows for direct export to ANSYS2 2.e ANSYS linking is one-way.

is means that there is no way todo aeroelastic analysis or couple theFAST results to ANSYS.

or FAST.

e widespread use of the NREL 5 MW turbine means that it’s verylikely that the Sandia 5 MW turbine will see a lot of future use in higherdelity aeroelastic modeling. e fact that it’s designed to give the sameperformance as the NREL 5 MW blade means that it’s possible to useexisting codes to verify the results of the aeroelastic analysis. Becauseof this, the choice was made to base the model on the Sandia 5 MWreference turbine and compare the results with those of the NREL 5MWreport.

5.2.2 Creation of modelDespite the fact that the model has an ANSYS output option, it was notpossible to simply remesh the output model to an all quad mesh3 3.e mesher had trouble with the

messy geometry.. To get

the geometry for the mesher the entire blade was remodeled with theprocedure described in Section 4.7.

e only dierence between the Sandia model and the model usedin TACS is the use of a straight spar. Spars used in wind turbines are nottwisted.e spar in the NuMADmodel is twisted due to limitations inthe NuMAD.e blade design should not be sensitive to this change4 4. Personal correspondance with

Brian Resor of Sandia NationalLabs.

.

5.2.3 Mesh renemente constitutive relationships which govern structural elements can cre-ate discretization errors if the elements are too large. To make sure thatthe discretization errors aren’t too large a number of simulations are run.e simulations compare the maximum deformation using xed aero-dynamic loads5 5.e loads are xed by freezing

the loads to those of the rst itera-tion (undeformed).is is imple-mented by xing the timestep atfor AeroDyn. AeroDyn does notrecalculate the loads unless there’s aprogression in time.

. e results are shown in Figure 5.3. Using these resultsa mesh size of 30 532 was selected for Section 5.3’s results. e choice forthis mesh size was a compromise of accuracy and computational time.

5.2.4 Mass comparisone simplest form of comparison is the mass. Table 5.3 shows a compari-son of the NREL data, ANSYS model, and TACS models.

e mass is around 2.6% percent less than that of the NuMADmodelon which it’s based.is is most likely due to small dierences caused byremodeling the blade, such as using straight spars instead of twisted etc.

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0 10 20 30 40 50 60 709

10

11

12

13

14

15

16

17

Amount of elements (thousands)

Figure 5.3: Deformation results for a variety of mesh sizes.

Mass [kg] NREL FAST ANSYS TACSOne blade 17 740 16 878 17 700 17 249

Table 5.3: Mass calculation results, ANSYS and FAST models are NuMADmodelresults.

5.2.5 Mass distributionSpan [m] mass [kg/m]1.5-4.2 741.84.2-7 604.487-9.7 406.749.7-13.8 409.513.8-17.9 362.9317.9-22 337.8322-26.1 320.5826.1-30.2 294.6430.2-34.3 262.334.3-38.4 234.138.4-42.5 192.4442.5-46.6 160.4746.6-50.7 134.2550.7-54.8 103.5154.8-57.53 86.1557.5-60.3 66.3260.3-63 40.62

Table 5.4: Mass prop-erties per element

e current implementation of the centrifugal forces applies them as apseudo-aerodynamic force. is means that the mass properties of eachaerodynamic element have to be set as an input to pyAeroDyn.e massdistribution of the original NREL blade is shown in Jonkman et al. (2009,tab. 2-1).e mass has been plotted and tted to a function in MATLABshown in gure Figure 5.4.e tted function is then integrated alongthe span of an element. e resulting mass properties are shown inTable 5.4.

5.3 results of turbine blade modelis section compares the results of the developed aeroelastic modelwith the NREL 5 MW data as well as data computed with the Sandiamodel using FAST.e FASTmodel has a controller implemented whichvaries the rotational speed and pitch to achieve a set power curve (Ashuri,2012).e current implementation of the framework does not includerotational speed and pitch control.

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0 10 20 30 40 50 60 700

100

200

300

400

500

600

700

800

Distance from hub (m)

Rela

tivem

ass(

kg/m

)

Fit curveData point

Figure 5.4: Mass distribution along the span of the blade with interpolation.

0 5 10 15 20 25−3000

−2000

−1000

0

1000

2000

3000

4000

5000

6000

Wind speed [m/s]

Power

[kW]

bCbC

bCbC

bC

bC

bC

bC

bC

bC

bC

bC

bC

bC

bC

bC

bC

bC

bC

bC

bC

bC

bCbC

rS rSrS

rS

rS

rS

rS

rS

rSrS rS rS rS rS rS rS rS rS rS rS rS rS rS rS

TACSFAST

bC

rS

Figure 5.5: Power curve with FAST control settings. Figure 5.6: Eect of blade pitch to the power coe-cient as a function of the tip speed ratio. From http://www.mathworks.com/help/physmod/powersys/ref/windturbine.html

Figure 5.5 shows the power curve of the FAST and TACSmodels.eplot uses the rotational speed and pitch settings of the FAST simulation6 6.e moment per blade is cal-

culated using Equation (4.18).e power output is calculatedas P = 3tshaftω.

.

e plot shows that the power is slightly increased below the ratedwind speed (11.4 m/s). e power drops aer reaching its peak at therated wind speed. To understand what’s happening it’s rst necessary to

73

get an understanding of the eect that pitch has on the power output ofa wind turbine.

Figure 5.6 shows how the pitch of the blade shis the power curve.For lower wind speeds the power is increased with pitch. For higherwind speeds the power output is decreased. In some cases the powercoecient even becomes negative77.is physically means that the

moment acting on the blade isopposite to the rotation of the

blade.is would cause the bladeto slow down its rotational speed.

e results indicate that the blade has less torsional stiness thanthe blade modeled by NuMAD. To conrm this the blade is "tuned" tothe performance curve of the FAST model using only the pitch control88. In real turbines the power

output is only controlled withpitch above the rated wind speed.

.e power curve and the blade pitch angles are shown in Figures 5.7and 5.8.

0 5 10 15 20 250

5

10

15

20

25

Wind speed [m/s]

Bladep

itch[deg]

bCbC bC bC bC bC bC bC bC

bCbC

bC

bC

bC

bC

bC

bC

bC

bC

bCbC

bCbC

bC

rS rS rS rS rS rS rS rS rS rS

rS

rS

rS

rS

rS

rS

rS

rS

rS

rS

rS

rSrS

rS

TACSFAST

bC

rS

Figure 5.7: Pitch angle setting of "tuned" TACS vs FASTcontrolled model.

0 5 10 15 20 25−3000

−2000

−1000

0

1000

2000

3000

4000

5000

6000

Wind speed [m/s]

Power

[kW]

bCbC

bCbC

bC

bC

bC

bC

bC

bC

bC

bC

bC

bC

bC

bC

bC

bC

bC

bC

bC

bC

bCbC

rS rSrS

rS

rS

rS

rS

rS

rSrS rS rS rS rS rS rS rS rS rS rS rS rS rS rS

ld ldld

ld

ld

ld

ld

ld

ldld ld ld ld ld ld ld ld ld ld ld ld ld ld ld

TACSFASTTACS ”tuned”

bC

rS

ld

Figure 5.8: Power output of "tuned" TACS compared toother models.

e results show that for very low wind speeds there is no pitch.epitch is increased and slowly decreased at the nominal rated wind speed.is is because the nominal maximum Cp has a fairly small gradient.Small changes in the pitch have less of an eect on the performance thanat other wind speeds.

A major source of error is the lack of blade twist degrees of freedomin FAST (Jonkman, 2013).e forces acting on the blade are very sensi-tive to even small changes in the angle of attack. A change in the angleof attack caused by twist will signicantly alter the dynamic performance.

Another possible source of errors is the calculation of the structuralproperties. NuMAD uses PreComp (Gunjit, 2005) to calculate the equiv-alent beam properties. Research done by Chen et al. (2010) shows thatthe stiness properties calculated by PreComp have an error of up to300%.e paper’s conclusion states:

74

Because of the poor, and inconsistent performance of PreComp, FAROB,and CROSTAB for simple cross-sections, their applicability to real complex,composite wind turbine blades is questionable.

e limited reliability of PreComp combined with the lack of twist de-grees of freedom make the FAST results computed somewhat question-able. It likely means that the aeroelastic solution computed with TACSis more realistic. To answer this denitively further validation has to bedone.e lack of test results suitable for model validation may necessi-tate wind tunnel tests. Such tests are out of the scope of this thesis.Other factors which can have an eect:

Geometrical dierences e blade was manually modeled to t thegeometry of the NuMAD blade (Resor, 2013). It’s possible thatthese things have an aect.

Dierences in delity e FAST model uses beam data as an input.e structural model used by FAST is not designed to handleanisotropic materials.e structure is modeled as isotropic9 9.e model used has a constant

ber orientation in the spanwisedirection.is should result inlimited bend-twist coupling.

.

Dynamic eects e currentmodel does have dynamic eects.e cur-rent implementation also does not take the tower shadow eectsinto account.ese create an oscillation in the forces on the bladeevery time the blade passes in front of the tower. Other eectssuch as gyroscopic forces should not have an eect for steady-statecases. Gravitational forces should not have much eect for steadystate performance analysis.e modeled blade should not havesignicant edgewise bend-twist coupling.

5.3.1 Displacement resultse deformation results given as an output by FAST are of a one dimen-sional model. e only output FAST gives are in the in-plane (y) andout-of-plane (x) deection. e structural twists is not an output inFAST.

Comparing the FAST output to the TACS output for the 11 m/s loadcase it is apparent that the apwise stiness in the TACS model is higher.e deection is 0.7 m compared to the 3.13 m of the FAST model.eedgewise stiness seems to be similar, 0.3 in TACS vs 0.24 in FAST.eresults are summarized in Table 5.5.

e lack of twist outputmakes it hard to directly compare the torsion.From the results of Section 5.3 it can be assumed that the FAST modelis torsionally stier. e signicant dierences in deection warrantfurther research into verication and validation.

75

u0 v0 w0 rx ry rzFAST 3.13 m 0.3 m - - - -TACS 0.7 m 0.24 m 9 mm 0.4 ° 1.3° 2.4 °

Table 5.5: Deection and twist of the blade at 11 m/s wind conditions FAST vs"tuned" TACS.

Plots of the tuned 11 m/s load case deected structure can be foundin Figures 5.9 to 5.14.

Figure 5.9: x-direction deec-tion [m].

Figure 5.10: y-direction deec-tion [m].

Figure 5.11: z-direction deec-tion [m].

76

Figure 5.12: x-rotation [deg]. Figure 5.13: y-rotation [deg]. Figure 5.14: z-rotation [deg].

77

PART III

CONCLUSIONS

6

CONCLUSIONS AND FUTUREWORK

6.1 conclusionse goal of this MSc research was to pave the road for high-delityaerostructural modeling of wind turbines by modifying and adaptingthe techniques used in aerospace research. is is done by couplingAeroDyn, an aerodynamic solver, to TACS, a high-delity structuralsolver.e coupling was not straight forward and easy.is was due totwo facts:

1. AeroDyn is written in Fortran and TACS in C++. Direct couplingbetween them is not possible.

2. TACSwas not developed to be used for bodies in rotating referenceframes.

e MDOlab makes extensive use of Python to couple aerostruc-tural codes.Due to this fact, pyAeroDyn was incorporated into existingaerostructural codes. It was therefore necessary to create a Python in-terface for AeroDyn. Using F2py, AeroDyn was compiled as a sharedobject1 1. A shared object on unix platforms

are similar to a Dynamic LinkLibrary (DLL) in the Windowsplatforms

which can be accessed as an external module at the Python level.pyAeroDyn, which is a piece of code developed during this researchfunctions as an aerodynamic module in the existing pyAeroStruct analy-sis framework. A similar technique is used for TACS to enable access toits functions using SWIG, a C++ to Python translator. pyTACS functionsas a structural module in the pyAeroStruct framework.

e developed code uses a body attached coordinate system. etransformation to the global coordinate system used by AeroDyn is donethrough a series of rotation matrices.ese make it possible to changethe azimuth, sha tilt, precone and pitch angles.

e above steps enabled TACS to be used also as a high-delity struc-tural solver for wind turbines. In contrast to many existing wind turbineaeroelastic codes based on low delity models, the code developed dur-ing this work can cope with the complex physical behavior of the rotor.

81

e code is currently limited to the analysis of steady state behavior.is limitation is in place because time-domain support inside TACSwas added in the late stages of this research.ere was no time to buildthis support into the pyTACS and pyAeroStruct codes. It also has thelimitations inherent to BEMmethods discussed in Section 2.2.

e pyAeroDyn code is relatively fast. e CPU usage of pyAero-Dyn during an aerostructural analysis is less than 1% of the total.ismakes it very suitable for fast aerodynamic analysis, CFD takes manymagnitudes of order more computational time.

Bend-twist coupling is of particular interest in the context of high-delity modeling and simulation. Most of the wind turbine aeroelasticcodes currently used do not support this feature. Bend-twist couplingcan be caused by an asymmetric stiness in the blade’s structure.is isimportant when using asymmetric composite ber orientation in bladeconstruction.

Research is being done in incorporating anisotropic beam modelsfor use in wind turbines (Zhou et al., 2012; Kim et al., 2013; Lago et al.,2013). e use of anisotropic beam models could potentially be usedto model anisotropic blades. e accuracy of these models partiallydepends on how the beam properties are calculated. e calculationmethods used tend to have questionable accuracy (Chen et al., 2010).Geometric bend-twist coupling can not modeled this way. is cancause inaccuracies for swept blades and similar structures.

e code during this project shows that the bend-twist coupling hasa signicant inuence on the blade deformation.is has a direct eecton the aerodynamic performance compared to low-delity code.isindicates that high-delity modeling and simulation is of great impor-tance. is is especially true when dealing with the design of exiblewind turbine blades made with composite materials. Incorporatinga high-delity structural solver turned out to be very eective whendealing with the composite materials.

6.2 future workough the model is currently limited to steady-state analysis, it ispossible to expand it to handle unsteady cases. Limited support fortime-domain analysis has recently been added to TACS (Kennedy andMartins, 2013). Future work should be focused on modifying pyTACSand pyAeroStruct codes. Changes in pyAeroDyn will likely be limited.Certain eects, such as out of plane translation must be incorporated22. Velocities can be computed as dp i

dtin the unsteady case.is shouldbe accurate if the time step is small

enough.

.ese changes can be implemented as an analysis option, allowing the

82

user to perform both steady and unsteady analyses.

High-delity optimization usually uses gradient-based algorithmsdue to the large amount of design variables3 3. In the order of hundreds up to a

few thousand.. Gradient calculation is

not supported in the current version of AeroDyn & pyAeroDyn. Aero-Dyn could be altered to compute gradients using the complex variablemethod4 4.ough the computational cost

of pyAeroDyn is limited, usingnite dierences to calculate thegradients of hundreds of designvariables can become quite costly.

. More advanced methods of gradient computation, such as thecoupled-adjoint method, (Mader et al., 2008) will require considerablymore work.

Future implementation of high-delity aerodynamics using CFDwill allow the analysis of unconventional turbine blade congurations.Examples of these are winglets and smart structures5 5. An example would be a blade

in which the airfoil shape changeswith deection. When designedcorrectly this could allow for pas-sive wind turbine control. It wouldalso allow for better performancein non-steady situations in whichthe optimal pitch angle is constantlychanging along the span of theblade (Engels et al., 2010).

. In contrast toaerospace applications, there are some diculties modeling the turbineblade in CFD because of its rotating nature. Future optimization studiesusing CFD may be able to use pyAeroDyn in the form of hybrid multi-delity optimization.is will allow for computationally more ecientmethods of investigating the design space.

is report’s results were generated by manually changing the bladepitch and generator rotational speed. Actual wind turbines use con-trollers to dynamically control these parameters.e accurate simula-tion of complete wind turbine design will require adding a controller.Future work can incorporate the controller.is will make aeroservoe-lastic analysis possible (Ashuri et al., 2010b).

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