State of the art in wind turbine aerodynamics and ...phyexp/uploads/EolienneFlexible/Eoli… ·...

ARTICLE IN PRESS

Progress in Aerospace Sciences 42 (2006) 285ndash330

State of the art in wind turbine aerodynamics and aeroelasticity

MOL Hansena JN Soslashrensena S Voutsinasb N Soslashrensencd HAa Madsenc

aDepartment of Mechanical Engineering Technical University of Denmark Fluids Section Nils Koppels AlleBuilding 403 DK-2800 Lyngby Denmark

bDepartment of Mechanical Engineering National Technical University of Athens Fluids Section 15780 Zografou GreececWind Energy Department Risoe National Laboratory Building VEA-762 PO Box 49 Frederiksborgvej 399 DK-4000 Roskilde Denmark

dDepartment of Civil Engineering Alborg University Sohngaardsholmsvej 57 DK 9000 Aalborg Denmark

Available online 29 December 2006

Abstract

A comprehensive review of wind turbine aeroelasticity is given The aerodynamic part starts with the simpleaerodynamic Blade Element Momentum Method and ends with giving a review of the work done applying CFD on windturbine rotors In between is explained some methods of intermediate complexity such as vortex and panel methods Alsothe different approaches to structural modelling of wind turbines are addressed Finally the coupling between theaerodynamic and structural modelling is shown in terms of possible instabilities and some examplesr 2006 Elsevier Ltd All rights reserved

Keywords Aeroelasticity Wind turbines

Contents

1 Introduction 2862 Predicting aerodynamic loads on a wind turbine 287

21 Blade Element Momentum Method 287211 Dynamic wakeinflow 288212 Yawtilt model 289213 Dynamic stall 289214 Airfoil data 290215 Wind simulation 290

22 Lifting line panel and vortex models 291221 Vortex methods 291222 Panel methods 292

23 Generalized actuator disc models 29524 NavierndashStokes solvers 297

241 Introduction to computational rotor aerodynamics 297242 Approaches 298243 Turbulence and transition 299

wwwelseviercomlocatepaerosci

0376-0421$ - see front matter r 2006 Elsevier Ltd All rights reserveddoi101016jpaerosci200610002

Corresponding author Tel +4545254316E-mail address molhmekdtudk (MOL Hansen)

244 Geometry and grid generation 299245 Numerical issues 299246 Application of CFD to wind turbine aerodynamics 300247 Future 302

3 Structural modelling of a wind turbine 30231 Principle of virtual work and use of modal shape functions 30232 FEM modelling of wind turbine components applying non-linear beam theory 304

4 Problems and solutions in wind turbine aeroelasticity 30841 Aeroelastic stability 30842 Aeroelastic coupling linear vs non-linear formulations 30943 Examples of time simulations and instabilities 310

431 Edgewise blade vibration instability 312432 Instability problems of parked rotors 317433 Flutter instability 317

5 Present and future developments of aeroelastic models 31851 Areas with influence on the development of aeroelastic models 318

511 Influence of up-scaling 318512 Siting of the turbines 319513 Future trends in turbine design and siting 319

52 Areas of development in present and new codes 319521 Non-linear structural dynamics 319522 Calculation of induction and its dynamics 320523 Wake operation 321524 Derivation of airfoil data for aeroelastic simulations 322525 Complex inflow 323526 Aerodynamics of parked rotors 324527 Offshore turbines including floating turbines 324

6 Discussion 325References 325

1 Introduction

The size of commercial wind turbines hasincreased dramatically in the last 25 years fromapproximately a rated power of 50 kW and a rotordiameter of 10ndash15m up to todayrsquos commerciallyavailable 5MW machines with a rotor diameter ofmore than 120m This development has forced thedesign tools to change from simple static calcula-tions assuming a constant wind to dynamic simula-tion software that from the unsteady aerodynamicloads models the aeroelastic response of the entirewind turbine construction including tower drivetrain rotor and control system The Danishstandard DS 472 [1] allows simplified load calcula-tions if the rotor diameter is less than 25m andsome other criteria are fulfilled A rotor diameter of25m corresponds approximately to a rated power of200ndash250 kW which is less than almost any moderncommercial wind turbine today Instead modernwind turbines are designed to fulfill the require-ments of the more comprehensive IEC 61 400-1 [2]standard At some time during the development of

larger and larger commercial wind turbines the needfor aeroelastic tools thus became necessary Aero-elastic tools were mainly developed at the univer-sities and research laboratories in parallel with theevolution of commercial wind turbines At the sametime governments and utility companies erectedlarge non-commercial prototypes for research pur-poses as the Nibe [3] and Tjaereborg machines [4]Measurement campaigns were undertaken on thesemachines and the results used to tune and validatethe aeroelastic programmes in order to developadvanced software for the rapidly growing industryEven today measurements from the Tjaereborgmachine is used as a benchmark when developingnew aeroelastic codes see eg [5] In [5] is alsocompiled a long list of available software that atdifferent levels of complexity can model the aero-elastic response of a wind turbine construction Allthe aeroelastic models need as input a time historyof the wind seen by the rotor which as a minimummust contain some physical correct properties suchas realistic power spectra and spatial coherenceApart from the wind input aeroelastic codes contain

ARTICLE IN PRESSMOL Hansen et al Progress in Aerospace Sciences 42 (2006) 285ndash330286

an aerodynamic part to determine the wind loadsand a structural part to describe the dynamicresponse of the wind turbine construction For theaerodynamic part most codes use the Blade ElementMomentum Method (BEM) as described byGlauert [6] since this method is very fast andprovided that reliable airfoil data exist yieldsaccurate results Therefore this method with allthe necessary engineering adds on is thoroughlydescribed later in this article However moreadvanced numerical models based on the Eulerand NavierndashStokes (NS) equations are becoming sofast that they now begun to replace the BEMmethod in some situations eg when analysing yawor interaction between wind turbines in parksThese models contain more physics and lessempirical input than the BEM method and areextensively described in this paper The discretiza-tion of the wind turbine structure is presently wherethe various available codes differ most Roughlythere exist three ways to model the structuraldynamics of a wind turbine One is a full FiniteElement Method (FEM) discretization and anotheris a multi-body formulation where different rigidparts are connected through springs and hingesFinally the description of blade and tower deflec-tions can be made as a linear combination of somephysical realistic modes typically the lowest eigen-modes The last method greatly reduces thecomputational time per time step as comparedwith a full FEM discretization All the various waysof discretizing the wind turbine structure will betreated in details later in the paper The verydetailed description of the aerodynamic and struc-tural models is where this paper differs mostly fromother review articles concerning wind turbineaeroelasticity such as eg [7ndash9]

2 Predicting aerodynamic loads on a wind turbine

Methods of various levels of complexity tocalculate the aerodynamic loads on a wind turbinerotor are given starting with the popular BEM andending with the solution of the NS equations

21 Blade Element Momentum Method

BEM is the most common tool for calculating theaerodynamic loads on wind turbine rotors since it iscomputationally cheap and thus very fast Furtherit provides very satisfactory results provided thatgood airfoil data are available for the lift and drag

coefficients as a function of the angle of attack andif possible the Reynolds number The method wasintroduced by Glauert [6] as a combination of one-dimensional (1D) momentum theory and bladeelement considerations to determine the loadslocally along the blade span The method assumesthat all sections along the rotor are independent andcan be treated separately typically in the order of10ndash20 radial sections are calculated At a givenradial section a difference in the wind speed isgenerated from far upstream to deep in the wakeThe resulting momentum loss is due to the axialloads produced locally by the flow passing theblades creating a pressure drop over the bladesection The local angle of attack at a given radialsection on a blade can be constructed provided thatthe induced velocity generated by the action of theloads is known see Fig 1 V0 is the undisturbedwind velocity W the induced velocity Vrot o rthe rotational speed of the blade section Vblade thevelocity of the blade section apart from the bladerotation and b is the local angle of the blade sectionto the rotor plane

Combining the global momentum loss with theloads generated locally at the blade section yieldsformulas for the induced velocity as

Wz BL cos f

4rprF V0 $ f g nn Wamp (211)

Wy BL sin f

B is the number of blades L the lift computed fromthe lift coefficient f is the flow angle r the densityof air r the radial position considered V0 the windvelocity W the induced velocity and n the normalvector to the rotor plane F is Prandtlrsquos tip losscorrection that corrects the equations to be valid fora finite number of blades see [6 10] If there is noyaw misalignment that is the normal vector to therotor plane n is parallel to the wind vector then

ARTICLE IN PRESS

W

Vo

VbladeVrot

Vrel

y x

rotor plane

z

$

Fig 1 Construction of angle of attack a

MOL Hansen et al Progress in Aerospace Sciences 42 (2006) 285ndash330 287

Eq (211) reduces to the well-known expression

CT 4aF 1 f g aamp (213)

where by definition for an annual element ofinfinitesimal thickness dr and area dA 2prdr

CT dT

1=2rV20dA

(214)

The axial interference factor is defined as

a Wz

V0(215)

and fg usually referred to as the Glauert correctionis an empirical relationship between CT and a in theturbulent wake state It may assume the form

f g 1 for ap03

145 3aamp for a403

(

(216)

Eqs (211) and (212) are also known to be validfor an extreme yaw misalignment of 901 that is theincoming wind is parallel to the rotor plane as ahelicopter in forward flight Without any proofGlauert therefore assumed that Eqs (211) and(212) are valid for any yaw angle

An aeroelastic code is running in the time domainand for every time step the aerodynamic loads mustbe calculated at all the chosen radial stations alongthe blades as input to the structural model For agiven time the local angle of attack is determined onevery point on the blades as indicated in Fig 1 Thelift and drag coefficients can now be found fromtable look-up and the lift can be determinedThe induced velocities can now be updated using

Eqs (211) and (212) simply assuming old valuesfor the induced velocities on the right-hand sides(RHS) Updating the RHS of Eqs (211) and(212) could continue until the equations are solvedwith all values at the same time step However thisis not necessary as this update takes place in thenext time step ie time acts as iteration Moreimportant the values of the induced velocitieschange very slowly in time due to the phenomenaof dynamic inflow or dynamic wake

211 Dynamic wakeinflowThe induced velocities calculated using Eqs (211)

and (212) are quasi-steady in the sense that theygive the correct values only when the wake is inequilibrium with the aerodynamic loads If the loadsare changed in time there is a time delay proportionalto the rotor diameter divided by the wind speedbefore a new equilibrium is achieved To take intoaccount this time delay a dynamic inflow modelmust be applied In two EU-sponsored projects([1112]) different engineering models were testedagainst measurements One of these models pro-posed by S Oslashye is a filter for the induced velocitiesconsisting of two first-order differential equations

W int $ t1dW int

dt Wqs $ k t1

dWqs

dt (217)

W $ t2dW

dt W int (218)

Wqs is the quasi-static value found by Eqs (211)and (212) Wint an intermediate value and W thefinal filtered value to be used as the induced velocityThe two time constants are calibrated using a simplevortex method as

t1 11

1 13aampR

V 0(219)

and

t2 039 026r

R

2$

t1 (2110)

where R is rotor radiusIn Fig 2 is shown for the Tjaereborg machine the

computed and measured response on the rotorshafttorque for a sudden change of the pitch angle Att 2 s the pitch is increased from 01 to 371decreasing the local angles of attack First therotorshaft torque drops from 260 to 150 kNm andnot until approximately 10 s later the inducedvelocities and thus the rotorshaft torque have settled

ARTICLE IN PRESS

400

350

300

250

200

150

1000 10 20 30 40 50 60

time [s]

Rot

orsh

aft t

orqu

e [k

Nm

]

BEMMeasurement

Fig 2 Comparison between measured and computed time seriesof the rotorshaft torque for the Tjaereborg machine during a stepinput of the pitch for a wind speed of 87ms

MOL Hansen et al Progress in Aerospace Sciences 42 (2006) 285ndash330288

at a new equilibrium At t 32 s the pitch ischanged back to 01 and a similar overshoot inrotorshaft torque is observed The decay of thespikes seen in Fig 2 can only be computed with adynamic inflow model and such a model is there-fore of utmost importance for a pitch-regulatedwind turbine

212 Yawtilt modelAnother engineering model for the induced

velocities concerns yaw or tilt When the rotor discis not perfectly aligned with the incoming wind thereis an angle different from zero between the rotornormal vector and the incoming wind see Fig 3 Ayawtilt model redistributes the induced velocities sothat the induced velocities are higher when a blade ispositioned deep in the wake than when it is pointingmore upstream An example of such a model takenfrom helicopter literature [13] is given below Herethe input is the induced velocity W0 calculatedusing Eqs (211) (212) (217) and (218) Theoutput is a redistributed value finally used whenestimating the local angle of attack W

W W0 1$r

Rtan

w2cosyb y0amp

(2111)

yb is the actual position of a blade y0 is the positionwhere the blade is furthest downstream and w is thewake skew angle see Fig 3 In some BEMimplementations W0 is the average value of allblades at the same radial position r and in othercodes it is the local value This difference inimplementation may cause a small difference fromcode to code Further there exist different mod-ifications of Eq (2111) from different codes see

[12] A yawtilt model increases the inducedvelocities on the downstream part of the rotor anddecreases similarly the induced velocity on theupstream part of the rotor disc This introduces ayaw moment that tries to align the rotor with theincoming wind hence tending to reduce yawmisalignment For a free yawing turbine such amodel is therefore of utmost importance whenestimating the yaw stability of the machine

213 Dynamic stallThe wind seen locally on a point on the blade

changes constantly due to wind shear yawtiltmisalignment tower passage and atmosphericturbulence This has a direct impact on the angleof attack that changes dynamically during therevolution The effect of changing the blades angleof attack will not appear instantaneously but willtake place with a time delay proportional to thechord divided with the relative velocity seen at theblade section The response on the aerodynamicload depends on whether the boundary layer isattached or partly separated In the case of attachedflow the time delay can be estimated usingTheodorsen theory for unsteady lift and aerody-namic moment [14] For trailing edge stall ie whenseparation starts at the trailing edge and graduallyincreases upstream at increasing angles of attackso-called dynamic stall can be modelled through aseparation function fs as described in [15] see laterThe BeddoesndashLeishman model [16] further takesinto account attached flow leading edge separationand compressibility effects and also corrects thedrag and moment coefficients For wind turbinestrailing edge separation is assumed to represent themost important phenomenon regarding dynamicairfoil data but also effects in the linear region maybe important see [17] It is shown in [15] that if adynamic stall model is not used one might computeflapwise vibrations especially for stall regulatedwind turbines which are non-existing on the realmachine For stability reasons it is thus highlyrecommended to at least include a dynamic stallmodel for the lift For trailing edge stall the degreeof stall is described through fs as

Claamp f sClinvaamp $ 1 f sampClfsaamp (2112)

where Clinv denotes the lift coefficient for inviscidflow without any separation and Clfs is the liftcoefficient for fully separated flow eg on a flatplate with a sharp leading edge Clinv is normally anextrapolation of the static airfoil data in the linear

ARTICLE IN PRESS

Rotor disc

Vo Von x

ampyawtilt

V Wn

Fig 3 Wind turbine rotor not aligned with the incoming windThe angle between the velocity in the wake (the sum of theincoming wind and the induced velocity normal to the rotorplane) is denoted the wake skew angle w

region and in [17] a way of estimating Clfs and fsst is

shown fsst is the value of fs that reproduces the static

airfoil data when applied in Eq (2112) Theassumption is that fs always will try to get backto the static value as

df sdt

f sts f s

t (2113)

that can be integrated analytically to give

f st$ Dtamp f sts $ f stamp f sts amp expDt=tamp (2114)

t is a time constant approximately equal to AcVrelwhere c denotes the local chord and Vrel is therelative velocity seen by the blade section A is aconstant that typically takes a value about 4Applying a dynamic stall model the airfoil data isalways chasing the static value at a given angle ofattack that is also changing in time If eg the angleof attack is suddenly increased from below to abovestall the unsteady airfoil data contains for a shorttime some of the inviscidunstalled value Clinv andan overshoot relative to the static data is seen It canthus been seen as a model of the time constant forthe viscous boundary layer to develop from onestate to another

214 Airfoil dataThe BEM as described above including all

engineering corrections is used in most aeroelasticcodes to compute the unsteady aerodynamic loadson wind turbine rotors The method is often quitesuccessful but depends on reliable airfoil data forthe different blade sections Three-dimensional (3D)effects from the tip vortices are taken into accountwhen applying Prandtlrsquos tip loss correction and afterthis correction the local flow around a given bladesection is assumed to be two-dimensional ie 2Dairfoil data from wind tunnel measurements areused However such measurements are oftenlimited to the maximum lift coefficient Clmax forairplanes that usually are operated at unstalled flowconditions Further at higher values it is difficult tomeasure the forces because of the unsteady and 3Dnature of stall In contrast to airplane wings a windturbine blade often operates in deep stall especiallyfor stall regulation For the inner part of the bladeseven data for low angles of attack might be difficultto find in literature since for structural reasons theairfoils used are much thicker than those used onairplanes Further because of rotation the bound-ary layer is subjected to Coriolis- and centrifugalforces which alter the 2D airfoil characteristics

This is especially pronounced in stall It is thus oftennecessary to extrapolate existing airfoil data intodeep stall and to include the effect of rotationMethods have been developed that from a CFDcalculation of the flow past a full wind turbine rotorcan extract 3D airfoil data [18] which then later canbe applied in aeroelastic calculations using the muchfaster BEM method In this method the inducedvelocity at the rotor plane is estimated from theazimuthally averaged velocity in very thin annularelements up- and downstream of the rotorplane In[1920] two engineering methods to correct 2Dairfoil data for 3D rotational effects are given as

Cx3D Cx2D $ ac=ramph cosn b DCx x ldm

(2115)

DCl Clinv Cl2D

DCd Cd2D Cd2Dmin

DCm Cm2D Cminv

c is the chord r the radial distance to rotational axisand b the twist

In [19] only the lift is corrected ie x l and theconstants are a 3 n 0 and h 2 whereas in [20]a 22 n 4 and h 1 In [21] another methodbased on correcting the pressure distribution alongthe airfoil is given One must however be veryaware that the choice of airfoil data directlyinfluences the results from the BEM method Forcertain airfoils a lot of experience has been gatheredregarding appropriate corrections to be used inorder to obtain good results and because of thisblade designers tend to be conservative in theirchoice of airfoils With maturing CFD algorithmsespecially for the transition and turbulence modelsand more wind tunnel tests the trend is now to useairfoils specially designed and dedicated to windturbine blades see eg [22]

215 Wind simulationBesides airfoil data also realistic spatial-temporal

varying wind fields must be generated as input to anaeroelastic calculation of a wind turbine As aminimum the simulated field must satisfy somestatistical requirements such as a specified powerspectre and spatial coherence see [2324] In thismethod each velocity component is generatedindependently from the others meaning that thereis no guarantee for obtaining correct cross-correla-tions In [25] a method ensuring this is developed

on the basis of the linearized NS equations In thefuture wind fields are expected to be generatednumerically from Large Eddy Simulations (LES) orDirect Numerical Simulations (DNS) of the NSequations for the flow on a landscape similar to theactual siting of a specific wind turbine

22 Lifting line panel and vortex models

In the present section 3D inviscid aerodynamicmodels are reviewed They have been developed inan attempt to obtain a more detailed description ofthe 3D flow that develops around a wind turbineThe fact that viscous effects are neglected iscertainly restrictive as regards the usage of suchmodels on wind turbines However they should begiven the credit of contributing to a better under-standing of dynamic inflow effects as well as thecredit of providing a better insight into the overallflow development [1112] There have been attemptsto introduce viscous effects using viscousndashinviscidinteraction techniques [2627] but they have not yetreached the required maturity so as to becomeengineering tools although they are full 3D modelsthat can be used in aeroelastic analyses

221 Vortex methodsIn vortex models the rotor blades trailing and

shed vorticity in the wake are represented by liftinglines or surfaces [28] On the blades the vortexstrength is determined from the bound circulationthat stems from the amount of lift created locally bythe flow past the blades The trailing wake isgenerated by the spanwise variation of the boundcirculation while the shed wake is generated by atemporal variation and ensures that the totalcirculation over each section along the bladeremains constant in time Knowing the strengthand position of the vortices the induced velocitycan be found in any point using the BiotndashSavartlaw see later In some models (namely the lifting-line models) the bound circulation is found fromairfoil data table-look up just as in the BEMmethod The inflow is determined as the sum ofthe induced velocity the blade velocity andthe undisturbed wind velocity see Fig 1 Therelationship between the bound circulation and thelift is denoted as the KuttandashJoukowski theorem(first part of Eq (221)) and using this togetherwith the definition of the lift coefficient (secondpart of Eq (221)) a simple relationship betweenthe bound circulation and the lift coefficient can

be derived

L rV relG 1=2rV2relcCl ) G 1=2V relcCl

(221)

Any velocity field can be decomposed in asolenoidal part and a rotational part as

V rC$ rF (222)

where C is a vector potential and F a scalarpotential [29] From Eq (222) and the definition ofvorticity a Poisson equation for the vector potentialis derived

r2C o (223)

In the absence of boundaries C can be expressed inconvolution form as

Cxamp 1

4p

Zo0

x x0j jdVol (224)

where x denotes the point where the potential iscomputed a prime denotes evaluation at the pointof integration x0 which is taken over the regionwhere the vorticity is non-zero designated by VolFrom its definition the resulting induced velocityfield is deduced from the induction law of BiotndashSavart

wxamp 1

4p

Zx x0amp o0

x x0j j3dVol (225)

In its simplest form the wake from one blade isprescribed as a hub vortex plus a spiralling tipvortex or as a series of ring vortices In this case thevortex system is assumed to consist of a number ofline vortices with vorticity distribution

oxamp Gdx x0amp (226)

where G is the circulation d is the line Dirac deltafunction and x0 is the curve defining the location ofthe vortex lines Combining (225) and (226)results in the following line integral for the inducedvelocity field

wxamp 1

4p

Z

S

Gx x0ampx x0j j3

qx0

qS0 qS0 (227)

where S is the curve defining the vortex line and S0 isthe parametric variable along the curve

Utilizing (227) simple vortex models can bederived to compute quite general flow fields aboutwind turbine rotors The first example of a simplevortex model is probably the one due to Joukowski[30] who proposed to represent the tip vortices byan array of semi-infinite helical vortices with

ARTICLE IN PRESSMOL Hansen et al Progress in Aerospace Sciences 42 (2006) 285ndash330 291

constant pitch see also [31] In [32] a system ofvortex rings was used to compute the flow past aheavily loaded wind turbine It is remarkable that inspite of the simplicity of the model it was possibleto simulate the vortex ringturbulent wake statewith good accuracy as compared to the empiricalcorrection suggested by Glauert see [6] Further asimilar simple vortex model was used in [33] tocalculate the relation between thrust and inducedvelocity at the rotor disc of a wind turbine in orderto validate basic features of the streamtube-momentum theory The model includes effects ofwake expansion and as in [32] simulates a rotorwith an infinite number of blades with the wakebeing described by vortex rings From the model itwas found that the axial induced velocities at therotor disc are smaller than those determined fromthe ordinary stream tube-momentum theory Asimilar approach has been utilized by Koh andWood [34] and Wood [35] for studying rotorsoperating at high tip speed ratios

222 Panel methodsThe inviscid and incompressible flow past the

blades themselves can be found by applying asurface distribution of sources s and dipoles m(Fig 4) The background is Greenrsquos theorem whichallows obtaining an integral representation of anypotential flow field in terms of the singularitydistribution [3637]

Vx tamp V0 $ rfx tamp

fx tamp Z

S

s0tamp4p x x0j j

m0tamp n0r

4p x x0j j3

$ dS0

228amp

V0 denotes a given (external) potential flow fieldpossibly varying in time and space and f is theperturbation scalar potential S stands for the activeboundary of the flow and includes the solidboundaries of the flow SB as well as the wake

surfaces of all lifting components SW In (228) m sare defined as jumps of f and its normal derivativeacross S m 1fU and s 1qnfU with n definedas the unit normal vector pointing towards the flowSource distributions are responsible for displacingthe unperturbed flow so that the solid boundariesare shaped as flow surfaces and therefore are definedon SB Dipoles are added so as to developcirculation into the flow to simulate lift They aredefined on SW and the part of SB referring to thelifting components In fact a surface distribution ofm is identified to minus the circulation around aclosed circuit which cuts the surface on which m isdefined at one point G m

An important result given by Hess [36] states thatthe flow induced by a dipole distribution m defined onSm is given by a generalization of the BiotndashSavart law

rZ

Sm

m0n0 x x0amp4p x x0j j3

dS0

Z

Sm

r0m n0 amp x x0amp4p x x0j j3

dS0

$I

Sm

m0 s0 x x0amp4p x x0j j3

dS0 229amp

where the line integral is taken along the boundary ofSm and s is the unit tangent vector to qSm in theanticlockwise sense If Sm is a closed surface the lineintegral vanishes whereas if m is piecewise constant asin the vortex lattice method the surface term willvanish leaving only the line term which correspondsto a closed-loop vortex filament present along all linesof m discontinuity on Sm The two terms on the RHSof (229) have the form as the BiotndashSavart law(225) From this analogy c rm n is calledsurface vorticity and ms line vorticity which justifiesthe term vortex sheet for the wake of lifting bodies

In potential theory a wake surface is the idealiza-tion of a shear layer in the limit of vanishingthickness For an incompressible flow the flow willexhibit a velocity jump 1VU(W rmW while1VU(W n 0 1pU(W 0 Using Bernoullirsquos equa-tion it follows that

1pU(Wr

qmWqt

$ VWm 1VU(W

qmWqt

$ VWm ramp

mW 0 2210amp

where VWm V$W $ V

W

amp =2 Since G mW

Kelvinrsquos theorem is obtained from (229) providedthat SW is a material surface moving with the mean

ARTICLE IN PRESS

nr

BS micro

W WS micro

( )Pmicro

F

extUr

C

C = W

Fig 4 Notations for the potential flow around a wing

flow VWm Circulation will be materially conservedand therefore mW is identified with its value at t 0For a lifting problem this means that mW is knownfrom the history of the wing loading Assuming thatSW starts at the trailing edge of the wing thegeneration of the wake can be viewed as acontinuous release of vorticity in the free flowThe streak line of a point along the trailing edge willreveal the history of the loading of the specific wingsection as indicated in Fig 7

The first model developed within the abovecontext is Prandtlrsquos lifting-line theory see eg [38]It concerns a lifting body of vanishing chord (or elselarge aspect ratio) and thickness (Fig 5) So s ) 0whereas SB becomes a line carrying the loading Gyamp(bound vorticity) which is the only unknown sincethe vorticity in the wake (trailing vorticity) is givenby yGyamp In Prandtlrsquos original model Gyamp isdetermined from airfoil data as equation (221)Then as an introduction to lifting-surface theorybound vorticity was placed along the c4 line whilealong the 3c4 the non-penetration condition wasapplied in order to determine Gy tamp The next stepwas to introduce the lifting body as a lifting surfaceThe most widely used model in this respect is thevortex-lattice model [39] It consisted of dividing SB

and SW into panels and defining on them piecewiseconstant m distributions (Fig 6) Then according to(229) the perturbation induced by the wing and itswake is generated by a set of closed-loop vortexfilaments each defined along the boundary of apanel The dipole intensities on the wing can bedetermined by the non-penetration condition at thepanel centres whereas along the trailing edge see(2210) m mW to ensure zero loading locally Inthe case of an unsteady flow the loading GB will

change so that the vorticity shed in the wake willalso have a cross component qGB=qtdt asindicated in Fig 6

Having determined m it is possible to calculate theinduced velocity and thus the angle of attack andfinally the loads from an airfoil data table look-upAnother option frequently used in propeller appli-cations is to determine lift by integrating thepressure jump along the section Then by consider-ing that the lift force is perpendicular to the effectiveinflow direction the angle of attack is determined Inthis case the pressure jump is obtained directly fromBernoullirsquos equation over the section except at theleading edge where the geometrical singularity of ablade with no thickness makes it necessary toinclude the so-called suction force In propellerapplications where this concept was first introducedthe suction force is estimated by means of semi-empirical modelling [40] Whenever used in windenergy applications the suction force has beendetermined as rV Gdl where V G are the localvalues at the leading edge and dl represents thevector length along the leading edge line In generalthe two schemes give comparable results It isdifficult to clearly state which scheme is better touse since deviations appear as the angle of attackincreases so that a theoretical justification based onmatched asymptotic expansions is difficult Anotherpoint of concern regarding both lifting theories isthe detail in which the flow can be recorded Thefact that the flow geometry is approximate suggeststhat only at some distance from the solid boundarythe flow could be meaningful Finally for the samereason viscous corrections based on boundary layertheory cannot be applied

In order to overcome these difficulties the exactgeometry of the flow had to be included This wasdone by Hess who first introduced the panel method

ARTICLE IN PRESS

= (y )y

(y)

w

y) )

Fig 5 The lifting-line model

emission line

Zero loading attrailing edge

Se

W B

B

= t

(

(t=

w

+

Fig 6 The lifting-surface model

in its full form [41] Now the panel grid is defined onthe true solid boundary and a piecewise constantsource distribution is introduced which can bedetermined by the no-penetration boundary condi-tion In order to account for lift dipole distributionsare added Because there is no kinematic conditionto determine m Hess defined m to vary linearly alongthe contour of each section (Fig 7) ms y tamp s Gy tamp=Lyamp and used (2210) at the trailing edgeas an extra condition for determining Gy tamp (Fig7) The resulting problem is non-linear and aniterative procedure must be used which penalizes thecomputational cost considerably as compared tothe lifting-surface model

An important aspect of potential flow modelsconcerns wake dynamics As discussed earlier thewake of a lifting body is a moving surface and SW

should be allowed to change in time Regarded as avortex sheet the evolution of SW in time will besubjected to convection and deformation Forexample in the case of the vortex lattice methodeach wake segment will conserve its intensity but itsvector length dl will satisfy the following equation

d

dtdl dl rampV (2211)

As time evolves wake deformations will generatesingularities such as intense roll-up along the wakeextremities and crossings In fact at finite time theflow will blow-up In order to avoid blow-upcorrective actions must be taken If the simulationretains the connectivity of the wake surface thelines defining the wake must be smoothenedregularly during the run time Alternatively onecan apply remeshing which consists in dividing thewake vortex segments so that they do not exceed a

prescribed upper limit Both schemes howeverrequire substantial bookkeeping and quite intenseprocedures With panel methods this problembecomes more complicated because the wake willalso contain a surface vorticity term A completelydifferent approach is to discard wake connectivityRehbach [42] was the first to note that for anincompressible flow vorticity concentrations dO ofthe type o dD g dS and Gdl behave similarly Theirkinematic analogy was already discussed withreference to (229) Dynamically they all satisfythe same evolution equation

D

DtdO )

qqt

dO$ urampdO dOrampu (2212)

where D=Dt denotes the total or material timederivative So Rehbach integrated the wake vorti-city into point vortices which subsequently movedfreely as fluid particles carrying vorticity Thisprocedure provides substantial flexibility in theevolution of the wake He also introduced theconcept of modifying the kernel of the BiotndashSavartlaw in order to cancel the r2 singularity Thetheoretical justification of Rehbachrsquos method camelater on leading to the development of the vortexblob method [43]

In vortex models the wake structure can either beprescribed or computed as a part of the overallsolution procedure In a prescribed vortex techni-que the position of the vortical elements is specifiedfrom measurements or semi-empirical rules Thismakes the technique fast to use on a computer butlimits its range of application to more or less well-known steady flow situations For unsteady flowsituations and complicated wake structures freewake analysis become necessary A free wakemethod is more straightforward to understand anduse as the vortex elements are allowed to convectand deform freely under the action of the velocityfield as in Eq (2212) The advantage of the methodlies in its ability to calculate general flow cases suchas yawed wake structures and dynamic inflow Thedisadvantage on the other hand is that the methodis far more computing expensive than the prescribedwake method since the BiotndashSavart law has to beevaluated for each time step taken Furthermorefree wake vortex methods tend to suffer fromstability problems owing to the intrinsic singularityin induced velocities that appears when vortexelements are approaching each other To a certainextent this problem can be remedied by introducinga vortex core model in which a cut-off parameter

ARTICLE IN PRESS

L

micro = L

W E emission timemicro micro=

TE tmicro TE t t

micro -2TE t t

micro - 3TE t tmicro

-

wake surface

streak line

( )=B TEy t micro

emiss

ion

line

s

T

Fig 7 The exact potential model of a wing

models the inner viscous part of the vortex filamentIn recent years much effort in the development ofmodels for helicopter rotor flow fields have beendirected towards free-wake modelling using ad-vanced pseudo-implicit relaxation schemes in orderto improve numerical efficiency and accuracy eg[4445]

To analyse wakes of horizontal axis windturbines prescribed wake models have been em-ployed by eg [46ndash48] Free vortex modellingtechniques have been utilized by eg [4950]A special version of the free vortex wake methodsis the method described in [51] where the wakemodelling is taken care of by vortex particles orvortex blobs

Recently the model of [52] was employed in theNREL blind comparison exercise [137] and themain conclusion from this was that the quality ofthe input blade sectional aerodynamic data stillrepresents the most central issue to obtaining high-quality predictions Nevertheless it is worth noti-cing that by introducing relaxing techniques in thewake evolution [53] it is nowadays possible to run alarge number of revolutions which is of importancein aeroelasticity with reference to fatigue andstability analysis see later

An alternative to panel methods is offered by theBoundary Integral Equation Methods (BIEM) Byassuming stagnant flow inside the blade m fand s nf the resulting integral equation is onlyweakly singular and so less expensive Within thefield of wind turbine aerodynamics BIEMs havebeen applied by eg [54ndash56] up to now howeveronly in simple flow situations

Vortex methods have been applied on windturbine rotors particularly in order to better under-stand wake dynamics The next and quite challen-ging step is to upgrade potential flow methods so asto also include viscous effects Examples of applyingviscousndashinviscid coupling within the context of 3Dboundary layer theory can be found in [2657] Alsoattempts to include separation were made in [58]All these works however cannot be consideredconclusive There are several unresolved issues suchas convergence at the inboard region wheresignificant radial flow develops as a result ofsubstantial separation and the end conditions atthe tip The fact that current trends in wind turbinedesign indicate preference to pitch regulated ma-chines could increase the interest in flow modelsbased on inviscid considerations Finally anotherapplication of potential flow models is to use them

in order to obtain far field conditions for RANScomputations in view of reducing their computa-tional cost [59]

23 Generalized actuator disc models

The actuator disc model is probably the oldestanalytical tool for analysing rotor performance Inthis model the rotor is represented by a permeabledisc that allows the flow to pass through the rotorat the same time as it is subject to the influence ofthe surface forces The lsquoclassicalrsquo actuator discmodel is based on conservation of mass momentumand energy and constitutes the main ingredient inthe 1D momentum theory as originally formulatedby Rankine [60] and Froude [61] Combining it witha blade-element analysis we end up with thecelebrated Blade-Element Momentum Technique[6] In its general form however the actuator discmight as well be combined with the Euler or NSequations Thus as will be shown in the followingno physical restrictions have to be imposed on thekinematics of the flow

A pioneering work in analysing heavily loadedpropellers using a non-linear actuator disc model isfound in [62] Although no actual calculations werecarried out this work demonstrated the opportu-nities for employing the actuator disc on compli-cated configurations such as ducted propellers andpropellers with finite hubs Later improvementsespecially on the numerical treatment of theequations are due to [6364] Recently Conway[6566] has developed further the analytical treat-ment of the method Within wind turbine aero-dynamics [67] developed a semi-analytical actuatorcylinder model to describe the flow field about avertical-axis wind turbine A thorough review oflsquoclassicalrsquo actuator disc models for rotors in generaland for wind turbines in particular can be found inthe dissertation [68] Later developments of themethod have mainly been directed towards the useof the NS or Euler equations

In a numerical actuator disc model the NS (orEuler) equations are typically solved by a second-order accurate finite differencevolume scheme as ina usual CFD computation However the geometryof the blades and the viscous flow around the bladesare not resolved Instead the swept surface of therotor is replaced by surface forces that act upon theincoming flow This can either be implemented at arate corresponding to the period-averaged mechan-ical work that the rotor extracts from the flow or by

using local instantaneous values of tabulated airfoildata

In the simple case of an actuator disc withconstant prescribed loading various fundamentalstudies can easily be carried out Comparisons withexperiments have demonstrated that the methodworks well for axisymmetric flow conditions andcan provide useful information regarding basicassumptions underlying the momentum approach[69ndash72] turbulent wake states occurring for heavilyloaded rotors [73] and rotors subject to coning[7475]

In Fig 8 an example of how various wake statescan be investigated by introducing a constantlyloaded actuator disc into the axisymmetric NSequations is shown By changing the thrust coeffi-cient all types of flow states can be simulatedranging from the wind turbine state through thechaotic wake state to the propeller state

The generalized actuator disc method resemblesthe BEM method in the sense that the aerodynamicforces has to be determined from measured airfoilcharacteristics corrected for 3D effects using ablade-element approach For airfoils subjected totemporal variations of the angle of attack thedynamic response of the aerodynamic forceschanges the static aerofoil data and dynamic stallmodels have to be included However correctionsfor 3D and unsteady effects are the same forgeneralized actuator disc models and the BEM

model hence the description of how to deriveaerofoil data is the same as in Section 21

In helicopter aerodynamics combined NSactua-tor disc models have been applied by eg [76] whosolved the flow about a helicopter employing achimera grid technique in which the rotor wasmodelled as an actuator disk and [77] whomodelled a helicopter rotor using time-averagedmomentum source terms in the momentum equa-tions

Computations of wind turbines employing nu-merical actuator disc models in combination with ablade-element approach have been carried out ineg [69707879] in order to study unsteadyphenomena Wakes from coned rotors have beenstudied by Madsen and Rasmussen [74] Mikkelsenet al [75] and Masson et al [78] rotors operating inenclosures such as wind tunnels or solar chimneyswere computed by Hansen et al [80] Phillips andSchaffarczyk [81] and Mikkelsen and Soslashrensen [82]and approximate models for yaw have beenimplemented by Mikkelsen and Soslashrensen [83] andMasson et al [78] Finally techniques for employ-ing the actuator disc model to study the wakeinteraction in wind farms and the influence ofthermal stratification in the atmospheric boundarylayer have been devised by Masson [84] andAmmara et al [85]

The main limitation of the axisymmetric assump-tion is that the forces are distributed evenly along

ARTICLE IN PRESS

Fig 8 Various wake states computed by actuator disc model with prescribed loading (a) wind turbine state (b) turbulent wake state (c)vortex ring state (d) hover state Reproduced from [7073]

the actuator disc hence the influence of the blades istaken as an integrated quantity in the azimuthaldirection To overcome this limitation an ex-tended 3D actuator disc model has recently beendeveloped [86] The model combines a 3D NSsolver with a technique in which body forcesare distributed radially along each of the rotorblades Thus the kinematics of the wake isdetermined by a full 3D NS simulation whereasthe influence of the rotating blades on the flowfield is included using tabulated airfoil data torepresent the loading on each blade As in theaxisymmetric model airfoil data and subsequentloading are determined iteratively by computinglocal angles of attack from the movement ofthe blades and the local flow field The conceptenables one to study in detail the dynamics ofthe wake and the tip vortices and their influenceon the induced velocities in the rotor plane A modelfollowing the same idea has recently been sug-gested by Leclerc and Masson [87] A mainmotivation for developing such types of model isto be able to analyse and verify the validity ofthe basic assumptions that are employed in thesimpler more practical engineering models Re-views of the basic modelling of actuator discand actuator line models can be found in [88] thatalso includes various examples of computationsRecently another PhD dissertation [89] carried outa simulation employing more than four millionmesh points in order to study the structure of tipvortices In the following we will give someexamples of how the actuator discline techniquemay help in understanding basic features of windturbine flows

Computed iso-contours of vorticity for a three-bladed rotor with airfoil characteristics corres-ponding to the Tjaeligreborg wind turbine is shownin Fig 9 In Fig 10 a similar computationshows the formation of the trailing tip vortices Itis remarkable that the vortices are clearly visiblemore than 3 turns downstream A new andinteresting application of the actuator line modelis to study the interaction between two or moreturbines especially for simulating park effects InFig 11 the outcome of a computation in which theinteraction between two wind turbines is simulatedby replacing the two rotors by actuator lines withforces obtained from airfoil data is shown Pre-sently this technique is used to investigate the effectof large wind farms including many up- anddownstream wind turbines

24 Navierndash Stokes solvers

241 Introduction to computational rotoraerodynamics

The first applications of CFD to wings and rotorconfigurations were studied back in the lateseventies and early eighties in connection withairplane wings and helicopter rotors [90ndash94] using

ARTICLE IN PRESS

Fig 9 Actuator line computation showing vorticity contoursand part of computational mesh around a three-bladed rotorReproduced from [88]

Fig 10 Iso-surface of constant vorticity showing the formationof tip and root vortices Reproduced from [89]

potential flow solvers To overcome some of thelimitations of potential flow solvers a shift towardsunsteady Euler solvers were seen through theeighties [95ndash98] When computing power allowedthe solution of full Reynolds Averaged NS equa-tions the first helicopter rotor computations in-cluding viscous effects were published in the lateeighties and early nineties [99ndash102]

In the late nineties with the CFD solvers capableof handling viscous flow around rotors applicationto wind turbine rotors became of practical interest

The first full NS computations of rotor aero-dynamics was reported in the literature in the latenineties [103ndash107] The European effort to apply NSsolvers to rotor aerodynamics had been madepossible through a series of National and Europeanproject through the nineties The European projectsdealing with development and application of the NSmethod to wind turbine rotor flows was the ViscousEffects on Wind turbine Blades (VISCWIND) from1995 to 1997 [108] Viscous and Aeroelastic effects onWind Turbine Blades (VISCEL) 1998 to 2000[109110] and Wind Turbine Blade Aerodynamicsand Aeroleasticity Closing Knowledge Gaps 2002 to2004 [111ndash115]

242 ApproachesAs a consequence of the origin of most CFD

rotor codes from the aerospace industry and relatedresearch many existing codes are solving thecompressible NS equations and are intended forhigh-speed aerodynamics in the subsonic andtransonic regime [116ndash120] For the helicopterapplications where compressibility plays an impor-

tant role this is the natural choice For wind turbineapplications however the choice is not as obviousone reason being the very low Mach numbers nearthe root of the rotor blades As the flow hereapproaches the incompressible limit Mach001 itis very difficult to solve the compressible flowequations One remedy to improve their capabilityis the so-called preconditioning that changes theeigenvalues of the system of the compressible flowequations by premultiplying the time derivatives bya matrix On the other hand the compressiblesolvers have many attractive features amongthese the ease of implementation of overlappingand sliding meshes application of high-order up-wind schemes and very well-developed solutionsmethods

Another very popular method especially in theUS is the Artificial Compressibility Method[121122] where an artificial sound speed is intro-duced to allow standard compressible solutionmethods and schemes to be applied for incompres-sible flows In case of transient computations sub-iterations are taken within each time step to enforceincompressibility [122] The method has severalattractive features Among these a similar ease ofimplementation of overlapping grids as the com-pressible codes Overlapping grids are a necessity tosolve rotorstator problems that are present whenthe rotor tower and nacelle are all included in thecomputations The main shortcoming of the methodmay be problems to enforce incompressibility intransient computations without the need for a hugeamount of sub-iterations and the problem ofdetermining the optimum artificial compressibilityparameter

Due to the low Mach number encountered inwind turbine aerodynamics an obvious choice isthus the incompressible NS equations Thesemethods are generally based on treating pressureas a primary variable [123ndash125] Extensions togeneral curvilinear coordinates can be made alongthe lines of [126] The method is not as easilyextended to overlapping grids as the compressibleand the artificial compressibility method due to theelliptical pressure correction equation But themethod is well suited for solving the nearlyincompressible problems often experienced in con-nection with wind energy In connection withsteady-state problems the method can be acceler-ated using local time stepping while the methodusing global time stepping still is well suited fortransient computations

ARTICLE IN PRESS

Fig 11 Interaction of the wake between two partly aligned windturbines Reproduced from [88]

243 Turbulence and transitionIt is well known that the NS equations cannot be

directly solved for any of the cases of practicalinterest to wind turbines and that some kind ofturbulence modelling are needed The standardapproach to derive turbulence models is by timeaveraging the NS equation resulting in the so-calledReynolds Averaged NS equations (RANS) Severaldifferent models have been used with good resultsfor wind turbine applications the most successfulones being the k-omega SST model of Menter [127]the SpalartndashAllmaras model [128] and the Bald-winndashBarth model [129] The BaldwinndashLomax [130]model often used in connection with helicopter andfix-wing applications are not very well suited forwind turbine applications where relatively highangles of attack are very common

Several studies performed for stall controlledwind turbines have shown that all RANS modelslack the capability to model the stalled flow regimeat high wind speeds One possible way around thisproblem the so-called Detached Eddy Simulation(DES) technique [131132] has shown some promis-ing results but still needs further validationAdditionally the DES technique is much morecomputationally expensive than the standardRANS approach as it needs much finer computa-tional meshes and the computations needs to becomputed with time accurate algorithms

From experiments it is known that laminarturbulent transition influences the flow over rotorblades for some cases It has been demonstratedfor 2D applications that transition models cangreatly improve the accuracy for cases wheretransition phenomena are important Even thoughnearly all rotor studies so far have been com-puted assuming fully turbulent conditions it isgenerally accepted that it is important to in-clude laminarturbulent transition to model thephysics as close as possible [133134] Predictingtransition in 3D is a much more complex task thandealing with 2D and 3D transition is an activeresearch field

244 Geometry and grid generationTo compute a rotor using CFD the first step is to

obtain a digitized description of the blade geometryOften the blade descriptions are given as spanwisesectional information listing the airfoil section thetwist the thickness and the position with respect tothe blade axis Often the blades are highly twistedand with a large taper in the spanwise direction

Depending on the flow solver different ap-proaches to the mesh generation process existCartesian cut cells unstructured and structuredand combinations of these So far the majority offlow solvers applied to wind turbine research haveutilized structured grids with hexahedral cells Inconnection with structured grids there are severalissues that need to be decided upon Generally theproblem of making a high quality grid around amodern rotor cannot be handled by a single blockconfiguration but needs some kind of multi blockmesh These can either be conforming at the blockboundaries non-conforming or overlapping Theoverlapping grids gives the highest degrees offreedom followed by the non-conforming and theconforming grids Firstly the grid needs to accu-rately resolve the blade shape with good resolutionof the leading edge and tip region Secondly thegrids also need to resolve the regions around theblade with sufficient resolutions to capture the flowphysics As the Reynolds numbers are quite high1ndash6 million the cells near the rotor blades becomevery thin as the non-dimensional distance y+ mustbe approximately 1 to resolve the laminar sub-layerand have accurate solutions The mesh generationprocess calls for some degree of experience and gridrefinement studies to verify that the grid is sufficientto resolve the desired physics Also the grids need toextend far away from the rotor in the order ofseveral rotor diameters to avoid disturbing theinduced velocity field near the rotor blades Foraxial flow conditions the flow solvers often takeadvantage of the rotational periodicity of the rotorsolving only for a single blade using periodicconditions

Using an unstructured flow solver with tetrahe-dral cells the grid generation process is lesscumbersome But the problem of resolving verythin boundary layers using tetrahedral cells is wellknown and it may be necessary to combine thesolver with some kind of prismatic grids near theblade surface to avoid this problem The use ofunstructured flow solvers is not wide spread inconnection with wind turbine aerodynamics prob-ably because of the limited geometrical complexityand the strength of unstructured solvers mainlybeing their ability to cope with complex geometries

245 Numerical issuesThe codes typically used for wind turbines are of

at least second-order accuracy in both time andspace often with an implicit time discretization

scheme to loosen the time step restriction inherentto explicit methods Typically the viscous terms arediscretized with central differences while the con-vective terms are discretized with second- or third-order upwind schemes To solve routinely for 5ndash10million grid points the solvers are often available ina parallelized version that allows for execution onseveral CPUrsquos in parallel The rotating nature of theproblem requires the use of either a moving frameincluding the non-inertial acceleration terms or amoving mesh option where so-called mesh fluxesmust be included in the code For a good overviewof the numerical issues in connection with incom-pressible flow see [135]

246 Application of CFD to wind turbineaerodynamics

The major part of wind turbine rotor computa-tions performed until now has been focused on zeroyaw rotor only configuration where the nacelle andtower have been neglected and the inflow to therotor has been assumed to be steady without shearThis is of course a great simplification but in manycases still a sufficiently good approximation Theeffect of the tower on the rotor on an upwindturbine is comparable to other unsteady effectssuch as incoming turbulence time variations of therotor and of the incoming flow A simulationworking with a full turbine geometry has been tried[106] This type of simulation is much moreexpensive and needs some kind of slidingover-lapping mesh to accommodate the movement of therotor with respect to the turbine tower and nacelleAdditionally the simulation needs to be timeaccurate and good resolution of the flow aroundthe tower is needed to capture the tower wake fardownstream of the turbine

One of the first real proofs that CFD for windturbine rotor applications can be useful came inconnection with the blind comparison organized bythe National Renewable Energy Laboratory inBoulder Colorado in December 2000 [136ndash138]Some of these results were later published in[139140] Here several wind turbine research groupswere asked to compute a series of differentoperational conditions for the NREL Phase-VIturbine corresponding to actual cases measured inthe NASA Ames 80 120 ft wind tunnel When theresults were made publicly available it proved thatone of the applied CFD codes were consistentlyreproducing the measured distribution of the aero-dynamic forces along the blade span even underhighly 3D and extreme stall conditions

The output extracted from typical CFD rotorcomputations are the low-speed shaft torque orpower production and root flap moments see Fig12 For modern full size commercial turbines thepower and root flap moment are typically the onlyavailable properties Besides the quantities normallymeasured the CFD simulations provide a hugeamount of detailed information that can be used toprovide more insight The data typically extractedare the spanwise distributions of force coefficientsFig 13 the limiting streamlines on the bladesurfaces Fig 14 and the sectional pressuredistributions along the blade span Fig 15 Forthe NREL Phase-VI turbine these detailed quan-tities can be compared to measurements but this isnot generally the case for commercially availableturbines

Another application of rotor CFD is the study ofdifferent aerodynamic details of the rotor such asthe blade tips the design of the root section etcHere the CFD technique can be used to supply

ARTICLE IN PRESS

700800900

10001100120013001400150016001700

6 8 10 12 14 16 18 20 22 24 26

Low

Spe

ed S

haft

Tor

que

[Nm

]

Wind Speed [m s]

MeasuredComputed

1000

1500

2000

2500

3000

3500

4000

4500

5000

6 8 10 12 14 16 18 20 22 24 26

Roo

t Fla

p M

omen

t [N

m]

Wind Speed [ms]

MeasuredComputed

Fig 12 Comparison of computed and measured low-speed-shaft torque and root flap moment for the NREL Phase-VI rotor during axialflow conditions The 7 one standard deviation of the measurements is indicated around the measured values

information that the engineering methods are notcapable of providing

With the increase in computational power it istoday both possible and affordable to do yawcomputations using CFD [133141ndash143] In contrastto the axial flow cases the total rotor must be

modelled in yaw simulations thereby increasing thenumber of mesh points typically by a factor ofthree Additionally the azimuthally variation in-herent in yaw simulations dictates a time accuratesimulation as no steady state solution can existthereby increasing the computing time severelyTypically a yaw computation will be 10ndash20 timesmore expensive compared to steady state axial flowcomputations Fig 16 shows the normal andtangential force at the rR 30 radius duringone revolution at 601 yaw operation

The release of the measurements on the NRELPhase-VI rotor has heavily influenced the CFDactivities dealing with wind turbine rotor aerody-namics This unique data set with several well-documented cases has given a new possibility to testdetails of state of the art CFD codes In the yearssince the release of the measurements nearly half ofall published CFD studies of wind turbine rotorsdeals with these measurements Besides the refer-ences mentioned other places in this paper thefollowing studies use the NRELNASA Amesmeasurements [144ndash147]

Recently DES simulations of rotors at realisticoperational conditions have been attempted [148]Again the NREL Phase-VI rotor was used to verifythe model The reason for this choice is besides theavailability of detailed measurements the fact thatthe rotor has a limited aspect ratio hence makingthe DES computations more affordable with respectto the number of grid cells The mesh for thissimulation consists of 15 million cells compared toaround 2 million cells for a standard RANS rotorcomputation The fact that DES computations mustbe performed using time accurate computationsmakes these types of simulations 20ndash40 times more

ARTICLE IN PRESS

0

05

1

15

2

25

3

02 03 04 05 06 07 08 09 1

CN

rR

MeasuredComputed

-008

-006

-004

-002

0

002

004

02 03 04 05 06 07 08 09 1

CT

rR

MeasuredComputed

Fig 13 Spanwise normal and tangential force coefficients for run S20000000 of the NRELNASA Ames measurements correspondingto a wind speed of 20ms

NREL PHASE-6 W=10 [m s] Limiting StreamlinesRISOE EllipSys 3D Computation

Fig 14 Limiting streamlines on the surface of the NREL Phase-VI rotor for the 10ms axial case The picture shows a suddenleading edge separation around rR 047

-15

-1

-05

0

05

1

15

2

25

3

0 02 04 06 08 1

-Cp

X Chord

rR=063

Fig 15 Comparison of the measured and computed pressuredistribution at the rR 063 section of the NREL Phase-VIblade the measurements are shown as circles while thecomputations are shown with lines

expensive than standard steady-state rotor compu-tations For modern-type rotors with large aspectratios the cell count would be even higher and thecomputations even more expensive For the NRELPhase-VI rotor the improvement using the DEStechnique is very limited but for other rotors wherethe RANS equations do not perform as well DESmay provide much more improvement The use ofpure Large Eddy Simulation has been demon-strated in [149] where a computational grid of 300million points is used to compute the initialtransient of the development of a tip vortex

247 FutureToday NS solvers are an important tool for

analysing different wind turbine rotor configura-tions and are used routinely along with measure-ments and other computational tools fordevelopment and investigation of wind turbinesNS solvers are especially well suited for detailedinvestigation of phenomena that cannot directly beaccessed by simpler and less computationallyexpensive methods Additionally NS methods canbe used as a supplement to measurements wherethey can be used both in the planning phase and tohave a better interpretation of the actual physics inconnection with the analysis of measurements

Finally one of the latest trends is to couple NSsolvers to structural codes to perform full elasticcomputations of wind turbine rotors

3 Structural modelling of a wind turbine

The main purpose of a structural model of a windturbine is to be able to determine the temporalvariation of the material loads in the variouscomponents This is accomplished by calculating

the dynamic response of the entire constructionsubject to the time-dependent load using an aero-dynamic model such as the BEM method Foroffshore wind turbines also wave loads and perhapsice loads on the bottom of the tower must beestimated Two different and frequently usedapproaches to set up a dynamic structural modelfor a wind turbine are described in the nextsubsections

31 Principle of virtual work and use of modal shapefunctions

The principle of virtual work is a method to setup the correct mass matrix M stiffness matrix Kand damping matrix C for a discretized mechanicalsystem as

M x$ C _x$ Kx Fg (311)

where Fg denotes the generalized force vectorassociated with the external loads p Eq (311) isof course nothing but Newtonrsquos second law assum-ing linear stiffness and damping and the method ofvirtual work is nothing but a method that helpssetting this up for a multibody system and that isespecially suited for a chain system Knowing theloads and appropriate conditions for the velocitiesand the deformations Eq (311) can be solved forthe accelerations wherefrom the velocities anddeformations can be determined for the next timestep The number of elements in x is called thenumber of DOF and the higher this number themore computational time is needed in each time stepto solve the matrix system Use of modal shapefunctions is a tool to reduce the number of DOFand thus reduce the size of the matrices to make thecomputations faster per time step A deflection

ARTICLE IN PRESS

-2

0

2

4

6

8

10

12

0 50 100 150 200 250 300 350

CN

Azimuth Angle [deg]

S1500600 r R=030

MeasurementsComputed

-04

-02

0

02

04

06

08

1

0 50 100 150 200 250 300 350

CT

Azimuth Angle [deg]

S1500600 rR=030

Fig 16 Azimuth variation of normal and tangential force coefficient for the NREL Phase-VI turbine at the rR 04 section during a601 yaw error at 15ms wind speed

shape is here described as a linear combination of afew but physical realistic basis functions which areoften the deflection shapes corresponding to thelowest eigenfrequencies (eigenmodes) For a windturbine such an approach is suited to describe thedeflection of the tower and the rotor blades and theassumption is that the combination of the PowerSpectral Density of the loads and the damping ofthe system do not excite the eigenmodes associatedwith higher frequencies In the commercially avail-able and widely used aeroelastic simulation toolFLEX see eg [150] only the first 3 or 4 (2 flapwiseand 1 or 2 edgewise) eigenmodes are used for theblades Results from this model are generally ingood agreement with measurements indicating thevalidity of the underlying assumption First one hasto decide on the DOF necessary to describe arealistic deformation of a wind turbine For instancein FLEX4 17ndash20 DOFs are used for a three bladedwind turbine with 3ndash4 DOFs per blade as describedabove 4 DOFs for the deformation of the shaft (1for torsion 2 for the hinges just before the firstbearing with associated angular stiffness to describebending and 1 for pure rotation) 1 DOF todescribe the tilt stiffness of the nacelle and finally3 DOFs for the tower (1 for torsion 1 for the firsteigenmode in the direction of the rotor normal and1 in the lateral direction)

The method of virtual work will only be brieflydescribed For a more rigorous explanation of themethod the reader is referred to textbooks ondynamics of structures The values in the vectordescribing the deformation of the construction xiare denoted the general coordinates To eachgeneralized coordinate is associated a deflectionshape ui that describes the deformation of theconstruction when only xi is different from zero andtypically has a unit value The element i in thegeneralized force corresponding to a small displace-ment in DOF number i dxi is calculated such thatthe work done by the generalized force equals thework done on the construction by the external loadson the associated deflection shape

Fgidxi Z

S

p uidS (312)

where S denotes the entire system Please note thatthe generalized force can be a moment and that thedisplacement can be angular All loads must beincluded ie also gravity and inertial loads such asCoriolis centrifugal and gyroscopic loads The non-linear centrifugal stiffening can be modelled as

equivalent loads calculated from the local centrifu-gal force and the actual deflection shape as shown in[151] The elements in the mass matrix mij can beevaluated as the generalized force from the inertialoads from an unit acceleration of DOF j for a unitdisplacement of DOF i The elements in the stiffnessmatrix kij correspond to the generalized force froman external force field which keeps the system inequilibrium for a unit displacement in DOF j andwhich then is displaced xi dij where dij isKroneckers delta The elements in the dampingmatrix can be found similarly For a chain systemthe method of virtual work as described herenormally gives a full mass matrix and diagonalmatrices for the stiffness and damping For oneblade rigidly clamped at the root (cantilever beam)it is relatively easy to estimate the lowest eigen-modes (first flapwise u1f(x) first edgewise u1e(x) andsecond flapwise u2f(x)) eg using an iterative methodas described in [151] The eigenmodes are normallydescribed in a coordinate system aligned with the tipchord as eg shown in Fig 1 It is practical tonormalize the deflection shapes so that the tipdeflection is unity It is now assumed that anydeflection can be described as a linear combinationof these modes as

uxamp x1u1f xamp $ x2u

1examp $ x3u2f xamp (313)

The velocity and accelerations can be calculatedrespectively as

_uxamp _x1u1f xamp $ _x2u

1examp $ _x3u2f xamp (314)

and

The advantage of the method using generalizedcoordinates and modal shape functions is that thenumber of DOF in the dynamic system can bereduced to a relatively small number Further somehigh eigenfrequencies are filtered away which isbeneficial for the allowable time step when comput-ing deformations xn$1

i and velocities _xn$1i at time

t (n+1)Dt from deformations xni velocities _xni and accelerations xni at time t nDt A goodchoice for the time integration scheme is theRungendashKuttandashNystrom method which requiresfor stability reasons that the time step shouldresolve the highest eigenfrequency with 4 pointsbut for accuracy reasons 10 points is preferredBy reducing the highest eigenfrequency using amodal description of eg the blades not only

reduces the number of DOFs but also larger timesteps can be taken

32 FEM modelling of wind turbine componentsapplying non-linear beam theory

Even though modal analysis offers a computa-tionally effective way to analyse wind turbinedynamics most of the recently developed aeroelasticcodes use a full FEM approach [152] which allowsa more complex deformation state of the windturbine The main features of such modellingprocedures together with some aspects of non-linearbeam theory are discussed in the present section

The structural modelling of wind turbines isbased on beam theory For a 1D structure (beam)subjected to bending in two directions torsion andaxial tension the formulation of the problemconsists of two parts (a) the elastic model whichreduces the 3D structure of each component into a1D structure concentrated along the elastic axis ofthe beam and (b) the derivation of the dynamicequations In classical beam theory (first order) thedeformed position r of any point initially at x0 x y zampT (Fig 17) can be described by the displace-ment vector u fu vw ygT consisted of the two

bending displacements u w the torsion angle y andthe tension v

r n0 $ S0 u$ S1 yu

S0 1 0 0 z

0 1 0 0

0 0 1 x

2

64

3

75

S1 0 0 0 0

x 0 z 0

0 0 0 0

2

64

3

75

(321)

Using Hookersquos law and assuming that shear stressesdo not produce net torsion the sectional elasticloads can be derived

Fy EAampqyv EAx+ampq2yyu EAz+ampq2yyw

My GItampqyy

Mx EIxzampq2yyu$ EIxxampq2yyw EAz+ampqyv

Mz EIzzampq2yyu EIxzampq2yyw$ EAx+ampqyv

322amp

where the terms in parentheses denote the averagedsectional properties of the structure By consideringthe balance of loads and moments on of a beam

ARTICLE IN PRESS

Undeformed geometryDeformed geometrydy

y

x

z

A

u

w

u+du

w+dwrDeformed elastic axis

M + dMz

z

M + dMyy

M + dMxx

zF + dF

yyF + dF

xxF + dF

Mx

Fx

Mz

FzMy

Fy dr

ra

)P

x

z

y

A(a) (b)

a

b

Fig 17 Kinematics and dynamics of a beam structure

element of length dy the beam equations arederived

Z

A

rdm

$ dy dF$

Z

A

g dm

$ dy$ dpdy

(323)

Z

A

r0 rdm

$ dy dM$ dr F$ dFamp

$Z

A

r0 gdm

$ dy$ ra dpdy 324amp

where m denotes the mass per unit length dF dMare the net elastic loads on dy g is the accelerationof gravity and dp the sectional aerodynamic loadsexerted at the aerodynamic centre ra xa 0 zaampwhile

r0 fdu$ x$ zy dv$ dydw$ z xygT

fx$ zy dy z xygT

In view of a more systematic and general frameworkfor formulating dynamic equations Hamiltonrsquosprinciple has been also used [153154]

By introducing (322) into (323) and (324) thebeam dynamic equations are obtainedZ

A

S0ampT r dm qyZ

A

S1ampT r dm qy-K11 qyu(

$ q2yy-K22 q2yyu( $ qy-K12 q2yyu(

$ q2yy-K21 qyu( $Z

A

S0ampT g dm$ Sa dP

325amp

K11

Fy 0 0 0

0 EA 0 0

0 0 Fy 0

0 0 0 GIt

2

666664

3

777775

K22

EIzz 0 EIxz 0

0 0 0 0

EIxz 0 EIxx 0

0 0 0 0

2

666664

3

777775

Sa

1 0 0

0 1 0

0 0 1

za 0 xa

2

666664

3

777775

K12

0 0 0 0

EAx+ 0 EAz+ 0

0 0 0 0

2

666664

3

777775

K21

0 EAx+ 0 0

0 0 0 0

0 EAz+ 0 0

0 0 0 0

2

666664

3

777775

In case of flexible blades if large deformations areexpected as in the case of helicopter blades second-order non-linear beam theory is to be used Therelevant developments originate from the work in[154] The beam axis again lies along the y-axis butin the undeformed state of the beam while x and zdenote the two bending directions of the beam Atits deformed state a local co-ordinate system O0xZzis introduced which follows the pre-twist of thebeam so that x and z coincide with the localstructural principle axes of the cross section(Fig 18)

Then (321) becomes

r u

y$ v

w

8gtlt

gt

9gt=

gt$ E

xlyy

z

8gtlt

gt

9gt=

gt (326)

where l denotes the warping function of the crosssection and E is the transformation matrix betweenOxyz and O0xZz In [154] E is given in terms of theelastic displacements up to second order Next thestrain tensor ij defined through (326) is introducedin Hookersquos law in order to define the strainndashdispla-cements relations which are used in the derivationof the resulting sectional elastic loads as in (322)Because they are derived in the O0xZz system beforeintroducing them in (323) and (324) they aretransformed into the Oxyz system using the

ARTICLE IN PRESS

x

z

yu

w

vO

Orsquoz

xz

xu

w amptamp

ampamp

amp +ˆ

Fig 18 Beam co-ordinate system definition

transformation matrix E The final expressions arequite lengthy and the reader is referred to [154] It isnoted that compared to the classical beam modelsecond-order theory will produce fully completestiffness matrices containing many non-linear termsThe above non-linear beam model is a specificexample amongst several models of varying com-plexity which have been gradually developed start-ing from the Timoshenko beam theory byeliminating the ad hoc simplifying assumptions ofthe original model [155ndash158]

Regarding wind turbines almost all structuralmodels are based on classical beam theory[150153159ndash162] This is partially due to the factthat wind turbine components are far more rigidcompared to helicopter blades for which non-lineartheory has been developed Furthermore most of thedifficulties in analysing wind turbine systems aregenerated by the aerodynamics and in particular theonset of stall which is certainly less pronounced onhelicopter rotors Current designs are quite stiff sothat non-linear beam modelling is not expected tochange drastically the quality of predictions besidesproviding a sound basis for including the geometricalnon-linearities [163] However if wind turbinesbecome more flexible it could become necessary toadopt such kind of structural modelling

Regardless the details the beam equations arefourth order with respect to bending and secondorder with respect to tension and torsion So for aFE approximation of the equations C1 shapefunctions are used for uw and C0 for v and y Atthe element level uw are approximated with third-order polynomials and the DOFs are the values andspace derivatives at the end nodes while v and y areapproximated with first-order polynomials and theDOFs again at the end nodes In some models andcertainly when the second-order beam theory isapplied second- and third-order polynomials areused respectively for v and y In this case the mid-point as well the points at 13 and 23 interior pointsare used as nodes So within an element lsquolsquoersquorsquo

uey tamp Neyamp uetamp (327)

where Neyamp is the matrix containing the shapefunctions and uetamp the vector of the DOFs at thenodes of the elements [164] As in Section 31concerning the principle of virtual work which inthe FEM terminology is the Galerkin formulationof the problem is used to generate the discreteequations With reference to (325) taken symboli-cally as Zu _u uamp 0 for any admissible virtual

displacement field du we require that

Z L

0duT Zu _u uampdy

)X

e

Z Le

0duTe Zue _ue ueampdy 0 8du 328amp

Because duey tamp Neyamp duetamp it follows that

X

e

Z Le

0NeT ZNeueN

e _ueNe ueampdy 0 (329)

which is a set of second-order ordinary equations intime with respect to uetamp Although Z can containnon-linear terms it can always be written in theusual form

Muamp u$ Cu

amp _u$ Ku

amp u Qu

amp (3210)

where the mass damping and stiffness matrices aswell as the generalized loads in the RHS in generalwill depend on the displacement field and its timeand space derivatives (noted by a tilde)

ARTICLE IN PRESS

0k

rGk

OG O

xG x

zG z

yG

y y

z

x

O

x

y

z

O

Fig 19 Multi-body representation of a wind turbine

The main components of a wind turbine are theblades the drive-train and the tower (Fig 19) Theyare all modelled as beam structures and typically thestructural properties are assumed for each compo-nent to continuously vary along the correspondingelastic axis However localized properties can beadded in the form of concentrated masses dampersor springs The gearbox (if present) the generatorthe hub are usually added in this way Otherexamples are the flexibility or damping character-istics of the yaw bearing or the pitch mechanismThe involvement of different body motions for eachcomponent in combination with the connectionswhere loads and displacements are communicatedfrom one component to the other calls for a globalformulation of the dynamic problem To this endmost works adopt a multi-body approach [165]which consists of considering each componentseparately subjected to appropriate boundary con-ditions which fit the different components into thecomplete configuration In such an approach theelastic DOF of each component are defined as localelastic displacements to which rigid body motionsare added through the kinematic boundary condi-tions see eg [161166167] for a more generaldiscussion It is also possible to introduce the globaldisplacements and rotations at each node instead ofthe local ones [153] allowing a unified description ofthe complete configuration In this case howeverthe non-linearities of the connections are introducedimplicitly and therefore linearization can only beperformed globally which is not always desirable

In multi-body modelling each component k isassigned a local coordinate system Oxyz with itsundeformed elastic axis along the y-axis Then theposition of a point with respect to the fixed systemrGk can be expressed with respect to its localposition rk (defined as in (321)) as follows

rGk qk $ Ak rk (3211)

where qk denotes the position of the origin of thelocal system with respect to the fixed system and Ak

denotes the local to global transformation matrixThe exact form of qk and Ak depends on thekinematic conditions introduced when joining (con-necting) the various components (ie blades-to-hubor drive-train-to-tower) Depending on the type ofconnection common global displacements androtations are assigned for the restricted DOF whichare identified to either an existing elastic DOF (shaftbending at the hub) or a rigid body motion (bladepitch) The component contributing the kinematics

will in response receive from the rest of theconnected components their internal (or reaction)loads This operation involves co-ordinate systemtransformations between the components The setof kinematical DOF involved in the definition of qkand Ak for all components is denoted collectively asq So qk qk(q t) and Ak Ak(q t) qk is defined asa mixed series of translations and rotations whereasAk is defined solely as a sequence of rotations Atypical example is shown in Fig 20 where q1ndashq6 arethe elastic DOF at the top of the tower q8ndashq13 arethe elastic DOF of the drive train at the hub and q7and q14 are the yaw angle and the pitch of the bladerespectively By introducing (3211) into (325) thecentrifugal and Coriolis terms of the inertia loadsappear as a result of the time derivatives of Ak whilein the Hamiltonrsquos principle approach they areproduced automatically

By combining the equations of all componentsthe complete system of the dynamic equations forthe wind turbine is obtained in the form of (3210)with respect to an extended vector of DOFuext u q

( ) By construction qk and Ak will depend

on unknown DOF while at the same time theycorrelate the state of the components in contact sothe terms they generate are non-linear with respectto the unknown DOF uk and q Linearization in thiscase is a tedious procedure which must be donecarefully and systematically Fortunately softwareusing symbolic mathematics such as Mathematica

ARTICLE IN PRESS

y

O

q8 q9 q10 q11q12 q13

yG y

z

x

O

x

y

z

OG O

xG x

zG z

q1 q2 q3 q4q5 q6

q7

q14

Fig 20 A typical definition of the q DOF on a HAWT

are available which are strongly recommended Theprocedure is based on Taylorrsquos expansions of allvariables and retaining of first-order terms Alter-natively the extended version of (3210) can besolved directly through an iterative procedure ineach time step which can use the linearized solutionas a starting predictor

4 Problems and solutions in wind turbineaeroelasticity

In the following section the use of aeroelasticcodes on wind turbine constructions includingstability studies will be illuminated mainly throughexamples

41 Aeroelastic stability

Wind turbines suffer from low structural damp-ing which can become critical under certain opera-tional conditions Most problems appear on theblades that receive almost 100 of the loading Inparticular the onset of stall plays a decisive role notonly on stall-regulated machines but also on pitch-regulated ones around rated conditions On theother hand the blades are made of compositematerials for which the knowledge on damping issubstantially inferior compared to steel or concreteconstructions The damping of composite structuresdepends on the ambient temperature So dependingon the season or the time during the day thestructural damping of the blades can decreasesubstantially Furthermore aging always degradesthe damping of composite structures There is adefinite need to increase structural damping andsignificant effort has been put recently on itsmodelling in composite structures as well as in theirappropriate design This is one direction of researchregarding the improvement of the stability char-acteristics of modern wind turbines see [168] for areview of the current developments Damping isrequired to suppress the onset of vibrationsgenerated by the unsteady aerodynamic loads thatinteract with the wind turbine structure First theaeroelastic modelling of wind turbines is consideredand then on this basis the aeroelastic stabilityproblem is formulated and the relevant availablemethods are reviewed

Aerodynamic modelling involves the calculationof the aerodynamic loads that also depend on thedynamics of the system The flap and lag vibrationsare interpreted by the incoming flow as a change in

the effective angle of attack indicating strongaeroelastic coupling All the currently availableaeroelastic tools for wind turbines use 2D dynamicairfoil data models such as the BeddoesndashLeishmannor the ONERA model They provide aerodynamiccoefficients as functions of the aerodynamic statevariables that satisfy appropriate dynamic equa-tions [17169] In either model the equations are 2Dand so they are applied on a strip-by-strip basismeaning that there is no interaction in the radialdirection This fits well with BEM modelling of theoverall aerodynamic analysis In fact the followingpresentation is referred to the BEM context sincemost existing stability models are based on it If apotential flow model is applied then the attachedpart of lift can be associated to potential lift whilethe rest is simply added [163] If the potential flowmodel is enhanced with a separation model thereshould be correlation on both aspects [58] How-ever the latter has not been applied yet inaeroelastic computations

By considering a blade section at a radial positionas shown in Fig 1 the local aerodynamic loadsacting on this section of the blade can be written asfollows since the lift is perpendicular and the dragparallel to the relative wind

py L sin fD cos f

pz L cos f$D sin fMy M 411amp

where L D are the lift and drag forces M is thepitching moment f is the local flow angle withrespect to the rotor plane Once the induced velocityhas been determined the local effective incidence aand relative flow velocity can be calculated as shownin Section 21 They will depend on the elasticdeformations and the velocities the rigid bodymotions and if appropriate orientation changes asin pitch and yaw variations In order to obtain aunified description of the coupled aeroelasticproblem a so-called aeroelastic finite element isdefined [170] which in addition to the elastic DOFthe aerodynamic state variables added For examplein the ONERA model there will be in total eightadditional DOF Because the dynamic equations forthe aerodynamic state variables only can be firstorder in time in order to produce a consistentcoupled system they can be differentiated as in[171] Therefore they can be assembled togetherwith the dynamic equations in the form of (3210)where the vector of the unknowns is further

extended to include the additional aerodynamicDOF Another approach used in [172] is tointroduce the modal expansion of the structuralDOF at stand still which better conditions thecoupled system

On the above basis it is possible to formulate theaeroelastic stability problem for wind turbinesWithin the context of linear theory stabilityboundaries are obtained by examining the evo-lution of small perturbations with respect to asteady state or a periodic solution acting asreference To this end the original equations arelinearized with respect to the reference state andthe resulting system solved with respect to thefluctuations

Muref amp u$ Curef amp _u$ Kuref amp u Q (412)

where M C and K depend only on the referencesolution If the matrices do not depend on time(412) is reformulated into a first-order system

_y D y$ b y u _un oT

(413)

The eigenvalue analysis of D provides the eigen-frequencies and damping of the system [173] This ispossible when the reference state corresponds to asteady solution as in the case of an isolated bladeHowever when analysing the complete windturbine configuration due to the rotation of theblades the reference state should correspond to aperiodic equilibrium state with reference to the rotorspeed which is assumed constant In this case thecorresponding theoretical framework is Floquetrsquostheory which for a large system is computationallyheavy [174] If the blades are identical and thenumber of blades NX3 which is the most frequentcase it is possible by means of multi-bladetransformation to eliminate the periodic coefficientsin the coefficients of (412) and therefore be ableto still use eigenvalue analysis [175] In this contextthe non-linear equations of the system are inte-grated in time until such a periodic state is attainedIn the case of an unstable situation because thetime domain response will contain significantcomponents in all of the basic eigenfrequencies ofthe system the reference state is obtained bytruncating the response so as to retain only its 1and Nrev components To this end all DOF in therotating system qm m 1 NX3 are reformedqm q0 $ qc cos cm $ qs sin cm where cm Ot$2p=Nampm 1amp is the corresponding azimuth loca-tion Next the equations for the blade DOFs are

rearranged by applying the operators 1=NampPN

m1

amp 2=NampPN

m1 amp cos cm and 2=NampPN

m1 ampsin cm This is performed after the periodic solutionhas been obtained over one period of rotation Theresulting equations will be in the non-rotating frameand contain only higher harmonics (3rev andhigher) By averaging over one period the finalconstant coefficient system is obtained in the formof (413) For the blades y will contain thecorresponding q0 qc and qs DOF and their timederivatives for both the purely structural and theaerodynamic DOF [171] In the above approach it isworth noticing that the averaging procedure willforce the reference state to appear in the equationsThis is an important aspect when dealing with non-linear systems

Stability of wind turbines has been consideredsystematically in Europe in the late 1990s [176] whiletoday there is an on-going activity under the EUfunded project STABCON Several linear stabilitytools have been produced along the lines describedabove and good correlation has been obtainedamongst the different codes see [177] and thereferences cited Inevitably linear theory is approx-imate so the results it produces are subject to crosschecking Clearly flutter measurements are difficultto obtain because of the involved risk therefore it isindispensable to rely on theoretical developmentswhich will be discussed in the next section

42 Aeroelastic coupling linear vs non-linearformulations

Linearization of the aeroelastic equations cer-tainly offers computational efficiency However it isnot always appropriate For the structure lineartheory requires that the deformations and displace-ments are small an assumption not always validFor the aerodynamics and its coupling with thestructure linearization will suppress some depen-dencies involved in (411) and therefore influencethe estimation of the aerodynamic damping In thisconnection the use of semi-empirical unsteadyaerodynamic models introduces still unresolveduncertainties Furthermore there are cases in whichthe multi-blade transformation is not applicable egwhen blades are non-identical or when the rota-tional speed varies

Retaining fully the non-linearities of the aero-elastic problem has the following consequences thestructure is considered at its deformed state thecoupling conditions among the components are

taken in their full form and the aerodynamicequations are not simplified The last point isimportant with respect to the best possible estima-tion of the aerodynamic damping within themargins offered by the semi-empirical models usedin the currently available aeroelastic tools In thisrespect multi-body analysis which has beenadopted in most of the existing stability tools[161178ndash181] offers this option by constructionprovided that the structural model considers theconstruction at its deformed state and the codetakes this into account in the definition of the localaerodynamics

Under such conditions the information of thestability characteristics of the wind turbine arecontained in the time signals of the loads calculatedthrough a non-linear time domain simulation Thereexist two types of methodologies for retrieving thisinformation the one is based on the work-basedapproach [176] while the other is using signal-processing techniques In the work-based methodany mode shape is excited and loads are recordeduntil a steady periodic state has been reached Thenthe aerodynamic damping is estimated by the workof the aerodynamic loads over the last period underthe assumption that there is no energy interchangebetween the modes through the aerodynamic loadsIn the signal processing approach the system isharmonically excited for a finite duration Then thetransient response following the abrupt terminationof the excitation is recorded wherefrom the aero-elastic damping and frequency of the specific modeare determined There are two different methodsboth well documented in the literature [182ndash184]that can be used the moving block method and themethod of Hilbert transform The moving blockmethod is a FFT based method commonly used inrotorcraft applications [182184] The transientresponse amplitude is computed on a block of datausing a FFT calculation The block is then movedforward by a single point in time and the computa-tion of the transient response amplitude is repeatedThe linear fit for the slope of the natural logarithmof the sequence of response amplitudes in time(peak plot) provides the damping (Fig 21) In theHilbert method the transient response y(t) and itsHilbert transform ~ytamp are used to define theenvelope signal Atamp

y2tamp $ ~y2tamp

p[183184]

For a transient response typical of a viscouslydamped system a line can be fitted to the logarithmof this envelope The slope of this line as in the caseof the moving block method gives the damping

(Fig 22) Comparing the two it has been found thatthe Hilbert method has certain advantages over themoving block method In particular the Hilbertdamping analysis provides an estimate of thedecaying envelope signal so it can be used inassessing the non-linear damping characteristics ofa mode Therefore it is not limited by the assump-tion of the viscous damping as the moving blockmethod does Moreover experience has shown thatthe Hilbert method is more efficient in calculatingthe damping of spectrally close modes [163]

43 Examples of time simulations and instabilities

In this section the challenges in aeroelastic design ofwind turbine rotors will be addressed and in particularthe phenomenon of aeroelastic instability of windturbine rotors will be explored by showing examplesof simulation results on a wind turbine design

Back from the beginning of development ofmodern wind turbine rotors in the late seventiesthere has been concern about the problems thatcould arise due to aeroelastic instability In parti-cular the use of the stall regulation principle wasuncertain as it was foreseen that a flapwiseinstability (so-called stall flutter) would occur whenoperating in the stall region due to the negativeslope of the CL vs a curve However it turned outthat the flapwise instability during operation in stallwas not the most critical problem but instead theedgewise instability resulting in edgewise bladevibrations The first experimental evidence of thisinstability was seen in the mid nineties and initiatedconsiderable research activities in order to explorethe phenomenon and provide practical solutionsBelow is mainly focused on this instability as thisserves as a good example to illustrate the problem ofaeroelastic instability in more general terms

In the more recent wind turbine designs theregulation of the turbines has shifted from stallregulation to pitch control where the operationalrange for the flow over the blade moves to low angleof attack at high wind and thus away from the stallregion This has almost removed the instabilityassociated with stall during operation but it is stillso that the edgewise blade modes are aerodynami-cally low damped on the pitch regulated turbines

A major instability problem on the modernturbines is seen when the rotor is parked or idlingat very low RPMs at wind speeds above stop windspeed which typically is around 25ms Again it istypically edgewise dominated instability problems

ARTICLE IN PRESS

3540

4550

tim

e (s

ec)

-35

E+0

6

-3E

+06

-25

E+0

6

-2E

+06

-15

E+0

6

-1E

+06

tower top tilt bending moment (Nm)

12

34

freq

uenc

y (H

z)

0

5E+

08

1E+

08

15E

+08

2E+

08

25E

+08

3E+

08

FFT of the tower top tilt bending moment (Nm)

-t

data

19

190

5

191

5

192

5

193

5

194

00

51

15

22

53

t (se

c)

- 1

n

log of response amplitude

Fig21Schem

atic

descriptionofthemovingblock

dam

pingan

alysis

related to stall of the blade A simulation examplewill be shown later to illustrate this problem

Finally the flutter instability will be addressed Itseems that the increased up scaling of the turbineshas led to rotor and blade designs where the flutterspeed is not so far away from normal operationalspeed Also this instability will shortly be illustrated

431 Edgewise blade vibration instabilityAs mentioned above the first experimental

evidence of this instability was seen in the mid

nineties on stall-regulated rotors with a diameter of35ndash40m An example is presented in Fig 23 and it isseen that the amplitude of the edgewise blade rootmoment (which at steady conditions vary sinusoidalwith 1p due to the gravity) increases 2ndash3 times dueto instability during operation in stall The experi-mental evidence of the edgewise instability led toconsiderable research on this subject and a majorEuropean project lsquolsquoPrediction of Dynamic Loadsand Induced Vibrations in Stallrsquorsquo funded by the EUwas carried out in the period from 1995ndash1998 [176]

ARTICLE IN PRESS

-15e+06

-1e+06

-500000

0

500000

1e+06

15e+06

2 4 6 8 10 12 14

tow

er to

p til

t ben

ding

mom

ent (

KN

m)

time (s)

transient signalenvelope signal

envelope signal exp fit

Fig 22 Envelope signal calculation of the tower top tilting moment transient response using the Hilbert transform method for the samesignal as in Fig 21

1000

800

600

400

200

0

-2000 50 100 150 200 250 300

Time [sec] Time [sec]

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

EDGEWISE BLADE VIBRATION INSTABILITY EDGEWISE BLADE VIBRATION INSTABILITY--MEASUREMENT

1000900800700600500400300200100

0-100

200 210 220 230 240 250 260

Fig 23 Measured edgewise blade root moment on a stall regulated rotor at high wind Detail of the time track to the right

The origin of the instability is simple If a rotatingairfoil section is harmonically translated along anaxis xB and the direction of this axis yRB relative tothe orientation xR of the rotor plane is varied theaerodynamic damping coefficient for the sectionvaries considerably see Fig 24 from [185] For lowyRB which means in-plane or edgewise vibrationdirection the damping coefficient is negative even atlow wind speed For vibrational directions close to901 which corresponds to out-of-plane or flapwisedirection the damping coefficient is strongly depen-dent on the inflow wind speed It is highly damped

at low wind but close to zero or negative damped athigh wind

On a complete rotating blade the direction ofvibration depends on the structural design of theblade as well as of the complete turbine structureThe movement of the tip section of the blade willnow no longer necessarily be along a straight lineHowever typically the tip section of the blade in thefirst flapwise mode will be on a path almostperpendicular to the rotor plane and in the 1stedgewise mode it will almost be in-plane It is thusexpected that the basic damping characteristics of

ARTICLE IN PRESS

Fig 24 The aerodynamic damping coefficient c_xx_b for a rotating airfoil section as function of vibration direction yRB and at threedifferent inflow velocities from [185]

Fig 25 Damping characteristics computed for a rotating isolated blade using the linear aeroelastic stability tool HAWCStab [186]

the whole blade will be comparable with thecharacteristics of a single airfoil section Using thelinear stability tool HAWCStab the dampingcharacteristics for a rotating blade can easily bederived and the computed damping characteristicsfor a rotating 19m blade (500 kW rotor) corre-sponding to the blades of the rotor where theexperimental instability was shown in Fig 23 areshown in Fig 25 from [186] It is seen that thedamping for the first flapwise mode varies fromhighly positive values at low wind speed and tovalues close to zero at 14ms and then increasingslightly at higher wind speeds It should be notedthat the influence of using unsteady blade sectionaerodynamics is shown (modelled with the BeddoesLeishman model in the present case) and that thishas considerable influence on the aeroelastic damp-ing Generally the unsteady aerodynamic effectsincrease the damping The damping for the firstedgewise mode is seen to decrease gradually withincreasing wind from being slightly positive at lowwind to slightly negative at high wind Finally itshould also be noted that the damping shown is thetotal damping including the structural dampingwhich in the present case is around 2 for the firstflapwise mode and 3 for the first edgewise mode

The final step in model complexity is nowobtained by going to a full aeroelastic model ofthe turbine comprising the dynamics of the shaftthe nacelle and the tower The aeroelastic stability

results using HAWCStab is for the complete windturbine see eg Fig 26 from [186] showing theaeroelastic damping for the first 10 mode shapes(numbered from lowest frequency) It is seen thattwo mode shapes are negatively damped at highwind speed and that one mode shape is close to zeroand slightly negatively damped at the highest windspeed For comparison time simulations with theaeroelastic code HAWC [187188] is performed onexactly the same turbine model at a wind speed of 8and 16ms respectively in order to see if theinstability at high wind speed predicted by the linearstability tool can be confirmed by the HAWCmodel with non-linear aerodynamics The com-puted edgewise blade root moment shown in Fig 27confirm the instability at high wind and comparesvery well with the measured instability shown inFig 23 It is thus demonstrated that the edgewiseinstability problem can be predicted in both timesimulations and using linearized stability analysisHowever as will be demonstrated below theedgewise vibration instability is much more complexthan just edgewise vibrations of the individualblades but comprises the dynamic characteristicsof the whole turbine This understanding is vital forthe development of design solutions preventing theinstabilities

In order to analyse the instability in more detailsthe use of parallel aeroelastic time simulations on aturbine with a stiff structure is made In this case the

ARTICLE IN PRESS

Fig 26 Aeroelastic damping computed with the aeroelastic stability HAWCStab [186] Three modes have negative damping at high windspeed

loads on the different components will mainlyreflect the direct load input from the externalaerodynamic forces and from gravity forces Itshould be mentioned that the simulations areperformed at two average wind speeds of 8 and16ms respectively and with a turbulence intensityof 12

A power spectral density analysis of the simulatededgewise blade root moment shows a distinct peakat 307Hz which is close to the frequency of the firstedgewise mode for a single rotating blade which is319Hz see Fig 28 The response at 16ms of theflexible turbine at this frequency is seen to be severalorders of magnitude bigger than the response of thestiff turbine indicating an instability situationSomewhat the same tendency is seen at 8ms but

with considerable less difference between the re-sponse of the stiff and flexible turbine indicating lowdamping but not an instability

Analysing further the results of the stabilityanalysis with HAWCStab as shown in Fig 26 itcan be derived from the results of the model that thethree mode shapes that are negatively damped orvery low damped at 16ms are

mode no 1 with frequency 077Hz and damping605

ARTICLE IN PRESS

Fig 28 Comparison of power spectra of edgewise blade root moment for a flexible and a stiff turbine respectively at 8 and 16ms

Fig 27 Time simulation with the aeroelastic code HAWC on a 500 kW stall regulated rotor at two wind speeds 8 and 16ms

The numbering in HAWCStab of the modes iswith increasing mode number with increasingfrequency starting with the lowest frequency Modeno 1 and no 3 are both modes where lateral towerbending is the main motion together with edgewisebending of the three blades and torsion of the mainaxis The edgewise movement of the blades are all inphase and the difference between the two modes isthe direction of the edgewise bending relative to thetower bending Comparing again the results of thelinear stability analysis with the time simulationresults the instability of the lateral tower bendingdominated modes is confirmed by the spectra of thetower top and tower bottom lateral bendingmoments Figs 29 and 30 A distinct peak at

079Hz is seen in both tower top and tower bottomspectra at 16ms and this peak cover probably alsoa considerable content at the 093Hz which was thefrequency of mode no 3

It can be noticed that the HAWCStab results donot contain an unstable mode with a frequencyaround 307Hz (the edgewise instability) as seen inthe time simulation results However mode no 7 isin fact the response on the non-rotating turbinestructure from the edgewise vibrations at 307HzThe edgewise vibrations in the present case is a so-called backward whirling edgewise mode wherethere is a phase shift of 1201 between the movementof the blades This whirling mode results in shaftand tower bending but with a frequency shifted 1p

ARTICLE IN PRESS

Fig 29 Comparison of power spectra of tower top lateral moment for a flexible and a stiff turbine respectively at 8 and 16ms

Fig 30 Comparison of power spectra of tower bottom lateral moment for a flexible and a stiff turbine respectively at 8 and 16ms

(p is the rotational frequency) up and downrespectively compared with the frequency in therotating system As 1p in the present case is 045Hza peak in the spectra of tower bending at afrequency of around 307ndash045Hz equal to 262Hzis expected which is also seen in Figs 28 and 29However in the HAWCStab results a slightlyhigher frequency 273Hz was seen for this mode

The presented example of the edgewise bladevibration instability has shown that the low aero-dynamic damping of an airfoil section undergoing amotion along a path close to the chordwise directioncan cause instability of quite different modes on theturbine at the same time The design challenges withthe objective to minimize the influence of lowaerodynamic damping or instability are thus big aseg changes in tower design could lead to egedgewise vibrations on the rotor A number ofdifferent methods to reduce the risk for edgewisevibrations on stall-regulated turbines have beeninvestigated [176186] and comprises eg changes ofthe stalling characteristics of the airfoils by so-calledlsquolsquostall stripsrsquorsquo as well as structural design of theblades to achieve optimal vibrational directions

432 Instability problems of parked rotorsThe turbines are normally designed to operate up

to a certain maximum wind speed and above thiswind speed the turbines are shut down In thestopped conditions the rotors can be completelyparked or they can idle with low rotational speeddepending on the actual design of the controlsystem As part of the certification of the turbineit must be shown that the turbine can withstand the

wind loads at extreme wind speed conditions withparked rotor and the wind coming from anydirection in the case that the yawing system of theturbine is not functioning

As an example the simulation of a rotor parkedat a wind speed of 50ms and a yaw error of 601 isshown in Fig 31 The turbine is the same as aboveand the time track of the edgewise blade rootmoment indicates low damping as the amplitudesvary much in time To the right in Fig 31 is shown apower spectrum of the same signal and it shows thatthe blade mainly vibrates in two modes mode 3 at afrequency of 093Hz which was described aboveand the first edgewise blade frequency at about319Hz For the present inflow direction thevibrations are not unstable but it has to bedocumented for all inflow directions

433 Flutter instabilityThe last example of an instability is flutter This is

a well-known instability from the aircraft industrybut has not yet been a problem on wind turbinesand has probably not been seen on commercialturbines However with the increasing size of theblades it seems that the flutter speed decreases dueto increasing structural flexibility of the blades andnot least the torsional frequency decreases There-fore it is a good idea to include a flutter speedcalculation in the design verification for eg 50mblades and above

Flutter involves two DOF of the blade torsionand translation The flutter speed decreases whenthe frequency of these two DOF approach eachother For 50m blades the frequency of the first

ARTICLE IN PRESS

Fig 31 Simulation of a parked rotor at 50ms and a yaw error of 601 Time track of the edgewise blade root moment to the left and apower spectrum of the same signal to the right

flapwise mode is typically slightly below 1Hz andthe frequency of the first torsional mode willtypically be in the range from 5 to 8Hz Howeverit has been seen that the flutter instability can occurby a coupling of the second flapwise mode and thefirst torsional mode of the blade and of course this isalso a reason for a decreasing flutter speed on thebigger blades Another important design parameterfor the flutter instability is the centre of mass in theblade sections relative to the centre of the elasticaxis As the centre of mass moves away from theelastic axis in the direction of the trailing edge theflutter speed decreases

An example of a simulated flutter instability isshown in Fig 32 The blade does not represent anindustrial blade but has selected structural para-meters which give a rather low flutter speed Thesecond flapwise mode for the blade is slightly above2Hz and the first torsional frequency is around6Hz Further the centre of mass is positioned 10of the chord length behind the elastic axis Theexample is mainly intended to show the character-istics of the flutter instability which is completelydifferent from the instabilities that have been seenso far In the simulation the rotor is free to speed upas the generator is disconnected When the flutterspeed is reached the instability develops within 1ndash2 sand this is completely different from eg theedgewise blade instability that can build up over aminute or more This characteristic means the bladeprobably will be damaged immediately if the flutterspeed is reached A further consequence of this fastdevelopment is that the flutter can occur at a lowerrotational speed of the rotor if there is a yaw errorThis is demonstrated in the right part of Fig 31

where there is a yaw error of 501 Here therotational speed at flutter instability is 13 lowerthan if there is no yaw error and the wind speed isjust 8ms If the for example the yaw error of 501occurred at around the stop wind speed of 25msthe rotational speed at the flutter instability woulddecrease even further

5 Present and future developments of aeroelasticmodels

The improvement of existing aeroelastic codesand the development of new aeroelastic models arehighly influenced by the design trends of new windturbines and trends in the siting of the turbinesbecause this determines the needs for new capabil-ities of the models

51 Areas with influence on the development ofaeroelastic models

511 Influence of up-scalingSo far the most important design trend has been

the up-scaling which within the last 10 years hasincreased the maximum size of the mass producedturbines with a factor of 10 from about 500 kW witha rotor diameter of about 40m to 5MW turbineswith a rotor diameter of 120m The newest turbinesare all pitch controlled with variable speed andtypically with some type of cyclic pitch for loadalleviation An accurate modelling of these newflexible turbines has increased the requirements tosimulate complex coupled modes where the inte-grated flexibility of the tower of the drive train andof the blades and in interaction with the control is

ARTICLE IN PRESS

Fig 32 Example of flutter instability on a 50m test blade where the rotor is free to speed up To the left normal inflow whereas to theright there is a yaw error of 501

important Also non-linear effects from consider-able deflections of the blades and the tower arebecoming important and the aeroelastic stabilitymust be predicted with a good accuracy Sometypical frequencies for a 5MW turbine could be

First tower bending frequency 02ndash025Hz

First flapwise frequency 08ndash10Hz

First edgewise frequency 09ndash11Hz

First blade torsional frequency 50ndash70Hz

The decrease in the fundamental turbine frequen-cies arising from the up-scaling has the effect thatexcitation from the turbulent inflow has increased asthe power spectrum of the turbulence peaks ataround 005Hz

512 Siting of the turbinesThe major part of all new turbines is placed in

small or bigger groups and in some cases in windfarms with more than 100 turbines This means thatwake operation is part of the inflow conditionswhich has to be simulated in the derivation of thetotal design loads of the turbines Load alleviationby cyclic or individual pitch controls is importantfor such inflow conditions and the aeroelasticmodels should be suited for this

Still the major part of turbines are set up on landbut the offshore part will increase at least if a 5ndash10years interval is considered On land the best siteshave been used in many countries and thereforemore and more complex sites will be used Thismeans that rather complex inflow situations must bemodelled as for example complex shear and non-uniform turbulence over the rotor disc

513 Future trends in turbine design and sitingIt seems that the speed of up-scaling might be

slowed down for some years as the industry wantsto be more focused on turbine reliability One wayto improve this is to have the individual turbinemodels on the market for more years than seen inthe past

However the increasing offshore developmentwill support the up-scaling tendency as the tran-sport and erection of big turbine components is nolonger a major problem Further the higherfoundation costs offshore will also support the up-scaling trend

Extending the possible offshore sites to deeperwater new foundation types typical with multipleframes will be developed and the aeroelastic codes

should be able to handle these structures A furtherstep in the offshore development is the floatingturbines either as single floating turbines or moreturbines on the same floating frame

Finally the offshore market could have the effectthat the two bladed turbine with a teetered down-wind rotor again will be considered as an alternativeto the three-bladed machines The main barrier forthe development of two bladed turbines has beenthe low frequency noise from the blade passage ofthe tower shadow This will be of less importanceoffshore as the noise restrictions are much lowerhere

52 Areas of development in present and new codes

Influenced by the continuously increasing require-ments to the capabilities of the aeroelastic models asdescribed above existing models are being furtherdeveloped and improved and new codes are writtenBelow some important areas of development will bediscussed

non-linear structural dynamics

calculation of induction and its dynamics

wake operation

derivation of airfoil data for aeroelastic simula-tions

complex inflow

aerodynamics of parked rotors and

off shore turbines including floating turbines

521 Non-linear structural dynamicsSo far almost all aeroelastic codes have contained

the model assumptions of small deformations androtations but the increased flexibility of the turbineshave made uncertainty about the validity of thisassumption for the new designs The influence ofnon-linear effects on the dynamic response of aturbine was treated in a recent paper [189] anddifferent effects were considered For examplecharacteristics (frequency and damping) of the firstedgewise mode will typically change as function ofincreased flapwise deflection because the edgewisebending will couple more and more with thetorsional mode of the blade Besides increasedinfluence to non-linear effects from the flexibilityof the turbine components the new foundationsused offshore and in particular floating foundationscan contribute significantly to big deflections andrelated non-linear effects

To overcome this uncertainty about the impor-tance of non-linear effects a few new non-linearaeroelastic codes have now been developed Siemenswind power has developed a non-linear codeBHawC based on co-rotaing elements [190] whilethe new non-linear code HAWC2 [191] developedby Risoe is based on multibody dynamics

The trend of including non-linear effects willcontinue in the future and codes based on modalexpansion techniques will be developed to includemore modes as eg the blade torsional mode

522 Calculation of induction and its dynamicsThe unsteady aerodynamic air loads are directly

dependent on the computed induction at the samepoint on the blade through the angle of attackTherefore an accurate prediction of induction is ofcrucial importance

As the rotor disc has become bigger and biggerrelative to the scales in the turbulent inflowincreasing variation in inflow conditions over therotor disc are seen Wind shear contributes to thisvariation in inflow The inflow variations results inlikewise considerable variation in the loadingexpressed through the local thrust coefficient CTlocal

over the rotor disc and thus also in induction Theinduction is therefore even in normal operationhighly dynamic and the induction model should beadapted to these conditions As an example Fig 33

(reproduced from [192]) shows the variation of thelocal thrust coefficient at the outer part of the bladeon an 80m diameter turbine in normal operation

Another source contributing to the load varia-tions and thus to the dynamic induction is theeigenmotion of the blades This part is alsoincreasing due to up-scaling and due to the moreflexible designs As an example a blade with aflapwise frequency of 1Hz as mentioned above andvibrating with an amplitude of 1m will experience arelative velocity component perpendicular to thechord of around 76ms in maximum This valuecan directly be compared with the variations ininflow and is thus considerable Different develop-ments to improve the computation of inductionhave been seen One type of modelling is thecoupling of a numerical actuator disc model to thestructural part of the aeroelastic model So far suchmodels have mainly been used to verify the accuracyof BEM modelling As an example the aeroelasticcode HAWC at Risoe with a BEM-type inductionmodelling in the standard version has been devel-oped in another version HAWC-3D where theinduction is computed with a 3D actuator discmodel and used eg to investigate the accuracy ofyaw modelling [142193] see Fig 34 from [142] Inthe coming years aeroelastic codes coupled to anactuator disc or actuator line model for computa-tion of induction will certainly be further developed

ARTICLE IN PRESS

Fig 33 Example of dynamic loading at a point close to the tip of a blade on an 80m diameter turbine in turbulent wind from [192]

but probably mainly used for research and forverification of simpler codes and not so much in theaeroelastic codes used by the industry due to themuch longer simulation time

However besides a direct coupling of an actuatordisc or line method to an aeroelastic code in order tocompute the induction at each time step there areother ways to utilize the capabilities of an actuatordisc model One example is to compute theinduction characteristics of a rotor within its windspeed operational interval with an actuator disc andthen afterwards use these quasi-steady inductioncharacteristics in aeroelastic simulations for therotor The BEM method in the new HAWC2aeroelastic code developed at Risoe has beenimplemented in such a way that this method canbe used The method is shortly first to compute thelocal induction as function of local loading for theactual rotor in a number of calculation points n mamp over the rotor with an actuator disc

arn ymamp f CT rn ym amp amp

Afterwards these induction functions are then usedin the BEM method during time simulations on therotor instead of the unique single relationship from1D momentum theory CT 4a1 aamp

In fact one of the main deficiencies of the BEMmethod is that this induction formula concerns thewhole rotor disc Deviations are seen in regions withconsiderable radial variation in the loading whichmeans in the tip or root region Another examplewhere the momentum induction formula does nothold is if the rotor disc is not plane eg conedrotors or rotors where the blades bend considerably

523 Wake operationMost turbines are now set up in clusters or wind

farms and this means that such turbines in part oftheir lifetime operate in wake from one or moreturbines In the international standard for design ofturbines IEC 61400-1 [2] the increased loading fromoperation in wake can be taken into account byusing an increased effective turbulence whichdepends on a number of parameters in theconsidered wind farm such as eg wind turbinespacing However it has turned out that in caseswhere more detailed knowledge of the increasedloading on the different turbine components fromwake operation is needed another more detailedaeroelastic modelling is needed than just increasingthe turbulence One such aeroelastic modelling hasbeen developed over the past few years [194195]The main components in the modelling is (1)

ARTICLE IN PRESS

Fig 34 Influence of yaw on the local inflow angle The curves show the difference in local inflow angle at yawed operation at 451compared with non-yawed operation The HAWC model has a BEM induction model HAWC-3D a 3D actuator disc model andEllipSys3D is a full 3D NavierndashStokes simulation from [142]

computation of the wake deficit from the up-streamturbine with an actuator disc model (2) meanderingof this deficit from the big scales in the turbulenceand (3) additional turbulence within the wake Oneof the advantages of this new modelling comparedto the method using an effective turbulence is thatboth the mean yaw loads as well as the yawdynamics compare well with measurement Fig 35from [194] shows the loads as function of winddirection and full wake operation at a direction ofaround 2101 The highest mean yaw loads occur inhalf wake operation at around 1951 and 2201Recently the wake resulting from the interaction ofseveral turbines in a row was computed in [196] bycombining large eddy simulation of the NS equa-tions with the actuator line methodology

It is expected that the more detailed aeroelasticmodelling of wake operation will be developedconsiderably in the coming years because suchmodelling is necessary for the development ofadvanced control algorithms adapted for loadreduction in wakes The development of the newdetailed models can be in the direction like themodel described above but also using actuator linemodels or full 3D rotor models to compute the wakecharacteristics which can be fed into an aeroelasticsimulation

524 Derivation of airfoil data for aeroelasticsimulations

The airfoil data used in aerodynamic and aero-elastic simulations are of crucial importance for the

ARTICLE IN PRESS

Fig 35 Measured and simulated loads at 8ms (crosses and full lines) and at 10ms (squares and dashed lines) on a 80m rotor duringwake operation (full wake at around 2101 on x-axis) using a new aeroelastic wake simulation method taking into account the wake deficitmeandering from [194]

accuracy of such simulations This was clearly seenin the result of the NREL blind test [137] where thedifference in the airfoil data used in the individualsimulation models probably were the most signifi-cant factor for the big variation between the outputof the different simulations However it should benoted that this result is not a representativeindicator of the uncertainty in the industry onaerodynamic and aeroelastic results on new turbinesas considerable experience has been learned fromolder rotor designs in order to adapt airfoil data setsso that computed rotor power and loads comparewell with measurements The drawback is that thisintroduces conservatism in the design and forexample can hinder the use of new airfoils There-fore methods and models to derive or correct airfoildata to be used in aeroelastic simulations have beena key issue for many years in the wind energyresearch community and will also be it in the nearfuture 2D wind tunnel airfoil data has so far beenthe common starting point to set up an airfoil dataset for simulations In order to correct for so-called3D flow effects and rotational effects some kind ofcorrection is typically applied [19ndash21197] Thecorrections reflect the increased lift and drag thatwas measured on different rotors in field rotormeasurements performed 10ndash15 years ago [198199]Full 3D CFD rotor computations later confirmedclearly the tendencies of the 3D flow effects androtational effetcts in the form of increased lift in theblade root region but also increased drag [137ndash139]

With the possibility of running full 3D simula-tions on a rotor a new source for derivation of 3Dairfoil data sets for input in aerodynamic and

aeroelastic codes has emerged Methods to extractthe airfoil data from 3D rotor computations havebeen developed [200201] One example of such aderived airfoil data set is shown in Fig 36 from[202] where a considerable increased maximum lifton the inboard stations are seen Within the nextfew years it is expected that a considerable effortwill be on further development of the methods toextract airfoil data from 2D wind tunnel data aswell as 3D CFD rotor data So far the CFDcomputations have mainly been used to extractquasi-steady data but they will in the future also beused to tune the parameters in the dynamic stallmodels

525 Complex inflowWind turbines are often set up in so-called

complex terrain which means some kind ofmountainous terrain because the wind potential isgood at such places However the consequence isthat the inflow conditions can vary considerablyfrom turbine to turbine due to local variations ofthe terrain As a result extreme wind shear has beenseen to occur over the rotor disc of turbines placedin such terrain and likewise the turbulence intensitycan be extremely high

The fast development of CFD codes has nowmade it possible to simulate the flow in detailsover the terrain spanned by a wind farm in suchcomplex terrain The flow data in the form of windshear and turbulence can be used as input inaeroelastic simulations There is thus a basis for asimulation tool to micro site turbines not only withrespect to energy production but also loads is

ARTICLE IN PRESS

2

18

16

14

12C1

Cd

1

08

06

040 5 10 15 20 25 30

r=14r=28r=42r=56r=69r=89r=94r=97

35 40$ $

05

045

04

035

03

025

02

015

01

005

00

5 10 15 20 25 30 35 40

Fig 36 Extracted 3D airfoil data in the form of CL CD data as function of angle of attack at different radial stations from a 3D CFDrotor computation Figure from [202]

expected Thus type of modelling is certainly ofhigh interest for the wind turbine industrybecause considerable costs to repair turbines dueto extreme inflow conditions have been experiencedin the past

526 Aerodynamics of parked rotorsIt is a common procedure to shut the turbines

down at high wind typically at wind speeds in therange from 20 to 25ms Ultimate loads on parts ofthe wind turbine can occur during standstill condi-tions at very high wind speed whereas ultimate loadson other components will occur during operationbelow the stop wind speed

The aerolelastic codes are therefore also used tocompute the turbine response during standstillconditions and it is particularly the computationof the aerodynamic loads on the blades that isuncertain This is due to the requirement that theloads on the parked rotor shall be investigated withthe wind from all directions The models musttherefore be able to compute aerodynamic loads forangle of attacks in the complete range from ndash180o to180o Furthermore the inflow is turbulent so that itis highly unsteady aerodynamic loads and a severecomplication is that some inflow conditions oftenlead to instability with the blade vibrating in a modewith negative aerodynamic damping In a recent EUfunded project lsquolsquoKNOWBLADErsquorsquo different CFDcodes were used to explore some basic character-istics of the blade standstill aerodynamics [115]Because of the complexity of the aerodynamics it isalso believed that the use of CFD will be the mainpath to explore the standstill aerodynamics in thecoming years

527 Offshore turbines including floating turbinesPresently there are considerable research and

development efforts on adapting aeroelastic modelsto simulate offshore wind turbines mounted on asub-structure standing on the sea-bed see Fig 37from [203] or on a floating foundation

Different approaches are seen in this develop-ment One line is characterized by a completeintegration of the standard aeroelastic model ofthe turbine with a hydroelastic model of the sub-merged supporting structure including the hydro-dynamic loads the wave loading and the soil forceson the part of the support structure in the sea bed[190191] In both these models the completestructure is described with finite elements and withengineering sub-models for computation of thehydrodynamic loads

Another line of modelling is to couple alsquolsquostandardrsquorsquo aeroelastic model to a separate modulesimulating the supporting structure including waveloads and hydro loads This approach has beenpresented by Repower [204] where the Flex5aeroelatic code is coupled with a standard packageASAS for computations of wave loads on off shorestructures In the third development line the mainidea is also to use a lsquolsquostandardrsquorsquo aeroelastic code andthen couple it to a super element of the foundationThis approach has been presented by Vestas [203]where the foundation is modelled with a finiteelement model but then decomposed to a superelement using the CraigndashBampton sub-structuringscheme Considerable efforts on further develop-ments of the aeroelastic models for off shoreapplications will be seen within the next years andwith more emphasis on deeper water installations

ARTICLE IN PRESS

Fig 37 Different support structure concepts from [203]

6 Discussion

On a wind turbine there is a strong couplingbetween the aerodynamic loads and the time-dependent structural behaviour of the constructionStatically a blade might change its twist and thus theangle of attack when deflected But also the anglesof attack are changed when the blades have avelocity relative to the fixed ground For instance ifthe tower is moving upstream and everything else isstiff it will be felt by the blades as an increased windspeed and thus higher angles of attack will bepresent along the blades The aerodynamic responsewill depend on how the lift and drag vary with theangle of attack In stall the lift will decrease yieldinga possible instability whereas for attached flow thelift will increase and thus creating a higher load seenby the blade opposite the movement indicatingstability On a real wind turbine not only the toweris vibrating but all other components as well whichdirectly feeds back to the angle of attacks and thusthe loads that again alter the motion To simulatethe aeroelastic response of a wind turbine a non-steady structural model including the inertia mustbe made Two methods ie the method of virtualwork applied on modal shape functions and theFEM are addressed To evaluate the aerodynamicloads the BEM is still the most widely used due toits simplicity and computing efficiency To obtainrealistic results some engineering adds-on arenecessary such as the Dynamic Wake DynamicStall and a yaw model which are thereforethoroughly described in this paper A BEM methodrelies however on airfoil data and the results aretherefore no better than the input To avoid theuncertainties of the engineering adds-on the Actua-tor Line model can be used since this methodresolves the physics of these models through the NSequations However the actual blades are notresolved and to estimate the aerodynamic lift anddrag airfoil data are still needed To avoid airfoildata one needs to solve the NS equations andresolve the blades and the possible boundary layersThis is extremely computationally costly and there-fore this model is not likely to replace the BEMmethod in the near future However the methodcan be used to extract airfoil data that can be usedin the less advanced models It is expected that theActuator Line model very soon will replace theBEM method since it contains less empirics Withthe AL model the wake is also a part of the solutionand therefore the effect from this wake on a wind

turbine placed further downstream can be calcu-lated These simulations are becoming very impor-tant as wind turbines are grouped in large (offshore)wind parks The inviscid flow models are included inthis paper mainly for historical reasons since theyhave played an important role in determining basicphysical features This type of model was eg usedto calibrate the time constants in the DynamicWake model Also a few Potential flow models existthat model the induced velocities and the wakebehind a wind turbine by shedding vortex blobsfrom the blade surface This model is similar to theAL model and may also in some aeroelastic codesreplace the BEM method

References

[1] Last og sikkerhed for vindmollekonstruktioner DANSKSTANDARD DS 472 [in Danish]

[2] IEC 61400-1 Ed3 CD 2 revision Wind turbines Part 1design requirements Edited by IEC TC88-MT1 25ndash26May 2004

[3] Hansen KS Performance Measurements on two Danish630 kW WECS In Proceedings of the WIND POWERrsquo85SERICP-217-2902 1985 p 130ndash5

[4] Hansen KS et al An evaluation of measured and predictedfatigue loads for the Tjaeligreborg wind turbine In Proceed-ings of ECWECrsquo93 Travemunde Germany 1993p 579ndash82

[5] Ahlstrom A Aeroelastic simulation of wind turbinedynamics Doctoral thesis in Structural Mechanics KTHSweden 2005

[6] Glauert H Airplane propellers In Durand WF editorAerodynamic theory New York Dover Publications 1963

[7] Rasmussen F Hansen MH Thomsen K Larsen TJBertagnolio F Johansen J et al Present status ofaeroelasticity of wind turbines Wind Energy 20036213ndash28

[8] Quarton DC The evolution of wind turbine designanalysismdasha twenty year progress review Wind Energy199815ndash24

[9] Hansen AC Butterfield CP Aerodynamics of horizontal-axis wind turbines Annu Rev Fluid Mech 199325115ndash49

[10] Shen WZ et al Tip loss corrections for wind turbinecomputations Wind Energy 20058(4)457ndash75

[11] Snel H Schepers JG Joint Investigation of dynamic infloweffects and implementation of an engineering methodECN-Cndash94-107 1995

[12] Schepers JG Snel H Dynamic inflow yawed conditionsand partial span pitch control ECN-C-95-056 1995

[13] Bramwell ARS Helicopter dynamics Paris EdwardArnold (Publishers) Ltd 1976

[14] Theodorsen T General theory of aerodynamic instabilityand the mechanism of flutter NACA report 496 1935p 413ndash33

[15] Oslashye S Dynamic stall simulated as a time lag of separationIn KF McAnulty editor Proceedings of the fourth IEAsymposium on the aerodynamics of wind turbines ETSU-N-118 1991

[16] Leishman JG Beddoes TS A semi-empirical model fordynamic stall J Am Helicopter Soc 198934(3)3ndash17

[17] Hansen MH Gaunaa M Madsen HA A BeddoesndashLeish-man type dynamic stall model in state-space and indicialformulations Risoe-R-1354(EN) 2004

[18] Johansen J Soslashrensen NN Aerofoil characteristics from 3DCFD rotor computations Wind Energy 20047(4)283ndash94

[19] Snel H Houwink B Bosschers J Piers WJ van BusselGJW Bruining A Sectional prediction of 3-D effects forstalled flow on rotating blades and comparison withmeasurements In Proceedings of the ECWEC 1993Travemunde 1993 p 395ndash9

[20] Chaviaropoulos PK Hansen MOL Investigating three-dimensional and rotational effects on wind turbine bladesby means of a quasi-3D NavierndashStokes solver J Fluids Eng2000122330ndash6

[21] Bak C Johansen J Three-dimensional corrections of airfoilcharacteristics for wind turbines based on pressure dis-tributions In Proceedings of the EWEC conferenceAthens 27 Februaryndash2 March 2006

[22] Fuglsang P Bak C Status of the Risoe wind turbineairfoils In Proceedings of the EWEC 2003 (CD-ROMavailable from EWEA) Madrid 2003

[23] Shinozuka M Jan C-M Digital simulation of randomprocess and its applications J Sound Vib 197225111ndash28

[24] Veers P Three-dimensional wind simulation SAND88-0152 UC261 Sandia National Laboratories NM 1988

[25] Mann J Wind field simulation Probl Eng Mech199813269ndash82

[26] Soslashrensen JN Three-level viscous-inviscid interaction tech-nique for the prediction of separated flow past rotatingwing PhD thesis AFM-83-03 Technical University ofDenmark 1986

[27] Karagiannis F Simandirakis G Chaviaropoulos P Papai-liou KD A 3D shear layer prediction method for steadyviscous flows around HAWTs In Proceedings of theECWEC 1993 Lubeck-Travemunde Germany p 391ndash4

[28] Milne-Thomson LM Theoretical aerodynamics NewYork Dover Publications 1966

[29] Richardson SM Cornish ARH Solution of three dimen-sional incompressible flow problems J Fluid Mech197782309ndash19

[30] Joukowski NE Vortex theory of a rowing screw TrudyOtdeleniya Fizicheskikh Nauk Obshchestva LubiteleiEstestvoznaniya 1912161

[31] Margoulis W Propeller theory of Professor Joukowski andhis pupils NACA Technical Memorandum No 79 1922

[32] Miller RH The aerodynamic and dynamic analysis ofhorizontal axis wind turbines J Wind Eng Ind Aerodyn198315329ndash40

[33] Oslashye S A simple vortex model In Proceedings of the thirdIEA symposium on the aerodynamics of wind turbinesETSU Harwell 1990 p 41ndash515

[34] Koh SG Wood DH Formulation of a vortex wake modelfor horizontal-axis wind turbines Wind Eng 199115(4)196ndash210

[35] Wood DH On wake modelling at high tip speed ratiosWind Eng 199216(5)291ndash303

[36] Hess JL Review of integral equation techniques for solvingpotential flow problems with emphasis on the surfacesource method Comput Methods Appl Mech Eng 19755145ndash96

[37] Katz J Plotkin A Low-speed aerodynamics New YorkMcGraw-Hill 1991

[38] Prandtl L Tietjens OG Applied hydro-and aeromechanicsNew York Dover 1934

[39] Landhal MT Stark VJE Numerical lifting surfacetheorymdashproblems and progress AIAA J 19776(11)2049ndash60

[40] Kerwin JE Lee CS Prediction of steady and unsteadymarine propeller performance by numerical lifting surfacetheory Trans SNAME 1978 86

[41] Hess JL Calculation of potential flow about arbitrarythree-dimensional lifting bodies McDonnell Douglasreport MDC J5679-01 1972

[42] Rehbach C Calcul drsquoecoulements autour drsquoailes sansepaisseur avec nappes tourbillonnaires evolutives Re-cherche Aerospatiale 1973253ndash61

[43] Cottet G-H Koumoutsakos PD Vortex methods theoryand practice Cambridge Cambridge University Press2000

[44] Bagai A Leishman JG Rotor free-wake modelling using apseudoimplicit relaxation algorithm J Aircraft 199532(6)1276ndash85

[45] Bhagwat M Leishman JG Accuracy of straight-linesegmentation applied to curvilinear vortex filaments JAm Helicopter Soc 200146(2)166ndash9

[46] Gould J Fiddes SP Computational methods for theperformance prediction of HAWTs In Hulle FVSmulders P Dragt J editors Wind energy technologyand implementation Amsterdam Elsevier Science Publish-ers 1991 p 29ndash33 [EWEC091]

[47] Robison DJ Coton FN Galbraith RAM Vezza MApplication of a prescribed wake aerodynamic predictionscheme to horizontal axis wind turbine in axial flow WindEng 199519(1)41ndash51

[48] Coton FN Wang T The prediction of horizontal axis windturbine performance in yawed flow using an unsteadyprescribed wake model Proc Inst Mech Eng Part AmdashJPower Energy 199921333ndash43

[49] Afjeh AA Keith TG A simplified free wake method forhorizontal axis wind turbine performance prediction TransASME J Fluid Eng 1986108303ndash9

[50] Simoes FJ Graham JMR Prediction of loading on ahorizontal axis wind turbine using a free vortex wakemodel In Proceedings of the BWEA conference 1991

[51] Voutsinas SG Beleiss MA Rados KG Investigation of theyawed operation of wind turbines by means of a vortexparticle method In AGARD conference proceedings vol552 1995 p 111ndash11

[52] Coton FN Wang T Galbraith RAM An examination ofkey aerodynamic modelling issues raised by the NRELblind comparison Wind Energy 20025199ndash212

[53] Voutsinas SG Vortex methods in aeronautics howto make things work Int J Comput Fluid Dynamics2006

[54] Preuss RD Suciu EO Morino L Unsteady potentialaerodynamics of rotors with applications to horizontal axiswindmills AIAA J 198018(4)385ndash93

[55] Arsuffi G A general formulation for aerodynamic analysisof wind turbine In Proceedings of the second IEAsymposium on the aerodynamics of wind turbines Depart-ment of Fluid Mechanics Technical University of Den-mark 1988

[56] Bareiss R Wagner S A hybrid wake model for hawt InMcAnulty K editor Proceedings of the sixth IEAsymposium on the aerodynamics of wind turbines ETSUHarwell 1993 p 71ndash10

[57] Mughal B Drela M A calculation method for the three-dimensional boundary-layer equations in integral formAIAA Paper 93-0786 1993 Reno NV January

[58] Riziotis VA Voutsinas SG Dynamic stall on wind turbinerotors comparative evaluation study of different modelsIn Proceedings of the EWECrsquo97 1997 Dublin

[59] Chaviaropoulos PK Nikolaou IG Aggelis KA SoslashrensenNN Johansen J Hansen MOL et al Viscous andaeroelastic effects on wind turbine blades The VISCELProject Part I 3D NavierndashStokes rotor simulations WindEnergy 20036(4)365ndash85

[60] Rankine WJM On the mechanical Principles of the Actionof Propellers Transactions of the Institution of NavalArchitects vol 6 1865

[61] Froude RE On the part played in propulsion by differenceof fluid pressure Trans R Inst Naval Arch 188930390ndash405

[62] Wu TY Flow through a heavily loaded actuator discSchiffstechnik 19629134ndash8

[63] Greenberg MD Powers SR Nonlinear actuator disc theoryand flow field calculations including nonuniform loadingNASA CR 1672 NASA 1970

[64] Greenberg MD Nonlinear actuator disc theory Z Flug-wissensch 197220(3)90ndash8

[65] Conway J Analytical solutions for the actuator disc withvariable radial distribution of load J Fluid Mech 1995297327ndash55

[66] Conway J Exact actuator disc solution for non-uniformheavy loading and slipstream contraction J Fluid Mech1998365235ndash67

[67] Madsen HA The actuator cylinder flow model for verticalaxis wind turbines PhD dissertation Aalborg UniversityCentre 1982

[68] van Kuik GAM On the limitations of Froudersquos actuatordisc concept PhD thesis Eindhoven University of Tech-nology Netherlands 1991

[69] Soslashrensen JN Myken A Unsteady actuator disc model forhorizontal axis wind turbines J Wind Eng Ind Aerodyn199239139ndash49

[70] Soslashrensen JN Kock CW A model for unsteady rotoraerodynamics J Wind Eng Ind Aerodyn 199558259ndash75

[71] Soslashrensen JN Mikkelsen R On the validity of the bladeelement momentum theory In Helm P Zervos A editorsProceedings of the 2001 European wind energy conferenceand exhibition WIP-renewable energies Munchen 2001p 362ndash6

[72] Madsen HA A CFD analysis for the actuator disc flowcompared with momentum theory results In Pedersen Beditor Proceedings of the 10th IEA symposium on theaerodynamics of wind turbines Department of FluidMechanics The Technical University of Denmark 1996p 109ndash24

[73] Soslashrensen JN Shen WZ Munduate X Analysis of wakestates by a full-field actuator disc model Wind Energy1998173ndash88

[74] Madsen HA Rasmussen F The influence of energyconversion and induction from large blade deflections In

Proceedings of the European wind energy conferenceJames amp James 1999 p 138ndash41

[75] Mikkelsen R Soslashrensen JN Shen WZ Modeling andanalysis of the flow field around a coned rotor WindEnergy 20014121ndash35

[76] Fejtek I Roberts L NavierndashStokes computation of wingrotor interaction for a tilt rotor in hover AIAA J199230(11)2595ndash603

[77] Rajagopalan RG Mathur ST Three dimensional analysisof a rotor in forward flight J Am Helicopter Soc199338(3)99

[78] Masson C Smaıli A Leclerc C Aerodynamic analysis ofHAWTs operating in unsteady conditions Wind Energy20014(1)1ndash22

[79] Masson C Ammara I Paraschivoiu I An aerodynamicmethod for the analysis of isolated horizontal-axis windturbines Int J Rotating Mach 1997321ndash32

[80] Hansen MOL Soslashrensen NN Flay RGJ Effect of placing adiffuser around a wind turbine Wind Energy 20004(3)207ndash13

[81] Phillips D Schaffarczyk AP Blade-element and actuatordisc models for a shrouded wind-turbine In Thor S-Eeditor Proceedings of the 15th IEA symposium onaerodynamics of wind turbines FOI Swedish DefenceResearch Agency 2001 p 95ndash105

[82] Mikkelsen R Soslashrensen JN Modelling of wind tunnelblockage In Thor S-E editor Proceedings of the 15th IEAsymposium on aerodynamics of wind turbines FOISwedish Defence Research Agency 2001 p 41ndash51

[83] Mikkelsen R Soslashrensen JN Yaw analysis using a numericalactuator disc model In Pedersen B editor Proceedings ofthe 14th IEA symposium on the aerodynamics of windturbines Department of Fluid Mechanics TechnicalUniversity of Denmark 2000 p 53ndash9

[84] Masson C Viscous differentialactuator disc method andits applications In Thor S-E editor Proceedings of the15th IEA symposium on aerodynamics of wind turbinesFOI Swedish Defence Research Agency 2001 p 65ndash80

[85] Ammara I Leclerc C Masson C A viscous three-dimensional differentialactuator disc method for aerody-namic analysis of wind farms J Solar EnergymdashTransASME 2002124(4)345ndash56

[86] Soslashrensen JN Shen WZ Numerical modelling of windturbine wakes J Fluids Eng 2002124(2)393ndash9

[87] Leclerc C Masson C Towards blade-tip vortex simulationwith an actuator-lifting surface model AIAA-2004-06672004

[88] Mikkelsen R Actuator disc methods applied to windturbines PhD dissertation Department of MechanicalEngineering DTU Lyngby 2003

[89] Ivanell SSA Numerical computations of wind turbinewakes PhD dissertation KTH Royal Institute of Tech-nology Stokholm 2005

[90] Arieli R Tauber ME Computation of subsonic andtransonic Flow about lifting rotor blades AIAA Paper79-1667 1979

[91] Borland C Rizzettaq D Yoshihara H Numerical solutionof three-dimensional unsteady transonic flow over sweptwings AIAA Paper 80-1369 1980

[92] Sankar NL Malone JB Tassa Y An implicit conservativealgorithm for steady and unsteady three-dimensionalpotential flows AIAA Paper 81-1016 1981

[93] Steger JL Caradonna FX Conservative implicit finitedifference algorithm for the unsteady transonic potentialequations AIAA Paper 80-1368 1980

[94] Caradonna FX Tung C Desopper A Finite differencemodeling of rotor flows including wake effects Journal ofthe American Helicopter Society April 1984 p 26ndash33

[95] Sankar NL Wake BE Lekoudis SG Solution of theunsteady Euler equations for fixed and rotor windconfigurations J Aircraft 198623(4)

[96] Jameson Baker TJ Solution of the Euler equations forcomplex configurations AIAA Paper 83-1919 1983

[97] Pulliam TH Euler and thin layer NavierndashStokes codesARC2D ARC3D notes for computational fluid dynamicsusers workshop The University of Tennessee SpaceInstitute March 12ndash16 1984

[98] Agarwal RK Deese JE Euler calculations for flowfield of ahelicopter rotor in hover J Aircraft 199724(4)

[99] Srinivasan GR McCroskey WJ NavierndashStokes calcula-tions of hovering rotor flowfields J Aircraft 198825(10)

[100] Srinivasan GR Raghavan V Duque EPN FlowfieldAnalysis of modern helicopter rotors in hover by Navierndash-Stokes method International Technical Specialists meetingRotorcraft Acoustics and Rotor Fluid Dynamics October15ndash17 1991 Philadelphia Pennsylvania USA

[101] Srinivasan GR Baeder JD Obayashi S McCroskey WJFlowfield of a lifting rotor in hover a NavierndashStokessimulation AIAA J 199230(10)

[102] Berkman ME Sankar LN Berezin CR Torok MSNavierndashStokesfull potentialfree-wake method for rotorflows J Aircraft 199734(5)

[103] Hansen MOL Soslashrensen JN Michelsen JA Soslashrensen NNA Global NavierndashStokes Rotor prediction model AIAA97-097 1997

[104] Soslashrensen NN Hansen MOL Rotor performance predic-tions using a NavierndashStokes method AIAA 98-0025 1998

[105] Xu G Sankar LN Computational study of horizontal axiswind turbines AIAA 99-0042 Reno NV January 1999

[106] Duque EPN van Dam CP Hughes S NavierndashStokesSimulations of the NREL combined experiment Phase IIrotor proceedings 1999 ASME wind energy symposium37th AIAA Aerospace Science Meeting and Exhibit AIAA99-0037 Reno NV January 1999

[107] Soslashrensen NN Michelsen JA Aerodynamic predictions forthe unsteady aerodynamics experiment phase-II rotor atthe National Renewable Energy Laboratory AIAA-2000-0037 2000

[108] Soslashrensen JN VISCWIND Viscous effects on wind turbineblades ET-AFM-9902 Department of Energy Engineer-ing Technical University of Denmark June 1999ISBN87-7475-218-9

[109] Chaviaropoulos PK Nikolaou IG Aggelis K SoslashrensenNN Montgomerie B von Geyr H et al Viscous andaeroelastic effects on wind turbine blades the ViscelProject European wind energy conference CopenhagenDenmark 2ndash6 July 2001

[110] Kang S Hirsch C Features of the 3D viscous flow aroundwind turbine blades based on numerical solutions Eur-opean Wind Energy Conference Copenhagen Denmark2ndash6 July 2001

[111] Johansen J Soslashrensen NN Reck M Hansen MOLStuermer A Ramboer J Hirsch C Ekaterinaris JVoutsinas S Perivolaris Y KNOW-BLADE task-33

report Rotor blade computations with 3D vortex gen-erators Risoslash-R-1486(EN) 2005 65p

[112] Soslashrensen NN Johansen J Conway S Voutsinas S HansenMOL Stuermer A KNOW-BLADE task-32 report tipshape study Risoslash-R-1495(EN) 2005 49p

[113] Politis ES Nikolaou IG Chaviaropoulos PK BertagnolioF Soslashrensen NN Johansen J KNOW-BLADE Task-4report NavierndashStokes aeroelasticity Risoslash-R-1492(EN)2005 39p

[114] Johansen J Soslashrensen NN Zahle F Kang S Nikolaou IPolitis ES et al KNOW-BLADE Task-2 report Aero-dynamic accessories Risoslash-R-1482(EN) 2004 33p

[115] Soslashrensen NN Johansen J Conway S CFD computations ofwind turbine blade loads during standstill operation KNOW-BLADE Task 31 report Risoslash-R-1465(EN) 2004 28p

[116] Buning P etal OVERFLOW user manual ver 16ap[117] Meaking R Moving grid overset grid methods for complete

aircraft tiltrotor simulations AIAA Paper 93-3350 July1993

[118] Rizzi A Eliasson P Lindblad I Hirsch C Lacor CHaeuser J The engineering of multiblockmultigrid soft-ware for NavierndashStokes flows on structured meshes JComput Fluids 199322341ndash67

[119] Kroll N Radespiel R Rossow CC Structured grid solversI accurate and efficient flow solvers for 3D applications onstructured meshes AGARD report 807 1995

[120] Eliasson P Edge a NavierndashStokes solver for unstructuredgrids FOI-R-0298-SE 2001

[121] Chorin AJ A numerical method for solving incompressibleviscous flow problems J Comput Phys 1967212ndash26

[122] Rogers SE Kwak D Steady and unsteady solutions of theincompressible NavierndashStokes equations AIAA J 199129(4)

[123] Harlow FH Welch JE Numerical calculations of time-dependent viscous incompressible flow of fluid with freesurface Phys Fluids 196582182

[124] Chorin AJ Numerical solution of the NavierndashStokesequations Math Comput 196822(104)745ndash62

[125] Patankar SV Spalding DB A calculation procedure forheat mass and momentum transfer in three-dimensionalparabolic flows Int J Heat Mass Transfer 1972(15)1787

[126] Rhie CM A Numerical study of the flow past an isolatedairfoil with separation PhD thesis University of IllinoisUrbane-Champaign 1981

[127] Menter FM Zonal Two equation kndasho Turbulence modelsfor aerodynamic flows AIAA-paper-932906 1993

[128] Spalart PR Allmaras SR A one-equation turbulencemodel for aerodynamic flows AIAA-92-0439 1992

[129] Baldwin BS Barth TJ A one-equation turbulence trans-port model for high Reynolds number wall-bounded flowsAIAA 91-0610 1991

[130] Baldwin BS Lomax H Thin layer approximation andalgebraic model for separated turbulent flow In AIAA16th aerospace science meeting Huntsvill Alabama 1978

[131] Travin A Shur M Strelets M Detached-eddy simulationpast a circular cylinder flow Turbulence Combust199963293ndash313

[132] Strelets M detached eddy simulations of massivelyseparated flows AIAA-2001-0879 2001

[133] Xu G Sankar LN effects of transition turbulence and yawon the performance of horizontal axis wind turbinesAIAA-2000-0048 2000

[134] Michelsen JA Soslashrensen NN Current developments inNavierndashStokes modelling of wind turbine rotor flow InEuropean wind energy conference Copenhagen Denmark2ndash6 July 2001

[135] Ferziger JH Milovan P Computational methods for fluiddynamics Berlin Heidelberg New York Springer 2002ISBN3-540-59434-5

[136] Fingersh LJ Simms D Hand M Jager D Contrell JRobinson M et al Wind tunnel testing of NRELrsquosunsteady aerodynamics experiment AIAA-2001-0035 Pa-per 39th aerospace sciences meeting amp exhibit 2001 Reno

[137] Simms D Schreck S Hand M Fingersh LJ NRELunsteady aerodynamics experiment in the NASA-Ameswind tunnel a comparison of predictions to measurementsNRELTP-500-29494 June 2001

[138] Soslashrensen NN Evaluation of 3D effects from 3D CFDcomputations In IEA Joint action aerodynamics of windturbines 14th symposium Boulder December 2000

[139] Soslashrensen NN Michelsen JA Schreck S NavierndashStokespredictions of the NREL phase VI rotor in the NASAAmes 80-by-120 wind tunnel AIAA-2002-0032 2002

[140] Xu G Sankar LN Application of a viscous flowmethodology to the NREL phase VI ROTOR AIAA-2002-0030 2002

[141] Soslashrensen NN Michelsen JA Schreck S Application ofCFD to wind turbine aerodynamics In Fourth GRACMcongress on computational mechanics GRACM 2002Patra 27ndash29 June 2002

[142] Madsen HA Soslashrensen NN Schreck S Yaw aerodynamicsanalyzed with three codes in comparison with experimentAIAA-2003-0519 2003

[143] Tongchitpakdee C Benjanirat S Sankar LN Numericalsimulation of the aeordynamics of horizontal axis windturbines under yawed flow conditions AIAA-2005-07732005

[144] Benjanirat S Sankar LN Recent improvements to acombined NavierndashStokes full potential methodology formodeling horizontal axis wind turbines AIAA-2004-08302004

[145] Duque EPN Burklund MD Johnson W NavierndashStokesand comprehensive analysis performance predictions of theNREL phase VI experiment AIAA-2003-0355 2003

[146] Benjanirat S Sankar LN Evaluation of turbulence modelsfor the prediction of wind turbine aerodynamics AIAA-2003-0517 2003

[147] Le Pape A Lecanu J 3D NavierndashStokes computations of astall regulated wind turbine In Proceedings the science ofmaking torque from wind 19ndash21 April 2004 DelftUniversity of Technology the Netherlands 2004

[148] Johansen J Soslashrensen NN Michelsen JA Schreck SDetached-eddy simulation of flow around the NRELphase-VI rotor In Proceedings CD-ROM CD 2 Eur-opean wind energy conference and exhibition 2003 (EWEC2003) Madrid (ES) 16ndash19 June 2003

[149] Fleig O Arakawa C Numerical simulation of wind turbinetip noise AIAA-2004-1190 2004

[150] Oslashye S FLEX4 simulation of wind turbine dynamics InProceedings of 28th IEA meeting of experts concerningState of the Art of Aeroelastic Codes for wind turbinecalculations (Available through IEA) 1996

[151] Hansen MOL Aerodynamics of Wind Turbines James ampJames (Science Publishers) Ltd 2000

[152] Schepers JG Verification of European wind turbine designcodes VEWTDC final report Technical report ECN-C-01-055 Netherlands Energy Research Foundation ECNPetten 2002

[153] Chaviaropoulos P Development of a state-of-the-artaeroelastic simulator for horizontal axis wind turbinesPart 1 structural aspects Wind Eng 199620(6)405ndash22

[154] Hodges DH Dowell EH Nonlinear equations of motionfor elastic bending and torsion of twisted non-uniformblades NASA report NASA TN D-7818 1975

[155] Hodges DH A review of composite rotor blade modellingAIAA J 199028(3)561ndash5

[156] Kunz DL Survey and comparison of engineering beamtheories for helicopter rotor blades J Aircraft 199431473ndash97

[157] Jung SN Nagaraj VT Chopra I Assessment of compositerotor blade modelling techniques J Am Helicopter Soc199944(3)188ndash205

[158] Cesnil CES Hodges DH Sutyrin VG Cross sectionalanalysis of composite beams including large initial twistand curvature effects AIAA J 199634(9)1913ndash20

[159] Nim E Coupling and reduction of the HAWC equationsRISOE report RISOE-R-1294 (EN) 2001

[160] Snel H Liendenburg C Aeroelastic rotor system codefor horizontal axis wind turbines PHATAS-II In Pro-ceedings of the ECWEC 1990 Madrid Spain 1990p 284ndash90

[161] Riziotis VA Voutsinas SG GAST A general aerodynamicand structural prediction tool for wind turbines InProceedings of the EWECrsquo97 1997 Dublin Ireland

[162] Hansen AC Laino DJ Validation study for AeroDyn andYawDyn using phase iii combined experiment data InAIAA-1997-943 aerospace sciences meeting and exhibit35th Reno NV January 6ndash9 1997

[163] Riziotis VA Voutsinas SG Advanced aeroelastic modelingof complete wind turbine configurations in view ofassessing stability characteristics In Proceedings of theEWECrsquo06 2006 Athens Greece

[164] Bazoune A Khulief YA Shape functions of three-dimensional Timoshenko beam element J Sound Vib 2003259(2)473ndash80

[165] Pfeiffer F Glocker C Multi-body dynamics with unilateralcontacts New York Wiley 1996

[166] Bauchau OA Computational schemes for flexible non-linear multi-body systems Multibody System Dynamics19982169ndash225

[167] Bauchau OA Hodges DH Analysis of non-linear multi-body systems with elastic couplings Multibody SystemDynamics 19993166ndash88

[168] Chaviaropoulos PK Politis ES Lekou DJ Sorensen NNHansen MH Bulder BH et al Enhancing the damping ofwind turbine rotor blades the Dampblade Project WindEnergy 20069163ndash77

[169] Petot D Differential equation modelling of dynamic stallLa Recherche Aerospatiale 1989559ndash72

[170] Chopra I Aeroelastic stability of an elastic circulationcontrol rotor blade in hover Vertice 19848(4)353ndash71

[171] Chaviaropoulos PK Flapleadndashlag aeroelastic stability ofwind turbine blades J Wind Energy 20014183ndash200

[172] Hansen MH Improved modal dynamics of wind turbinesto avoid stall-induced vibrations J Wind Energy 20036(2)179ndash95

[173] Friedman PP Hammond CE Woo TH Efficient numericaltreatment of periodic systems with application to stabilityproblems Int J Numer Methods Eng 1977111117ndash36

[174] Peters DA Fast Floquet theory and trim for multi-bladedrotorcraft J Am Helicopter Soc 199439(4)82ndash9

[175] Coleman RP Feingold AM Theory of self-excite mechan-ical oscillations of helicopter rotors with hinged bladesNASA report NASA-TN-3844 1957

[176] Petersen JT Madsen HA Bjorck A Enevoldsen P Oye SGanander H et al Prediction of dynamic loads andinduced vibrations in stall RISO report RISOE-R-1045(EN) 1998

[177] Politis ES editor Benchmark calculations on the NM80wind turbine CRES Technical report STABCON project2005

[178] Kirchgassner B ARLISmdasha program system for analysis ofrotating linear systems In Proceedings of the EWEC 1984Hamburg Germany 1984 p 22ndash6

[179] Hansen MH Stability Analysis of three-bladed turbinesusing an eigenvalue approach In 42 AIAA aerospacesciences meeting and exhibit 2004 ASME wind energysymposium Reno NV January 2004

[180] van Engelen TG Braam H TURBU offshore computerprogramme for frequency domain analysis of horizontalaxis offshore wind turbines implementation ECN reportECN-Cmdash04-0479 Petten The Netherlands 2004

[181] van Holten Th van Overbeek K Automatic simulation ofcomplex non-linear dynamic systems In 26th EuropeanRotorcraft Forum Paper 65 2000

[182] Tasker FA Chopra I Assessment of transient analysistechniques for rotor stability J Am Helicopter Soc19908839ndash50

[183] Simon M Tomlinson GR Use of the Hilbert transform inmodal analysis of linear and non-linear structures J SoundVib 198496421ndash36

[184] Smith CB Wereley NM Transient analysis for dampingidentification in rotating composite beams with integraldamping layers Smart Mater Struct 19965540ndash50

[185] Madsen HA Petersen JT Bjorck A Ganander HWinkelaar D Brand A et al Prediction of Dynamic Loadsand Induced Vibrations in StallmdashSTALLVIB Finalpublishable report on Contract JOR3-CT95-0047 RisoslashNational Laboratory 1998

[186] Hansen MH HAWCStab aeroelastic stability tool forwind turbinesmdashuserrsquos guide Risoslash-I-2232(EN) Risoslash Na-tional Laboratory August 2004

[187] Petersen JT Kinematically nonlinear finite element modelof a horizontal axis wind turbine PhD thesis Part 1 and 2Risoslash National Laboratory Roskilde Denmark July 1990

[188] Petersen JT The Aeroelastic Code HawCmdashmodel andcomparison In Pedersen BM editor Proceedings of stateof the art of aeroelastic codes for wind turbine calculations28th meeting of experts International Energy AgencyAnnex XI Technical University of Denmark LyngbyApril 11ndash12 1996 p 129ndash35

[189] Larsen TJ Hansen AM Buhl T Aeroelastic effects of largeblade deflections for wind turbines In Proceedings of theconference the science of making torque from windSpecial topic conference Delft 19ndash21 April 2004

[190] Rubak R Petersen JT Monopile as part of aeroelasticwind turbine simulation code In Proceedings of Copenha-gen Offshore Wind 2005 conference amp exhibition 26ndash28October 2005

[191] Larsen JT Madsen HA Hansen AM Thomsen KInvestigation of stability effects of an offshore wind turbineusing the new aeroelastic code HAWC2 In Proceedings ofCopenhagen Offshore Wind 2005 2005

[192] Madsen HA Rasmussen F A near wake model for trailingvorticity compared with the blade element momentumtheory Wind Energy 20047325ndash41

[193] Madsen HA Yaw simulation using a 3D actuator discmodel coupled to the aeroelastic code HawC In PedersenBM editor Proceedings 13th symposium IEA joint actionon aerodynamics of wind turbines Stockholm 1999

[194] Thomsen K Madsen HA A new simulation method forturbine in wakesmdashapplied to extreme response duringoperation Wind Energy 2005835ndash47

[195] Madsen HA Larsen GC Thomsen K Wake flowcharacteristics in low ambient turbulence conditionsIn Proceedings (CD-ROM) Copenhagen offshore windconference 2005 Copenhagen (DK) 25ndash28 September2005

[196] Troldborg N Soslashrensen JN Mikkelsen R Numericalsimulations of wakes of wind turbines in wind farms InProceedings of the EWEC 2006 European wind energyconference Athens 2006

[197] Madsen HA Rasmussen F Derivation of three-dimen-sional airfoil data on the basis of experiments and theoryIn Proceedings Windpower 088 Honolulu Hawaii 1988

[198] Schepers JG et al Final report of IEA Annex XIV Fieldrotor aerodynamics Report ECN-C-97-027 ECN June1997

[199] Schepers JG et al Final report of IEA Annex XVIIIenhanced field rotor aerodynamics database Report ECN-Cndash02-016 ECN February 2002

[200] Hansen MOL Soslashrensen NN Soslashrensen JN Michelsen JAExtraction of lift drag and angle of attack from computed3-D viscous flow around a rotating blade In Proceedingsof the European wind energy conference EWEC-1997Dublin Ireland October 1997

[201] Johansen J Soslashrensen NN Method for extractingairfoil data using 3D CFD computations In Thor SEeditor IEA Joint Action Committee on AerodynamicsAnnex IV Aero experts meeting Boulder CO (US) 5ndash6May 2003 FFA The Aeronautical Institute of Sweden2003

[202] Johansen J Soslashrensen NN Mikkelsen R Rotoraerodyna-mik In the report Risoslash-R-1434(DA) Forskning i Aero-elasticitet edited by Christian Bak February 2004

[203] Hald T Hoslashgedal M Implementation of a finite elementfoundation module in Flex5 using CraigndashBampton sub-structuring In Proceedings of Copenhagen offshore wind2005 conference amp exhibition 26ndash28 October 2005

[204] Seidel M von Mutius M Ris D Steudel D Integratedanalysis of wind and wave loading for complex supportstructures of offshore wind turbines In Proceedings ofCopenhagen Offshore Wind 2005 conference amp exhibition26ndash28 October 2005

- Introduction
- Predicting aerodynamic loads on a wind turbine
- - Blade Element Momentum Method
  - - Dynamic wakeinflow
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
    - - Lifting line panel and vortex models
      - Vortex methods
        
        Panel methods
        
        Generalized actuator disc models
        
        Navier-Stokes solvers
        
        Introduction to computational rotor aerodynamics
        
        Approaches
        
        Turbulence and transition
        
        Geometry and grid generation
        
        Numerical issues
        
        Application of CFD to wind turbine aerodynamics
        
        Future
        
        Structural modelling of a wind turbine
        
        Principle of virtual work and use of modal shape functions
        
        FEM modelling of wind turbine components applying non-linear beam theory
        
        Problems and solutions in wind turbine aeroelasticity
        
        Aeroelastic stability
        
        Aeroelastic coupling linear vs non-linear formulations
        
        Examples of time simulations and instabilities
        
        Edgewise blade vibration instability
        
        Instability problems of parked rotors
        
        Flutter instability
        
        Present and future developments of aeroelastic models
        
        Areas with influence on the development of aeroelastic models
        
        Influence of up-scaling
        
        Siting of the turbines
        
        Future trends in turbine design and siting
        
        Areas of development in present and new codes
        
        Non-linear structural dynamics
        
        Calculation of induction and its dynamics
        
        Wake operation
        
        Derivation of airfoil data for aeroelastic simulations
        
        Complex inflow
        
        Aerodynamics of parked rotors
        
        Offshore turbines including floating turbines
        
        Discussion
        
        References

Page 2: State of the art in wind turbine aerodynamics and ...phyexp/uploads/EolienneFlexible/Eoli… · ARTICLE IN PRESS Progress in Aerospace Sciences 42 (2006) 285–330 State of the art

244 Geometry and grid generation 299245 Numerical issues 299246 Application of CFD to wind turbine aerodynamics 300247 Future 302

3 Structural modelling of a wind turbine 30231 Principle of virtual work and use of modal shape functions 30232 FEM modelling of wind turbine components applying non-linear beam theory 304

4 Problems and solutions in wind turbine aeroelasticity 30841 Aeroelastic stability 30842 Aeroelastic coupling linear vs non-linear formulations 30943 Examples of time simulations and instabilities 310

431 Edgewise blade vibration instability 312432 Instability problems of parked rotors 317433 Flutter instability 317

5 Present and future developments of aeroelastic models 31851 Areas with influence on the development of aeroelastic models 318

511 Influence of up-scaling 318512 Siting of the turbines 319513 Future trends in turbine design and siting 319

52 Areas of development in present and new codes 319521 Non-linear structural dynamics 319522 Calculation of induction and its dynamics 320523 Wake operation 321524 Derivation of airfoil data for aeroelastic simulations 322525 Complex inflow 323526 Aerodynamics of parked rotors 324527 Offshore turbines including floating turbines 324

6 Discussion 325References 325

1 Introduction

The size of commercial wind turbines hasincreased dramatically in the last 25 years fromapproximately a rated power of 50 kW and a rotordiameter of 10ndash15m up to todayrsquos commerciallyavailable 5MW machines with a rotor diameter ofmore than 120m This development has forced thedesign tools to change from simple static calcula-tions assuming a constant wind to dynamic simula-tion software that from the unsteady aerodynamicloads models the aeroelastic response of the entirewind turbine construction including tower drivetrain rotor and control system The Danishstandard DS 472 [1] allows simplified load calcula-tions if the rotor diameter is less than 25m andsome other criteria are fulfilled A rotor diameter of25m corresponds approximately to a rated power of200ndash250 kW which is less than almost any moderncommercial wind turbine today Instead modernwind turbines are designed to fulfill the require-ments of the more comprehensive IEC 61 400-1 [2]standard At some time during the development of

larger and larger commercial wind turbines the needfor aeroelastic tools thus became necessary Aero-elastic tools were mainly developed at the univer-sities and research laboratories in parallel with theevolution of commercial wind turbines At the sametime governments and utility companies erectedlarge non-commercial prototypes for research pur-poses as the Nibe [3] and Tjaereborg machines [4]Measurement campaigns were undertaken on thesemachines and the results used to tune and validatethe aeroelastic programmes in order to developadvanced software for the rapidly growing industryEven today measurements from the Tjaereborgmachine is used as a benchmark when developingnew aeroelastic codes see eg [5] In [5] is alsocompiled a long list of available software that atdifferent levels of complexity can model the aero-elastic response of a wind turbine construction Allthe aeroelastic models need as input a time historyof the wind seen by the rotor which as a minimummust contain some physical correct properties suchas realistic power spectra and spatial coherenceApart from the wind input aeroelastic codes contain

Wz BL cos f

Wy BL sin f

ARTICLE IN PRESS

W

Vo

VbladeVrot

Vrel

y x

rotor plane

z

$

CT dT

1=2rV20dA

(214)

a Wz

V0(215)

f g 1 for ap03

145 3aamp for a403

(

(216)

W int $ t1dW int

dt Wqs $ k t1

dWqs

dt (217)

W $ t2dW

dt W int (218)

t1 11

1 13aampR

V 0(219)

and

t2 039 026r

R

2$

t1 (2110)

ARTICLE IN PRESS

400

350

300

250

200

150

1000 10 20 30 40 50 60

time [s]

Rot

orsh

aft t

orqu

e [k

Nm

]

BEMMeasurement

W W0 1$r

Rtan

w2cosyb y0amp

(2111)

ARTICLE IN PRESS

Rotor disc

Vo Von x

ampyawtilt

V Wn

df sdt

f sts f s

t (2113)

(2115)

DCl Clinv Cl2D

DCd Cd2D Cd2Dmin

DCm Cm2D Cminv

be derived

(221)

V rC$ rF (222)

r2C o (223)

Cxamp 1

4p

Zo0

x x0j jdVol (224)

wxamp 1

4p

Zx x0amp o0

x x0j j3dVol (225)

wxamp 1

4p

Z

S

Gx x0ampx x0j j3

qx0

qS0 qS0 (227)

fx tamp Z

S

s0tamp4p x x0j j

m0tamp n0r

4p x x0j j3

$ dS0

228amp

rZ

Sm

dS0

Z

Sm

dS0

$I

Sm

dS0 229amp

1pU(Wr

qmWqt

$ VWm 1VU(W

qmWqt

$ VWm ramp

mW 0 2210amp

where VWm V$W $ V

W

amp =2 Since G mW

ARTICLE IN PRESS

nr

BS micro

W WS micro

( )Pmicro

F

extUr

C

C = W

ARTICLE IN PRESS

= (y )y

(y)

w

y) )

emission line

Se

W B

B

= t

(

(t=

w

+

d

D

DtdO )

qqt

ARTICLE IN PRESS

L

micro = L

TE tmicro TE t t

micro -2TE t t

-

wake surface

streak line

( )=B TEy t micro

emiss

ion

line

s

T

ARTICLE IN PRESS

700800900

10001100120013001400150016001700

6 8 10 12 14 16 18 20 22 24 26

Low

Spe

ed S

haft

Tor

que

[Nm

]

Wind Speed [m s]

MeasuredComputed

1000

1500

2000

2500

3000

3500

4000

4500

5000

6 8 10 12 14 16 18 20 22 24 26

Roo

t Fla

p M

omen

t [N

m]

Wind Speed [ms]

MeasuredComputed

ARTICLE IN PRESS

0

05

1

15

2

25

3

02 03 04 05 06 07 08 09 1

CN

rR

MeasuredComputed

-008

-006

-004

-002

0

002

004

02 03 04 05 06 07 08 09 1

CT

rR

MeasuredComputed

-15

-1

-05

0

05

1

15

2

25

3

0 02 04 06 08 1

-Cp

X Chord

rR=063

M x$ C _x$ Kx Fg (311)

ARTICLE IN PRESS

-2

0

2

4

6

8

10

12

0 50 100 150 200 250 300 350

CN

Azimuth Angle [deg]

S1500600 r R=030

-04

-02

0

02

04

06

08

1

0 50 100 150 200 250 300 350

CT

Azimuth Angle [deg]

S1500600 rR=030

Fgidxi Z

S

p uidS (312)

and

r n0 $ S0 u$ S1 yu

S0 1 0 0 z

0 1 0 0

0 0 1 x

2

64

3

75

S1 0 0 0 0

x 0 z 0

0 0 0 0

2

64

3

75

(321)

My GItampqyy

322amp

ARTICLE IN PRESS

y

x

z

A

u

w

u+du

M + dMz

z

M + dMyy

M + dMxx

zF + dF

yyF + dF

xxF + dF

Mx

Fx

Mz

FzMy

Fy dr

ra

)P

x

z

y

A(a) (b)

a

b

Z

A

rdm

$ dy dF$

Z

A

g dm

$ dy$ dpdy

(323)

Z

A

r0 rdm

$Z

A

r0 gdm

fx$ zy dy z xygT

A

S0ampT r dm qyZ

A

$ q2yy-K21 qyu( $Z

A

S0ampT g dm$ Sa dP

325amp

K11

Fy 0 0 0

0 EA 0 0

0 0 Fy 0

0 0 0 GIt

2

666664

3

777775

K22

EIzz 0 EIxz 0

0 0 0 0

EIxz 0 EIxx 0

0 0 0 0

2

666664

3

777775

Sa

1 0 0

0 1 0

0 0 1

za 0 xa

2

666664

3

777775

K12

0 0 0 0

EAx+ 0 EAz+ 0

0 0 0 0

2

666664

3

777775

K21

0 EAx+ 0 0

0 0 0 0

0 EAz+ 0 0

0 0 0 0

2

666664

3

777775

Then (321) becomes

r u

y$ v

w

8gtlt

gt

9gt=

gt$ E

xlyy

z

8gtlt

gt

9gt=

gt (326)

ARTICLE IN PRESS

x

z

yu

w

vO

Orsquoz

xz

xu

w amptamp

ampamp

amp +ˆ

Z L

0duT Zu _u uampdy

)X

e

Z Le

X

e

Z Le

0NeT ZNeueN

Muamp u$ Cu

amp _u$ Ku

amp u Qu

amp (3210)

ARTICLE IN PRESS

0k

rGk

OG O

xG x

zG z

yG

y y

z

x

O

x

y

z

O

ARTICLE IN PRESS

y

O

q8 q9 q10 q11q12 q13

yG y

z

x

O

x

y

z

OG O

xG x

zG z

q1 q2 q3 q4q5 q6

q7

q14

py L sin fD cos f

(413)

m1

amp 2=NampPN

y2tamp $ ~y2tamp

p[183184]

ARTICLE IN PRESS

3540

4550

tim

e (s

ec)

-35

E+0

6

-3E

+06

-25

E+0

6

-2E

+06

-15

E+0

6

-1E

+06

12

34

freq

uenc

y (H

z)

0

5E+

08

1E+

08

15E

+08

2E+

08

25E

+08

3E+

08

-t

data

19

190

5

191

5

192

5

193

5

194

00

51

15

22

53

t (se

c)

- 1

n

Fig21Schem

atic

dam

pingan

alysis

ARTICLE IN PRESS

-15e+06

-1e+06

-500000

0

500000

1e+06

15e+06

2 4 6 8 10 12 14

tow

er to

p til

t ben

ding

mom

ent (

KN

m)

time (s)

1000

800

600

400

200

0

-2000 50 100 150 200 250 300

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

1000900800700600500400300200100

0-100

200 210 220 230 240 250 260

ARTICLE IN PRESS

wake operation

complex inflow

ARTICLE IN PRESS

2

18

16

14

12C1

Cd

1

08

06

040 5 10 15 20 25 30

r=14r=28r=42r=56r=69r=89r=94r=97

35 40$ $

05

045

04

035

03

025

02

015

01

005

00

5 10 15 20 25 30 35 40

ARTICLE IN PRESS

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

Page 3: State of the art in wind turbine aerodynamics and ...phyexp/uploads/EolienneFlexible/Eoli… · ARTICLE IN PRESS Progress in Aerospace Sciences 42 (2006) 285–330 State of the art

Wz BL cos f

Wy BL sin f

ARTICLE IN PRESS

W

Vo

VbladeVrot

Vrel

y x

rotor plane

z

$

CT dT

1=2rV20dA

(214)

a Wz

V0(215)

f g 1 for ap03

145 3aamp for a403

(

(216)

W int $ t1dW int

dt Wqs $ k t1

dWqs

dt (217)

W $ t2dW

dt W int (218)

t1 11

1 13aampR

V 0(219)

and

t2 039 026r

R

2$

t1 (2110)

ARTICLE IN PRESS

400

350

300

250

200

150

1000 10 20 30 40 50 60

time [s]

Rot

orsh

aft t

orqu

e [k

Nm

]

BEMMeasurement

W W0 1$r

Rtan

w2cosyb y0amp

(2111)

ARTICLE IN PRESS

Rotor disc

Vo Von x

ampyawtilt

V Wn

df sdt

f sts f s

t (2113)

(2115)

DCl Clinv Cl2D

DCd Cd2D Cd2Dmin

DCm Cm2D Cminv

be derived

(221)

V rC$ rF (222)

r2C o (223)

Cxamp 1

4p

Zo0

x x0j jdVol (224)

wxamp 1

4p

Zx x0amp o0

x x0j j3dVol (225)

wxamp 1

4p

Z

S

Gx x0ampx x0j j3

qx0

qS0 qS0 (227)

fx tamp Z

S

s0tamp4p x x0j j

m0tamp n0r

4p x x0j j3

$ dS0

228amp

rZ

Sm

dS0

Z

Sm

dS0

$I

Sm

dS0 229amp

1pU(Wr

qmWqt

$ VWm 1VU(W

qmWqt

$ VWm ramp

mW 0 2210amp

where VWm V$W $ V

W

amp =2 Since G mW

ARTICLE IN PRESS

nr

BS micro

W WS micro

( )Pmicro

F

extUr

C

C = W

ARTICLE IN PRESS

= (y )y

(y)

w

y) )

emission line

Se

W B

B

= t

(

(t=

w

+

d

D

DtdO )

qqt

ARTICLE IN PRESS

L

micro = L

TE tmicro TE t t

micro -2TE t t

-

wake surface

streak line

( )=B TEy t micro

emiss

ion

line

s

T

ARTICLE IN PRESS

700800900

10001100120013001400150016001700

6 8 10 12 14 16 18 20 22 24 26

Low

Spe

ed S

haft

Tor

que

[Nm

]

Wind Speed [m s]

MeasuredComputed

1000

1500

2000

2500

3000

3500

4000

4500

5000

6 8 10 12 14 16 18 20 22 24 26

Roo

t Fla

p M

omen

t [N

m]

Wind Speed [ms]

MeasuredComputed

ARTICLE IN PRESS

0

05

1

15

2

25

3

02 03 04 05 06 07 08 09 1

CN

rR

MeasuredComputed

-008

-006

-004

-002

0

002

004

02 03 04 05 06 07 08 09 1

CT

rR

MeasuredComputed

-15

-1

-05

0

05

1

15

2

25

3

0 02 04 06 08 1

-Cp

X Chord

rR=063

M x$ C _x$ Kx Fg (311)

ARTICLE IN PRESS

-2

0

2

4

6

8

10

12

0 50 100 150 200 250 300 350

CN

Azimuth Angle [deg]

S1500600 r R=030

-04

-02

0

02

04

06

08

1

0 50 100 150 200 250 300 350

CT

Azimuth Angle [deg]

S1500600 rR=030

Fgidxi Z

S

p uidS (312)

and

r n0 $ S0 u$ S1 yu

S0 1 0 0 z

0 1 0 0

0 0 1 x

2

64

3

75

S1 0 0 0 0

x 0 z 0

0 0 0 0

2

64

3

75

(321)

My GItampqyy

322amp

ARTICLE IN PRESS

y

x

z

A

u

w

u+du

M + dMz

z

M + dMyy

M + dMxx

zF + dF

yyF + dF

xxF + dF

Mx

Fx

Mz

FzMy

Fy dr

ra

)P

x

z

y

A(a) (b)

a

b

Z

A

rdm

$ dy dF$

Z

A

g dm

$ dy$ dpdy

(323)

Z

A

r0 rdm

$Z

A

r0 gdm

fx$ zy dy z xygT

A

S0ampT r dm qyZ

A

$ q2yy-K21 qyu( $Z

A

S0ampT g dm$ Sa dP

325amp

K11

Fy 0 0 0

0 EA 0 0

0 0 Fy 0

0 0 0 GIt

2

666664

3

777775

K22

EIzz 0 EIxz 0

0 0 0 0

EIxz 0 EIxx 0

0 0 0 0

2

666664

3

777775

Sa

1 0 0

0 1 0

0 0 1

za 0 xa

2

666664

3

777775

K12

0 0 0 0

EAx+ 0 EAz+ 0

0 0 0 0

2

666664

3

777775

K21

0 EAx+ 0 0

0 0 0 0

0 EAz+ 0 0

0 0 0 0

2

666664

3

777775

Then (321) becomes

r u

y$ v

w

8gtlt

gt

9gt=

gt$ E

xlyy

z

8gtlt

gt

9gt=

gt (326)

ARTICLE IN PRESS

x

z

yu

w

vO

Orsquoz

xz

xu

w amptamp

ampamp

amp +ˆ

Z L

0duT Zu _u uampdy

)X

e

Z Le

X

e

Z Le

0NeT ZNeueN

Muamp u$ Cu

amp _u$ Ku

amp u Qu

amp (3210)

ARTICLE IN PRESS

0k

rGk

OG O

xG x

zG z

yG

y y

z

x

O

x

y

z

O

ARTICLE IN PRESS

y

O

q8 q9 q10 q11q12 q13

yG y

z

x

O

x

y

z

OG O

xG x

zG z

q1 q2 q3 q4q5 q6

q7

q14

py L sin fD cos f

(413)

m1

amp 2=NampPN

y2tamp $ ~y2tamp

p[183184]

ARTICLE IN PRESS

3540

4550

tim

e (s

ec)

-35

E+0

6

-3E

+06

-25

E+0

6

-2E

+06

-15

E+0

6

-1E

+06

12

34

freq

uenc

y (H

z)

0

5E+

08

1E+

08

15E

+08

2E+

08

25E

+08

3E+

08

-t

data

19

190

5

191

5

192

5

193

5

194

00

51

15

22

53

t (se

c)

- 1

n

Fig21Schem

atic

dam

pingan

alysis

ARTICLE IN PRESS

-15e+06

-1e+06

-500000

0

500000

1e+06

15e+06

2 4 6 8 10 12 14

tow

er to

p til

t ben

ding

mom

ent (

KN

m)

time (s)

1000

800

600

400

200

0

-2000 50 100 150 200 250 300

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

1000900800700600500400300200100

0-100

200 210 220 230 240 250 260

ARTICLE IN PRESS

wake operation

complex inflow

ARTICLE IN PRESS

2

18

16

14

12C1

Cd

1

08

06

040 5 10 15 20 25 30

r=14r=28r=42r=56r=69r=89r=94r=97

35 40$ $

05

045

04

035

03

025

02

015

01

005

00

5 10 15 20 25 30 35 40

ARTICLE IN PRESS

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

Page 4: State of the art in wind turbine aerodynamics and ...phyexp/uploads/EolienneFlexible/Eoli… · ARTICLE IN PRESS Progress in Aerospace Sciences 42 (2006) 285–330 State of the art

CT dT

1=2rV20dA

(214)

a Wz

V0(215)

f g 1 for ap03

145 3aamp for a403

(

(216)

W int $ t1dW int

dt Wqs $ k t1

dWqs

dt (217)

W $ t2dW

dt W int (218)

t1 11

1 13aampR

V 0(219)

and

t2 039 026r

R

2$

t1 (2110)

ARTICLE IN PRESS

400

350

300

250

200

150

1000 10 20 30 40 50 60

time [s]

Rot

orsh

aft t

orqu

e [k

Nm

]

BEMMeasurement

W W0 1$r

Rtan

w2cosyb y0amp

(2111)

ARTICLE IN PRESS

Rotor disc

Vo Von x

ampyawtilt

V Wn

df sdt

f sts f s

t (2113)

(2115)

DCl Clinv Cl2D

DCd Cd2D Cd2Dmin

DCm Cm2D Cminv

be derived

(221)

V rC$ rF (222)

r2C o (223)

Cxamp 1

4p

Zo0

x x0j jdVol (224)

wxamp 1

4p

Zx x0amp o0

x x0j j3dVol (225)

wxamp 1

4p

Z

S

Gx x0ampx x0j j3

qx0

qS0 qS0 (227)

fx tamp Z

S

s0tamp4p x x0j j

m0tamp n0r

4p x x0j j3

$ dS0

228amp

rZ

Sm

dS0

Z

Sm

dS0

$I

Sm

dS0 229amp

1pU(Wr

qmWqt

$ VWm 1VU(W

qmWqt

$ VWm ramp

mW 0 2210amp

where VWm V$W $ V

W

amp =2 Since G mW

ARTICLE IN PRESS

nr

BS micro

W WS micro

( )Pmicro

F

extUr

C

C = W

ARTICLE IN PRESS

= (y )y

(y)

w

y) )

emission line

Se

W B

B

= t

(

(t=

w

+

d

D

DtdO )

qqt

ARTICLE IN PRESS

L

micro = L

TE tmicro TE t t

micro -2TE t t

-

wake surface

streak line

( )=B TEy t micro

emiss

ion

line

s

T

ARTICLE IN PRESS

700800900

10001100120013001400150016001700

6 8 10 12 14 16 18 20 22 24 26

Low

Spe

ed S

haft

Tor

que

[Nm

]

Wind Speed [m s]

MeasuredComputed

1000

1500

2000

2500

3000

3500

4000

4500

5000

6 8 10 12 14 16 18 20 22 24 26

Roo

t Fla

p M

omen

t [N

m]

Wind Speed [ms]

MeasuredComputed

ARTICLE IN PRESS

0

05

1

15

2

25

3

02 03 04 05 06 07 08 09 1

CN

rR

MeasuredComputed

-008

-006

-004

-002

0

002

004

02 03 04 05 06 07 08 09 1

CT

rR

MeasuredComputed

-15

-1

-05

0

05

1

15

2

25

3

0 02 04 06 08 1

-Cp

X Chord

rR=063

M x$ C _x$ Kx Fg (311)

ARTICLE IN PRESS

-2

0

2

4

6

8

10

12

0 50 100 150 200 250 300 350

CN

Azimuth Angle [deg]

S1500600 r R=030

-04

-02

0

02

04

06

08

1

0 50 100 150 200 250 300 350

CT

Azimuth Angle [deg]

S1500600 rR=030

Fgidxi Z

S

p uidS (312)

and

r n0 $ S0 u$ S1 yu

S0 1 0 0 z

0 1 0 0

0 0 1 x

2

64

3

75

S1 0 0 0 0

x 0 z 0

0 0 0 0

2

64

3

75

(321)

My GItampqyy

322amp

ARTICLE IN PRESS

y

x

z

A

u

w

u+du

M + dMz

z

M + dMyy

M + dMxx

zF + dF

yyF + dF

xxF + dF

Mx

Fx

Mz

FzMy

Fy dr

ra

)P

x

z

y

A(a) (b)

a

b

Z

A

rdm

$ dy dF$

Z

A

g dm

$ dy$ dpdy

(323)

Z

A

r0 rdm

$Z

A

r0 gdm

fx$ zy dy z xygT

A

S0ampT r dm qyZ

A

$ q2yy-K21 qyu( $Z

A

S0ampT g dm$ Sa dP

325amp

K11

Fy 0 0 0

0 EA 0 0

0 0 Fy 0

0 0 0 GIt

2

666664

3

777775

K22

EIzz 0 EIxz 0

0 0 0 0

EIxz 0 EIxx 0

0 0 0 0

2

666664

3

777775

Sa

1 0 0

0 1 0

0 0 1

za 0 xa

2

666664

3

777775

K12

0 0 0 0

EAx+ 0 EAz+ 0

0 0 0 0

2

666664

3

777775

K21

0 EAx+ 0 0

0 0 0 0

0 EAz+ 0 0

0 0 0 0

2

666664

3

777775

Then (321) becomes

r u

y$ v

w

8gtlt

gt

9gt=

gt$ E

xlyy

z

8gtlt

gt

9gt=

gt (326)

ARTICLE IN PRESS

x

z

yu

w

vO

Orsquoz

xz

xu

w amptamp

ampamp

amp +ˆ

Z L

0duT Zu _u uampdy

)X

e

Z Le

X

e

Z Le

0NeT ZNeueN

Muamp u$ Cu

amp _u$ Ku

amp u Qu

amp (3210)

ARTICLE IN PRESS

0k

rGk

OG O

xG x

zG z

yG

y y

z

x

O

x

y

z

O

ARTICLE IN PRESS

y

O

q8 q9 q10 q11q12 q13

yG y

z

x

O

x

y

z

OG O

xG x

zG z

q1 q2 q3 q4q5 q6

q7

q14

py L sin fD cos f

(413)

m1

amp 2=NampPN

y2tamp $ ~y2tamp

p[183184]

ARTICLE IN PRESS

3540

4550

tim

e (s

ec)

-35

E+0

6

-3E

+06

-25

E+0

6

-2E

+06

-15

E+0

6

-1E

+06

12

34

freq

uenc

y (H

z)

0

5E+

08

1E+

08

15E

+08

2E+

08

25E

+08

3E+

08

-t

data

19

190

5

191

5

192

5

193

5

194

00

51

15

22

53

t (se

c)

- 1

n

Fig21Schem

atic

dam

pingan

alysis

ARTICLE IN PRESS

-15e+06

-1e+06

-500000

0

500000

1e+06

15e+06

2 4 6 8 10 12 14

tow

er to

p til

t ben

ding

mom

ent (

KN

m)

time (s)

1000

800

600

400

200

0

-2000 50 100 150 200 250 300

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

1000900800700600500400300200100

0-100

200 210 220 230 240 250 260

ARTICLE IN PRESS

wake operation

complex inflow

ARTICLE IN PRESS

2

18

16

14

12C1

Cd

1

08

06

040 5 10 15 20 25 30

r=14r=28r=42r=56r=69r=89r=94r=97

35 40$ $

05

045

04

035

03

025

02

015

01

005

00

5 10 15 20 25 30 35 40

ARTICLE IN PRESS

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

Page 5: State of the art in wind turbine aerodynamics and ...phyexp/uploads/EolienneFlexible/Eoli… · ARTICLE IN PRESS Progress in Aerospace Sciences 42 (2006) 285–330 State of the art

W W0 1$r

Rtan

w2cosyb y0amp

(2111)

ARTICLE IN PRESS

Rotor disc

Vo Von x

ampyawtilt

V Wn

df sdt

f sts f s

t (2113)

(2115)

DCl Clinv Cl2D

DCd Cd2D Cd2Dmin

DCm Cm2D Cminv

be derived

(221)

V rC$ rF (222)

r2C o (223)

Cxamp 1

4p

Zo0

x x0j jdVol (224)

wxamp 1

4p

Zx x0amp o0

x x0j j3dVol (225)

wxamp 1

4p

Z

S

Gx x0ampx x0j j3

qx0

qS0 qS0 (227)

fx tamp Z

S

s0tamp4p x x0j j

m0tamp n0r

4p x x0j j3

$ dS0

228amp

rZ

Sm

dS0

Z

Sm

dS0

$I

Sm

dS0 229amp

1pU(Wr

qmWqt

$ VWm 1VU(W

qmWqt

$ VWm ramp

mW 0 2210amp

where VWm V$W $ V

W

amp =2 Since G mW

ARTICLE IN PRESS

nr

BS micro

W WS micro

( )Pmicro

F

extUr

C

C = W

ARTICLE IN PRESS

= (y )y

(y)

w

y) )

emission line

Se

W B

B

= t

(

(t=

w

+

d

D

DtdO )

qqt

ARTICLE IN PRESS

L

micro = L

TE tmicro TE t t

micro -2TE t t

-

wake surface

streak line

( )=B TEy t micro

emiss

ion

line

s

T

ARTICLE IN PRESS

700800900

10001100120013001400150016001700

6 8 10 12 14 16 18 20 22 24 26

Low

Spe

ed S

haft

Tor

que

[Nm

]

Wind Speed [m s]

MeasuredComputed

1000

1500

2000

2500

3000

3500

4000

4500

5000

6 8 10 12 14 16 18 20 22 24 26

Roo

t Fla

p M

omen

t [N

m]

Wind Speed [ms]

MeasuredComputed

ARTICLE IN PRESS

0

05

1

15

2

25

3

02 03 04 05 06 07 08 09 1

CN

rR

MeasuredComputed

-008

-006

-004

-002

0

002

004

02 03 04 05 06 07 08 09 1

CT

rR

MeasuredComputed

-15

-1

-05

0

05

1

15

2

25

3

0 02 04 06 08 1

-Cp

X Chord

rR=063

M x$ C _x$ Kx Fg (311)

ARTICLE IN PRESS

-2

0

2

4

6

8

10

12

0 50 100 150 200 250 300 350

CN

Azimuth Angle [deg]

S1500600 r R=030

-04

-02

0

02

04

06

08

1

0 50 100 150 200 250 300 350

CT

Azimuth Angle [deg]

S1500600 rR=030

Fgidxi Z

S

p uidS (312)

and

r n0 $ S0 u$ S1 yu

S0 1 0 0 z

0 1 0 0

0 0 1 x

2

64

3

75

S1 0 0 0 0

x 0 z 0

0 0 0 0

2

64

3

75

(321)

My GItampqyy

322amp

ARTICLE IN PRESS

y

x

z

A

u

w

u+du

M + dMz

z

M + dMyy

M + dMxx

zF + dF

yyF + dF

xxF + dF

Mx

Fx

Mz

FzMy

Fy dr

ra

)P

x

z

y

A(a) (b)

a

b

Z

A

rdm

$ dy dF$

Z

A

g dm

$ dy$ dpdy

(323)

Z

A

r0 rdm

$Z

A

r0 gdm

fx$ zy dy z xygT

A

S0ampT r dm qyZ

A

$ q2yy-K21 qyu( $Z

A

S0ampT g dm$ Sa dP

325amp

K11

Fy 0 0 0

0 EA 0 0

0 0 Fy 0

0 0 0 GIt

2

666664

3

777775

K22

EIzz 0 EIxz 0

0 0 0 0

EIxz 0 EIxx 0

0 0 0 0

2

666664

3

777775

Sa

1 0 0

0 1 0

0 0 1

za 0 xa

2

666664

3

777775

K12

0 0 0 0

EAx+ 0 EAz+ 0

0 0 0 0

2

666664

3

777775

K21

0 EAx+ 0 0

0 0 0 0

0 EAz+ 0 0

0 0 0 0

2

666664

3

777775

Then (321) becomes

r u

y$ v

w

8gtlt

gt

9gt=

gt$ E

xlyy

z

8gtlt

gt

9gt=

gt (326)

ARTICLE IN PRESS

x

z

yu

w

vO

Orsquoz

xz

xu

w amptamp

ampamp

amp +ˆ

Z L

0duT Zu _u uampdy

)X

e

Z Le

X

e

Z Le

0NeT ZNeueN

Muamp u$ Cu

amp _u$ Ku

amp u Qu

amp (3210)

ARTICLE IN PRESS

0k

rGk

OG O

xG x

zG z

yG

y y

z

x

O

x

y

z

O

ARTICLE IN PRESS

y

O

q8 q9 q10 q11q12 q13

yG y

z

x

O

x

y

z

OG O

xG x

zG z

q1 q2 q3 q4q5 q6

q7

q14

py L sin fD cos f

(413)

m1

amp 2=NampPN

y2tamp $ ~y2tamp

p[183184]

ARTICLE IN PRESS

3540

4550

tim

e (s

ec)

-35

E+0

6

-3E

+06

-25

E+0

6

-2E

+06

-15

E+0

6

-1E

+06

12

34

freq

uenc

y (H

z)

0

5E+

08

1E+

08

15E

+08

2E+

08

25E

+08

3E+

08

-t

data

19

190

5

191

5

192

5

193

5

194

00

51

15

22

53

t (se

c)

- 1

n

Fig21Schem

atic

dam

pingan

alysis

ARTICLE IN PRESS

-15e+06

-1e+06

-500000

0

500000

1e+06

15e+06

2 4 6 8 10 12 14

tow

er to

p til

t ben

ding

mom

ent (

KN

m)

time (s)

1000

800

600

400

200

0

-2000 50 100 150 200 250 300

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

1000900800700600500400300200100

0-100

200 210 220 230 240 250 260

ARTICLE IN PRESS

wake operation

complex inflow

ARTICLE IN PRESS

2

18

16

14

12C1

Cd

1

08

06

040 5 10 15 20 25 30

r=14r=28r=42r=56r=69r=89r=94r=97

35 40$ $

05

045

04

035

03

025

02

015

01

005

00

5 10 15 20 25 30 35 40

ARTICLE IN PRESS

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

Page 6: State of the art in wind turbine aerodynamics and ...phyexp/uploads/EolienneFlexible/Eoli… · ARTICLE IN PRESS Progress in Aerospace Sciences 42 (2006) 285–330 State of the art

df sdt

f sts f s

t (2113)

(2115)

DCl Clinv Cl2D

DCd Cd2D Cd2Dmin

DCm Cm2D Cminv

be derived

(221)

V rC$ rF (222)

r2C o (223)

Cxamp 1

4p

Zo0

x x0j jdVol (224)

wxamp 1

4p

Zx x0amp o0

x x0j j3dVol (225)

wxamp 1

4p

Z

S

Gx x0ampx x0j j3

qx0

qS0 qS0 (227)

fx tamp Z

S

s0tamp4p x x0j j

m0tamp n0r

4p x x0j j3

$ dS0

228amp

rZ

Sm

dS0

Z

Sm

dS0

$I

Sm

dS0 229amp

1pU(Wr

qmWqt

$ VWm 1VU(W

qmWqt

$ VWm ramp

mW 0 2210amp

where VWm V$W $ V

W

amp =2 Since G mW

ARTICLE IN PRESS

nr

BS micro

W WS micro

( )Pmicro

F

extUr

C

C = W

ARTICLE IN PRESS

= (y )y

(y)

w

y) )

emission line

Se

W B

B

= t

(

(t=

w

+

d

D

DtdO )

qqt

ARTICLE IN PRESS

L

micro = L

TE tmicro TE t t

micro -2TE t t

-

wake surface

streak line

( )=B TEy t micro

emiss

ion

line

s

T

ARTICLE IN PRESS

700800900

10001100120013001400150016001700

6 8 10 12 14 16 18 20 22 24 26

Low

Spe

ed S

haft

Tor

que

[Nm

]

Wind Speed [m s]

MeasuredComputed

1000

1500

2000

2500

3000

3500

4000

4500

5000

6 8 10 12 14 16 18 20 22 24 26

Roo

t Fla

p M

omen

t [N

m]

Wind Speed [ms]

MeasuredComputed

ARTICLE IN PRESS

0

05

1

15

2

25

3

02 03 04 05 06 07 08 09 1

CN

rR

MeasuredComputed

-008

-006

-004

-002

0

002

004

02 03 04 05 06 07 08 09 1

CT

rR

MeasuredComputed

-15

-1

-05

0

05

1

15

2

25

3

0 02 04 06 08 1

-Cp

X Chord

rR=063

M x$ C _x$ Kx Fg (311)

ARTICLE IN PRESS

-2

0

2

4

6

8

10

12

0 50 100 150 200 250 300 350

CN

Azimuth Angle [deg]

S1500600 r R=030

-04

-02

0

02

04

06

08

1

0 50 100 150 200 250 300 350

CT

Azimuth Angle [deg]

S1500600 rR=030

Fgidxi Z

S

p uidS (312)

and

r n0 $ S0 u$ S1 yu

S0 1 0 0 z

0 1 0 0

0 0 1 x

2

64

3

75

S1 0 0 0 0

x 0 z 0

0 0 0 0

2

64

3

75

(321)

My GItampqyy

322amp

ARTICLE IN PRESS

y

x

z

A

u

w

u+du

M + dMz

z

M + dMyy

M + dMxx

zF + dF

yyF + dF

xxF + dF

Mx

Fx

Mz

FzMy

Fy dr

ra

)P

x

z

y

A(a) (b)

a

b

Z

A

rdm

$ dy dF$

Z

A

g dm

$ dy$ dpdy

(323)

Z

A

r0 rdm

$Z

A

r0 gdm

fx$ zy dy z xygT

A

S0ampT r dm qyZ

A

$ q2yy-K21 qyu( $Z

A

S0ampT g dm$ Sa dP

325amp

K11

Fy 0 0 0

0 EA 0 0

0 0 Fy 0

0 0 0 GIt

2

666664

3

777775

K22

EIzz 0 EIxz 0

0 0 0 0

EIxz 0 EIxx 0

0 0 0 0

2

666664

3

777775

Sa

1 0 0

0 1 0

0 0 1

za 0 xa

2

666664

3

777775

K12

0 0 0 0

EAx+ 0 EAz+ 0

0 0 0 0

2

666664

3

777775

K21

0 EAx+ 0 0

0 0 0 0

0 EAz+ 0 0

0 0 0 0

2

666664

3

777775

Then (321) becomes

r u

y$ v

w

8gtlt

gt

9gt=

gt$ E

xlyy

z

8gtlt

gt

9gt=

gt (326)

ARTICLE IN PRESS

x

z

yu

w

vO

Orsquoz

xz

xu

w amptamp

ampamp

amp +ˆ

Z L

0duT Zu _u uampdy

)X

e

Z Le

X

e

Z Le

0NeT ZNeueN

Muamp u$ Cu

amp _u$ Ku

amp u Qu

amp (3210)

ARTICLE IN PRESS

0k

rGk

OG O

xG x

zG z

yG

y y

z

x

O

x

y

z

O

ARTICLE IN PRESS

y

O

q8 q9 q10 q11q12 q13

yG y

z

x

O

x

y

z

OG O

xG x

zG z

q1 q2 q3 q4q5 q6

q7

q14

py L sin fD cos f

(413)

m1

amp 2=NampPN

y2tamp $ ~y2tamp

p[183184]

ARTICLE IN PRESS

3540

4550

tim

e (s

ec)

-35

E+0

6

-3E

+06

-25

E+0

6

-2E

+06

-15

E+0

6

-1E

+06

12

34

freq

uenc

y (H

z)

0

5E+

08

1E+

08

15E

+08

2E+

08

25E

+08

3E+

08

-t

data

19

190

5

191

5

192

5

193

5

194

00

51

15

22

53

t (se

c)

- 1

n

Fig21Schem

atic

dam

pingan

alysis

ARTICLE IN PRESS

-15e+06

-1e+06

-500000

0

500000

1e+06

15e+06

2 4 6 8 10 12 14

tow

er to

p til

t ben

ding

mom

ent (

KN

m)

time (s)

1000

800

600

400

200

0

-2000 50 100 150 200 250 300

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

1000900800700600500400300200100

0-100

200 210 220 230 240 250 260

ARTICLE IN PRESS

wake operation

complex inflow

ARTICLE IN PRESS

2

18

16

14

12C1

Cd

1

08

06

040 5 10 15 20 25 30

r=14r=28r=42r=56r=69r=89r=94r=97

35 40$ $

05

045

04

035

03

025

02

015

01

005

00

5 10 15 20 25 30 35 40

ARTICLE IN PRESS

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

Page 7: State of the art in wind turbine aerodynamics and ...phyexp/uploads/EolienneFlexible/Eoli… · ARTICLE IN PRESS Progress in Aerospace Sciences 42 (2006) 285–330 State of the art

be derived

(221)

V rC$ rF (222)

r2C o (223)

Cxamp 1

4p

Zo0

x x0j jdVol (224)

wxamp 1

4p

Zx x0amp o0

x x0j j3dVol (225)

wxamp 1

4p

Z

S

Gx x0ampx x0j j3

qx0

qS0 qS0 (227)

fx tamp Z

S

s0tamp4p x x0j j

m0tamp n0r

4p x x0j j3

$ dS0

228amp

rZ

Sm

dS0

Z

Sm

dS0

$I

Sm

dS0 229amp

1pU(Wr

qmWqt

$ VWm 1VU(W

qmWqt

$ VWm ramp

mW 0 2210amp

where VWm V$W $ V

W

amp =2 Since G mW

ARTICLE IN PRESS

nr

BS micro

W WS micro

( )Pmicro

F

extUr

C

C = W

ARTICLE IN PRESS

= (y )y

(y)

w

y) )

emission line

Se

W B

B

= t

(

(t=

w

+

d

D

DtdO )

qqt

ARTICLE IN PRESS

L

micro = L

TE tmicro TE t t

micro -2TE t t

-

wake surface

streak line

( )=B TEy t micro

emiss

ion

line

s

T

ARTICLE IN PRESS

700800900

10001100120013001400150016001700

6 8 10 12 14 16 18 20 22 24 26

Low

Spe

ed S

haft

Tor

que

[Nm

]

Wind Speed [m s]

MeasuredComputed

1000

1500

2000

2500

3000

3500

4000

4500

5000

6 8 10 12 14 16 18 20 22 24 26

Roo

t Fla

p M

omen

t [N

m]

Wind Speed [ms]

MeasuredComputed

ARTICLE IN PRESS

0

05

1

15

2

25

3

02 03 04 05 06 07 08 09 1

CN

rR

MeasuredComputed

-008

-006

-004

-002

0

002

004

02 03 04 05 06 07 08 09 1

CT

rR

MeasuredComputed

-15

-1

-05

0

05

1

15

2

25

3

0 02 04 06 08 1

-Cp

X Chord

rR=063

M x$ C _x$ Kx Fg (311)

ARTICLE IN PRESS

-2

0

2

4

6

8

10

12

0 50 100 150 200 250 300 350

CN

Azimuth Angle [deg]

S1500600 r R=030

-04

-02

0

02

04

06

08

1

0 50 100 150 200 250 300 350

CT

Azimuth Angle [deg]

S1500600 rR=030

Fgidxi Z

S

p uidS (312)

and

r n0 $ S0 u$ S1 yu

S0 1 0 0 z

0 1 0 0

0 0 1 x

2

64

3

75

S1 0 0 0 0

x 0 z 0

0 0 0 0

2

64

3

75

(321)

My GItampqyy

322amp

ARTICLE IN PRESS

y

x

z

A

u

w

u+du

M + dMz

z

M + dMyy

M + dMxx

zF + dF

yyF + dF

xxF + dF

Mx

Fx

Mz

FzMy

Fy dr

ra

)P

x

z

y

A(a) (b)

a

b

Z

A

rdm

$ dy dF$

Z

A

g dm

$ dy$ dpdy

(323)

Z

A

r0 rdm

$Z

A

r0 gdm

fx$ zy dy z xygT

A

S0ampT r dm qyZ

A

$ q2yy-K21 qyu( $Z

A

S0ampT g dm$ Sa dP

325amp

K11

Fy 0 0 0

0 EA 0 0

0 0 Fy 0

0 0 0 GIt

2

666664

3

777775

K22

EIzz 0 EIxz 0

0 0 0 0

EIxz 0 EIxx 0

0 0 0 0

2

666664

3

777775

Sa

1 0 0

0 1 0

0 0 1

za 0 xa

2

666664

3

777775

K12

0 0 0 0

EAx+ 0 EAz+ 0

0 0 0 0

2

666664

3

777775

K21

0 EAx+ 0 0

0 0 0 0

0 EAz+ 0 0

0 0 0 0

2

666664

3

777775

Then (321) becomes

r u

y$ v

w

8gtlt

gt

9gt=

gt$ E

xlyy

z

8gtlt

gt

9gt=

gt (326)

ARTICLE IN PRESS

x

z

yu

w

vO

Orsquoz

xz

xu

w amptamp

ampamp

amp +ˆ

Z L

0duT Zu _u uampdy

)X

e

Z Le

X

e

Z Le

0NeT ZNeueN

Muamp u$ Cu

amp _u$ Ku

amp u Qu

amp (3210)

ARTICLE IN PRESS

0k

rGk

OG O

xG x

zG z

yG

y y

z

x

O

x

y

z

O

ARTICLE IN PRESS

y

O

q8 q9 q10 q11q12 q13

yG y

z

x

O

x

y

z

OG O

xG x

zG z

q1 q2 q3 q4q5 q6

q7

q14

py L sin fD cos f

(413)

m1

amp 2=NampPN

y2tamp $ ~y2tamp

p[183184]

ARTICLE IN PRESS

3540

4550

tim

e (s

ec)

-35

E+0

6

-3E

+06

-25

E+0

6

-2E

+06

-15

E+0

6

-1E

+06

12

34

freq

uenc

y (H

z)

0

5E+

08

1E+

08

15E

+08

2E+

08

25E

+08

3E+

08

-t

data

19

190

5

191

5

192

5

193

5

194

00

51

15

22

53

t (se

c)

- 1

n

Fig21Schem

atic

dam

pingan

alysis

ARTICLE IN PRESS

-15e+06

-1e+06

-500000

0

500000

1e+06

15e+06

2 4 6 8 10 12 14

tow

er to

p til

t ben

ding

mom

ent (

KN

m)

time (s)

1000

800

600

400

200

0

-2000 50 100 150 200 250 300

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

1000900800700600500400300200100

0-100

200 210 220 230 240 250 260

ARTICLE IN PRESS

wake operation

complex inflow

ARTICLE IN PRESS

2

18

16

14

12C1

Cd

1

08

06

040 5 10 15 20 25 30

r=14r=28r=42r=56r=69r=89r=94r=97

35 40$ $

05

045

04

035

03

025

02

015

01

005

00

5 10 15 20 25 30 35 40

ARTICLE IN PRESS

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

Page 8: State of the art in wind turbine aerodynamics and ...phyexp/uploads/EolienneFlexible/Eoli… · ARTICLE IN PRESS Progress in Aerospace Sciences 42 (2006) 285–330 State of the art

fx tamp Z

S

s0tamp4p x x0j j

m0tamp n0r

4p x x0j j3

$ dS0

228amp

rZ

Sm

dS0

Z

Sm

dS0

$I

Sm

dS0 229amp

1pU(Wr

qmWqt

$ VWm 1VU(W

qmWqt

$ VWm ramp

mW 0 2210amp

where VWm V$W $ V

W

amp =2 Since G mW

ARTICLE IN PRESS

nr

BS micro

W WS micro

( )Pmicro

F

extUr

C

C = W

ARTICLE IN PRESS

= (y )y

(y)

w

y) )

emission line

Se

W B

B

= t

(

(t=

w

+

d

D

DtdO )

qqt

ARTICLE IN PRESS

L

micro = L

TE tmicro TE t t

micro -2TE t t

-

wake surface

streak line

( )=B TEy t micro

emiss

ion

line

s

T

ARTICLE IN PRESS

700800900

10001100120013001400150016001700

6 8 10 12 14 16 18 20 22 24 26

Low

Spe

ed S

haft

Tor

que

[Nm

]

Wind Speed [m s]

MeasuredComputed

1000

1500

2000

2500

3000

3500

4000

4500

5000

6 8 10 12 14 16 18 20 22 24 26

Roo

t Fla

p M

omen

t [N

m]

Wind Speed [ms]

MeasuredComputed

ARTICLE IN PRESS

0

05

1

15

2

25

3

02 03 04 05 06 07 08 09 1

CN

rR

MeasuredComputed

-008

-006

-004

-002

0

002

004

02 03 04 05 06 07 08 09 1

CT

rR

MeasuredComputed

-15

-1

-05

0

05

1

15

2

25

3

0 02 04 06 08 1

-Cp

X Chord

rR=063

M x$ C _x$ Kx Fg (311)

ARTICLE IN PRESS

-2

0

2

4

6

8

10

12

0 50 100 150 200 250 300 350

CN

Azimuth Angle [deg]

S1500600 r R=030

-04

-02

0

02

04

06

08

1

0 50 100 150 200 250 300 350

CT

Azimuth Angle [deg]

S1500600 rR=030

Fgidxi Z

S

p uidS (312)

and

r n0 $ S0 u$ S1 yu

S0 1 0 0 z

0 1 0 0

0 0 1 x

2

64

3

75

S1 0 0 0 0

x 0 z 0

0 0 0 0

2

64

3

75

(321)

My GItampqyy

322amp

ARTICLE IN PRESS

y

x

z

A

u

w

u+du

M + dMz

z

M + dMyy

M + dMxx

zF + dF

yyF + dF

xxF + dF

Mx

Fx

Mz

FzMy

Fy dr

ra

)P

x

z

y

A(a) (b)

a

b

Z

A

rdm

$ dy dF$

Z

A

g dm

$ dy$ dpdy

(323)

Z

A

r0 rdm

$Z

A

r0 gdm

fx$ zy dy z xygT

A

S0ampT r dm qyZ

A

$ q2yy-K21 qyu( $Z

A

S0ampT g dm$ Sa dP

325amp

K11

Fy 0 0 0

0 EA 0 0

0 0 Fy 0

0 0 0 GIt

2

666664

3

777775

K22

EIzz 0 EIxz 0

0 0 0 0

EIxz 0 EIxx 0

0 0 0 0

2

666664

3

777775

Sa

1 0 0

0 1 0

0 0 1

za 0 xa

2

666664

3

777775

K12

0 0 0 0

EAx+ 0 EAz+ 0

0 0 0 0

2

666664

3

777775

K21

0 EAx+ 0 0

0 0 0 0

0 EAz+ 0 0

0 0 0 0

2

666664

3

777775

Then (321) becomes

r u

y$ v

w

8gtlt

gt

9gt=

gt$ E

xlyy

z

8gtlt

gt

9gt=

gt (326)

ARTICLE IN PRESS

x

z

yu

w

vO

Orsquoz

xz

xu

w amptamp

ampamp

amp +ˆ

Z L

0duT Zu _u uampdy

)X

e

Z Le

X

e

Z Le

0NeT ZNeueN

Muamp u$ Cu

amp _u$ Ku

amp u Qu

amp (3210)

ARTICLE IN PRESS

0k

rGk

OG O

xG x

zG z

yG

y y

z

x

O

x

y

z

O

ARTICLE IN PRESS

y

O

q8 q9 q10 q11q12 q13

yG y

z

x

O

x

y

z

OG O

xG x

zG z

q1 q2 q3 q4q5 q6

q7

q14

py L sin fD cos f

(413)

m1

amp 2=NampPN

y2tamp $ ~y2tamp

p[183184]

ARTICLE IN PRESS

3540

4550

tim

e (s

ec)

-35

E+0

6

-3E

+06

-25

E+0

6

-2E

+06

-15

E+0

6

-1E

+06

12

34

freq

uenc

y (H

z)

0

5E+

08

1E+

08

15E

+08

2E+

08

25E

+08

3E+

08

-t

data

19

190

5

191

5

192

5

193

5

194

00

51

15

22

53

t (se

c)

- 1

n

Fig21Schem

atic

dam

pingan

alysis

ARTICLE IN PRESS

-15e+06

-1e+06

-500000

0

500000

1e+06

15e+06

2 4 6 8 10 12 14

tow

er to

p til

t ben

ding

mom

ent (

KN

m)

time (s)

1000

800

600

400

200

0

-2000 50 100 150 200 250 300

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

1000900800700600500400300200100

0-100

200 210 220 230 240 250 260

ARTICLE IN PRESS

wake operation

complex inflow

ARTICLE IN PRESS

2

18

16

14

12C1

Cd

1

08

06

040 5 10 15 20 25 30

r=14r=28r=42r=56r=69r=89r=94r=97

35 40$ $

05

045

04

035

03

025

02

015

01

005

00

5 10 15 20 25 30 35 40

ARTICLE IN PRESS

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

ARTICLE IN PRESS

= (y )y

(y)

w

y) )

emission line

Se

W B

B

= t

(

(t=

w

+

d

D

DtdO )

qqt

ARTICLE IN PRESS

L

micro = L

TE tmicro TE t t

micro -2TE t t

-

wake surface

streak line

( )=B TEy t micro

emiss

ion

line

s

T

ARTICLE IN PRESS

700800900

10001100120013001400150016001700

6 8 10 12 14 16 18 20 22 24 26

Low

Spe

ed S

haft

Tor

que

[Nm

]

Wind Speed [m s]

MeasuredComputed

1000

1500

2000

2500

3000

3500

4000

4500

5000

6 8 10 12 14 16 18 20 22 24 26

Roo

t Fla

p M

omen

t [N

m]

Wind Speed [ms]

MeasuredComputed

ARTICLE IN PRESS

0

05

1

15

2

25

3

02 03 04 05 06 07 08 09 1

CN

rR

MeasuredComputed

-008

-006

-004

-002

0

002

004

02 03 04 05 06 07 08 09 1

CT

rR

MeasuredComputed

-15

-1

-05

0

05

1

15

2

25

3

0 02 04 06 08 1

-Cp

X Chord

rR=063

M x$ C _x$ Kx Fg (311)

ARTICLE IN PRESS

-2

0

2

4

6

8

10

12

0 50 100 150 200 250 300 350

CN

Azimuth Angle [deg]

S1500600 r R=030

-04

-02

0

02

04

06

08

1

0 50 100 150 200 250 300 350

CT

Azimuth Angle [deg]

S1500600 rR=030

Fgidxi Z

S

p uidS (312)

and

r n0 $ S0 u$ S1 yu

S0 1 0 0 z

0 1 0 0

0 0 1 x

2

64

3

75

S1 0 0 0 0

x 0 z 0

0 0 0 0

2

64

3

75

(321)

My GItampqyy

322amp

ARTICLE IN PRESS

y

x

z

A

u

w

u+du

M + dMz

z

M + dMyy

M + dMxx

zF + dF

yyF + dF

xxF + dF

Mx

Fx

Mz

FzMy

Fy dr

ra

)P

x

z

y

A(a) (b)

a

b

Z

A

rdm

$ dy dF$

Z

A

g dm

$ dy$ dpdy

(323)

Z

A

r0 rdm

$Z

A

r0 gdm

fx$ zy dy z xygT

A

S0ampT r dm qyZ

A

$ q2yy-K21 qyu( $Z

A

S0ampT g dm$ Sa dP

325amp

K11

Fy 0 0 0

0 EA 0 0

0 0 Fy 0

0 0 0 GIt

2

666664

3

777775

K22

EIzz 0 EIxz 0

0 0 0 0

EIxz 0 EIxx 0

0 0 0 0

2

666664

3

777775

Sa

1 0 0

0 1 0

0 0 1

za 0 xa

2

666664

3

777775

K12

0 0 0 0

EAx+ 0 EAz+ 0

0 0 0 0

2

666664

3

777775

K21

0 EAx+ 0 0

0 0 0 0

0 EAz+ 0 0

0 0 0 0

2

666664

3

777775

Then (321) becomes

r u

y$ v

w

8gtlt

gt

9gt=

gt$ E

xlyy

z

8gtlt

gt

9gt=

gt (326)

ARTICLE IN PRESS

x

z

yu

w

vO

Orsquoz

xz

xu

w amptamp

ampamp

amp +ˆ

Z L

0duT Zu _u uampdy

)X

e

Z Le

X

e

Z Le

0NeT ZNeueN

Muamp u$ Cu

amp _u$ Ku

amp u Qu

amp (3210)

ARTICLE IN PRESS

0k

rGk

OG O

xG x

zG z

yG

y y

z

x

O

x

y

z

O

ARTICLE IN PRESS

y

O

q8 q9 q10 q11q12 q13

yG y

z

x

O

x

y

z

OG O

xG x

zG z

q1 q2 q3 q4q5 q6

q7

q14

py L sin fD cos f

(413)

m1

amp 2=NampPN

y2tamp $ ~y2tamp

p[183184]

ARTICLE IN PRESS

3540

4550

tim

e (s

ec)

-35

E+0

6

-3E

+06

-25

E+0

6

-2E

+06

-15

E+0

6

-1E

+06

12

34

freq

uenc

y (H

z)

0

5E+

08

1E+

08

15E

+08

2E+

08

25E

+08

3E+

08

-t

data

19

190

5

191

5

192

5

193

5

194

00

51

15

22

53

t (se

c)

- 1

n

Fig21Schem

atic

dam

pingan

alysis

ARTICLE IN PRESS

-15e+06

-1e+06

-500000

0

500000

1e+06

15e+06

2 4 6 8 10 12 14

tow

er to

p til

t ben

ding

mom

ent (

KN

m)

time (s)

1000

800

600

400

200

0

-2000 50 100 150 200 250 300

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

1000900800700600500400300200100

0-100

200 210 220 230 240 250 260

ARTICLE IN PRESS

wake operation

complex inflow

ARTICLE IN PRESS

2

18

16

14

12C1

Cd

1

08

06

040 5 10 15 20 25 30

r=14r=28r=42r=56r=69r=89r=94r=97

35 40$ $

05

045

04

035

03

025

02

015

01

005

00

5 10 15 20 25 30 35 40

ARTICLE IN PRESS

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

Page 10: State of the art in wind turbine aerodynamics and ...phyexp/uploads/EolienneFlexible/Eoli… · ARTICLE IN PRESS Progress in Aerospace Sciences 42 (2006) 285–330 State of the art

d

D

DtdO )

qqt

ARTICLE IN PRESS

L

micro = L

TE tmicro TE t t

micro -2TE t t

-

wake surface

streak line

( )=B TEy t micro

emiss

ion

line

s

T

ARTICLE IN PRESS

700800900

10001100120013001400150016001700

6 8 10 12 14 16 18 20 22 24 26

Low

Spe

ed S

haft

Tor

que

[Nm

]

Wind Speed [m s]

MeasuredComputed

1000

1500

2000

2500

3000

3500

4000

4500

5000

6 8 10 12 14 16 18 20 22 24 26

Roo

t Fla

p M

omen

t [N

m]

Wind Speed [ms]

MeasuredComputed

ARTICLE IN PRESS

0

05

1

15

2

25

3

02 03 04 05 06 07 08 09 1

CN

rR

MeasuredComputed

-008

-006

-004

-002

0

002

004

02 03 04 05 06 07 08 09 1

CT

rR

MeasuredComputed

-15

-1

-05

0

05

1

15

2

25

3

0 02 04 06 08 1

-Cp

X Chord

rR=063

M x$ C _x$ Kx Fg (311)

ARTICLE IN PRESS

-2

0

2

4

6

8

10

12

0 50 100 150 200 250 300 350

CN

Azimuth Angle [deg]

S1500600 r R=030

-04

-02

0

02

04

06

08

1

0 50 100 150 200 250 300 350

CT

Azimuth Angle [deg]

S1500600 rR=030

Fgidxi Z

S

p uidS (312)

and

r n0 $ S0 u$ S1 yu

S0 1 0 0 z

0 1 0 0

0 0 1 x

2

64

3

75

S1 0 0 0 0

x 0 z 0

0 0 0 0

2

64

3

75

(321)

My GItampqyy

322amp

ARTICLE IN PRESS

y

x

z

A

u

w

u+du

M + dMz

z

M + dMyy

M + dMxx

zF + dF

yyF + dF

xxF + dF

Mx

Fx

Mz

FzMy

Fy dr

ra

)P

x

z

y

A(a) (b)

a

b

Z

A

rdm

$ dy dF$

Z

A

g dm

$ dy$ dpdy

(323)

Z

A

r0 rdm

$Z

A

r0 gdm

fx$ zy dy z xygT

A

S0ampT r dm qyZ

A

$ q2yy-K21 qyu( $Z

A

S0ampT g dm$ Sa dP

325amp

K11

Fy 0 0 0

0 EA 0 0

0 0 Fy 0

0 0 0 GIt

2

666664

3

777775

K22

EIzz 0 EIxz 0

0 0 0 0

EIxz 0 EIxx 0

0 0 0 0

2

666664

3

777775

Sa

1 0 0

0 1 0

0 0 1

za 0 xa

2

666664

3

777775

K12

0 0 0 0

EAx+ 0 EAz+ 0

0 0 0 0

2

666664

3

777775

K21

0 EAx+ 0 0

0 0 0 0

0 EAz+ 0 0

0 0 0 0

2

666664

3

777775

Then (321) becomes

r u

y$ v

w

8gtlt

gt

9gt=

gt$ E

xlyy

z

8gtlt

gt

9gt=

gt (326)

ARTICLE IN PRESS

x

z

yu

w

vO

Orsquoz

xz

xu

w amptamp

ampamp

amp +ˆ

Z L

0duT Zu _u uampdy

)X

e

Z Le

X

e

Z Le

0NeT ZNeueN

Muamp u$ Cu

amp _u$ Ku

amp u Qu

amp (3210)

ARTICLE IN PRESS

0k

rGk

OG O

xG x

zG z

yG

y y

z

x

O

x

y

z

O

ARTICLE IN PRESS

y

O

q8 q9 q10 q11q12 q13

yG y

z

x

O

x

y

z

OG O

xG x

zG z

q1 q2 q3 q4q5 q6

q7

q14

py L sin fD cos f

(413)

m1

amp 2=NampPN

y2tamp $ ~y2tamp

p[183184]

ARTICLE IN PRESS

3540

4550

tim

e (s

ec)

-35

E+0

6

-3E

+06

-25

E+0

6

-2E

+06

-15

E+0

6

-1E

+06

12

34

freq

uenc

y (H

z)

0

5E+

08

1E+

08

15E

+08

2E+

08

25E

+08

3E+

08

-t

data

19

190

5

191

5

192

5

193

5

194

00

51

15

22

53

t (se

c)

- 1

n

Fig21Schem

atic

dam

pingan

alysis

ARTICLE IN PRESS

-15e+06

-1e+06

-500000

0

500000

1e+06

15e+06

2 4 6 8 10 12 14

tow

er to

p til

t ben

ding

mom

ent (

KN

m)

time (s)

1000

800

600

400

200

0

-2000 50 100 150 200 250 300

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

1000900800700600500400300200100

0-100

200 210 220 230 240 250 260

ARTICLE IN PRESS

wake operation

complex inflow

ARTICLE IN PRESS

2

18

16

14

12C1

Cd

1

08

06

040 5 10 15 20 25 30

r=14r=28r=42r=56r=69r=89r=94r=97

35 40$ $

05

045

04

035

03

025

02

015

01

005

00

5 10 15 20 25 30 35 40

ARTICLE IN PRESS

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

ARTICLE IN PRESS

700800900

10001100120013001400150016001700

6 8 10 12 14 16 18 20 22 24 26

Low

Spe

ed S

haft

Tor

que

[Nm

]

Wind Speed [m s]

MeasuredComputed

1000

1500

2000

2500

3000

3500

4000

4500

5000

6 8 10 12 14 16 18 20 22 24 26

Roo

t Fla

p M

omen

t [N

m]

Wind Speed [ms]

MeasuredComputed

ARTICLE IN PRESS

0

05

1

15

2

25

3

02 03 04 05 06 07 08 09 1

CN

rR

MeasuredComputed

-008

-006

-004

-002

0

002

004

02 03 04 05 06 07 08 09 1

CT

rR

MeasuredComputed

-15

-1

-05

0

05

1

15

2

25

3

0 02 04 06 08 1

-Cp

X Chord

rR=063

M x$ C _x$ Kx Fg (311)

ARTICLE IN PRESS

-2

0

2

4

6

8

10

12

0 50 100 150 200 250 300 350

CN

Azimuth Angle [deg]

S1500600 r R=030

-04

-02

0

02

04

06

08

1

0 50 100 150 200 250 300 350

CT

Azimuth Angle [deg]

S1500600 rR=030

Fgidxi Z

S

p uidS (312)

and

r n0 $ S0 u$ S1 yu

S0 1 0 0 z

0 1 0 0

0 0 1 x

2

64

3

75

S1 0 0 0 0

x 0 z 0

0 0 0 0

2

64

3

75

(321)

My GItampqyy

322amp

ARTICLE IN PRESS

y

x

z

A

u

w

u+du

M + dMz

z

M + dMyy

M + dMxx

zF + dF

yyF + dF

xxF + dF

Mx

Fx

Mz

FzMy

Fy dr

ra

)P

x

z

y

A(a) (b)

a

b

Z

A

rdm

$ dy dF$

Z

A

g dm

$ dy$ dpdy

(323)

Z

A

r0 rdm

$Z

A

r0 gdm

fx$ zy dy z xygT

A

S0ampT r dm qyZ

A

$ q2yy-K21 qyu( $Z

A

S0ampT g dm$ Sa dP

325amp

K11

Fy 0 0 0

0 EA 0 0

0 0 Fy 0

0 0 0 GIt

2

666664

3

777775

K22

EIzz 0 EIxz 0

0 0 0 0

EIxz 0 EIxx 0

0 0 0 0

2

666664

3

777775

Sa

1 0 0

0 1 0

0 0 1

za 0 xa

2

666664

3

777775

K12

0 0 0 0

EAx+ 0 EAz+ 0

0 0 0 0

2

666664

3

777775

K21

0 EAx+ 0 0

0 0 0 0

0 EAz+ 0 0

0 0 0 0

2

666664

3

777775

Then (321) becomes

r u

y$ v

w

8gtlt

gt

9gt=

gt$ E

xlyy

z

8gtlt

gt

9gt=

gt (326)

ARTICLE IN PRESS

x

z

yu

w

vO

Orsquoz

xz

xu

w amptamp

ampamp

amp +ˆ

Z L

0duT Zu _u uampdy

)X

e

Z Le

X

e

Z Le

0NeT ZNeueN

Muamp u$ Cu

amp _u$ Ku

amp u Qu

amp (3210)

ARTICLE IN PRESS

0k

rGk

OG O

xG x

zG z

yG

y y

z

x

O

x

y

z

O

ARTICLE IN PRESS

y

O

q8 q9 q10 q11q12 q13

yG y

z

x

O

x

y

z

OG O

xG x

zG z

q1 q2 q3 q4q5 q6

q7

q14

py L sin fD cos f

(413)

m1

amp 2=NampPN

y2tamp $ ~y2tamp

p[183184]

ARTICLE IN PRESS

3540

4550

tim

e (s

ec)

-35

E+0

6

-3E

+06

-25

E+0

6

-2E

+06

-15

E+0

6

-1E

+06

12

34

freq

uenc

y (H

z)

0

5E+

08

1E+

08

15E

+08

2E+

08

25E

+08

3E+

08

-t

data

19

190

5

191

5

192

5

193

5

194

00

51

15

22

53

t (se

c)

- 1

n

Fig21Schem

atic

dam

pingan

alysis

ARTICLE IN PRESS

-15e+06

-1e+06

-500000

0

500000

1e+06

15e+06

2 4 6 8 10 12 14

tow

er to

p til

t ben

ding

mom

ent (

KN

m)

time (s)

1000

800

600

400

200

0

-2000 50 100 150 200 250 300

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

1000900800700600500400300200100

0-100

200 210 220 230 240 250 260

ARTICLE IN PRESS

wake operation

complex inflow

ARTICLE IN PRESS

2

18

16

14

12C1

Cd

1

08

06

040 5 10 15 20 25 30

r=14r=28r=42r=56r=69r=89r=94r=97

35 40$ $

05

045

04

035

03

025

02

015

01

005

00

5 10 15 20 25 30 35 40

ARTICLE IN PRESS

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

ARTICLE IN PRESS

700800900

10001100120013001400150016001700

6 8 10 12 14 16 18 20 22 24 26

Low

Spe

ed S

haft

Tor

que

[Nm

]

Wind Speed [m s]

MeasuredComputed

1000

1500

2000

2500

3000

3500

4000

4500

5000

6 8 10 12 14 16 18 20 22 24 26

Roo

t Fla

p M

omen

t [N

m]

Wind Speed [ms]

MeasuredComputed

ARTICLE IN PRESS

0

05

1

15

2

25

3

02 03 04 05 06 07 08 09 1

CN

rR

MeasuredComputed

-008

-006

-004

-002

0

002

004

02 03 04 05 06 07 08 09 1

CT

rR

MeasuredComputed

-15

-1

-05

0

05

1

15

2

25

3

0 02 04 06 08 1

-Cp

X Chord

rR=063

M x$ C _x$ Kx Fg (311)

ARTICLE IN PRESS

-2

0

2

4

6

8

10

12

0 50 100 150 200 250 300 350

CN

Azimuth Angle [deg]

S1500600 r R=030

-04

-02

0

02

04

06

08

1

0 50 100 150 200 250 300 350

CT

Azimuth Angle [deg]

S1500600 rR=030

Fgidxi Z

S

p uidS (312)

and

r n0 $ S0 u$ S1 yu

S0 1 0 0 z

0 1 0 0

0 0 1 x

2

64

3

75

S1 0 0 0 0

x 0 z 0

0 0 0 0

2

64

3

75

(321)

My GItampqyy

322amp

ARTICLE IN PRESS

y

x

z

A

u

w

u+du

M + dMz

z

M + dMyy

M + dMxx

zF + dF

yyF + dF

xxF + dF

Mx

Fx

Mz

FzMy

Fy dr

ra

)P

x

z

y

A(a) (b)

a

b

Z

A

rdm

$ dy dF$

Z

A

g dm

$ dy$ dpdy

(323)

Z

A

r0 rdm

$Z

A

r0 gdm

fx$ zy dy z xygT

A

S0ampT r dm qyZ

A

$ q2yy-K21 qyu( $Z

A

S0ampT g dm$ Sa dP

325amp

K11

Fy 0 0 0

0 EA 0 0

0 0 Fy 0

0 0 0 GIt

2

666664

3

777775

K22

EIzz 0 EIxz 0

0 0 0 0

EIxz 0 EIxx 0

0 0 0 0

2

666664

3

777775

Sa

1 0 0

0 1 0

0 0 1

za 0 xa

2

666664

3

777775

K12

0 0 0 0

EAx+ 0 EAz+ 0

0 0 0 0

2

666664

3

777775

K21

0 EAx+ 0 0

0 0 0 0

0 EAz+ 0 0

0 0 0 0

2

666664

3

777775

Then (321) becomes

r u

y$ v

w

8gtlt

gt

9gt=

gt$ E

xlyy

z

8gtlt

gt

9gt=

gt (326)

ARTICLE IN PRESS

x

z

yu

w

vO

Orsquoz

xz

xu

w amptamp

ampamp

amp +ˆ

Z L

0duT Zu _u uampdy

)X

e

Z Le

X

e

Z Le

0NeT ZNeueN

Muamp u$ Cu

amp _u$ Ku

amp u Qu

amp (3210)

ARTICLE IN PRESS

0k

rGk

OG O

xG x

zG z

yG

y y

z

x

O

x

y

z

O

ARTICLE IN PRESS

y

O

q8 q9 q10 q11q12 q13

yG y

z

x

O

x

y

z

OG O

xG x

zG z

q1 q2 q3 q4q5 q6

q7

q14

py L sin fD cos f

(413)

m1

amp 2=NampPN

y2tamp $ ~y2tamp

p[183184]

ARTICLE IN PRESS

3540

4550

tim

e (s

ec)

-35

E+0

6

-3E

+06

-25

E+0

6

-2E

+06

-15

E+0

6

-1E

+06

12

34

freq

uenc

y (H

z)

0

5E+

08

1E+

08

15E

+08

2E+

08

25E

+08

3E+

08

-t

data

19

190

5

191

5

192

5

193

5

194

00

51

15

22

53

t (se

c)

- 1

n

Fig21Schem

atic

dam

pingan

alysis

ARTICLE IN PRESS

-15e+06

-1e+06

-500000

0

500000

1e+06

15e+06

2 4 6 8 10 12 14

tow

er to

p til

t ben

ding

mom

ent (

KN

m)

time (s)

1000

800

600

400

200

0

-2000 50 100 150 200 250 300

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

1000900800700600500400300200100

0-100

200 210 220 230 240 250 260

ARTICLE IN PRESS

wake operation

complex inflow

ARTICLE IN PRESS

2

18

16

14

12C1

Cd

1

08

06

040 5 10 15 20 25 30

r=14r=28r=42r=56r=69r=89r=94r=97

35 40$ $

05

045

04

035

03

025

02

015

01

005

00

5 10 15 20 25 30 35 40

ARTICLE IN PRESS

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

ARTICLE IN PRESS

700800900

10001100120013001400150016001700

6 8 10 12 14 16 18 20 22 24 26

Low

Spe

ed S

haft

Tor

que

[Nm

]

Wind Speed [m s]

MeasuredComputed

1000

1500

2000

2500

3000

3500

4000

4500

5000

6 8 10 12 14 16 18 20 22 24 26

Roo

t Fla

p M

omen

t [N

m]

Wind Speed [ms]

MeasuredComputed

ARTICLE IN PRESS

0

05

1

15

2

25

3

02 03 04 05 06 07 08 09 1

CN

rR

MeasuredComputed

-008

-006

-004

-002

0

002

004

02 03 04 05 06 07 08 09 1

CT

rR

MeasuredComputed

-15

-1

-05

0

05

1

15

2

25

3

0 02 04 06 08 1

-Cp

X Chord

rR=063

M x$ C _x$ Kx Fg (311)

ARTICLE IN PRESS

-2

0

2

4

6

8

10

12

0 50 100 150 200 250 300 350

CN

Azimuth Angle [deg]

S1500600 r R=030

-04

-02

0

02

04

06

08

1

0 50 100 150 200 250 300 350

CT

Azimuth Angle [deg]

S1500600 rR=030

Fgidxi Z

S

p uidS (312)

and

r n0 $ S0 u$ S1 yu

S0 1 0 0 z

0 1 0 0

0 0 1 x

2

64

3

75

S1 0 0 0 0

x 0 z 0

0 0 0 0

2

64

3

75

(321)

My GItampqyy

322amp

ARTICLE IN PRESS

y

x

z

A

u

w

u+du

M + dMz

z

M + dMyy

M + dMxx

zF + dF

yyF + dF

xxF + dF

Mx

Fx

Mz

FzMy

Fy dr

ra

)P

x

z

y

A(a) (b)

a

b

Z

A

rdm

$ dy dF$

Z

A

g dm

$ dy$ dpdy

(323)

Z

A

r0 rdm

$Z

A

r0 gdm

fx$ zy dy z xygT

A

S0ampT r dm qyZ

A

$ q2yy-K21 qyu( $Z

A

S0ampT g dm$ Sa dP

325amp

K11

Fy 0 0 0

0 EA 0 0

0 0 Fy 0

0 0 0 GIt

2

666664

3

777775

K22

EIzz 0 EIxz 0

0 0 0 0

EIxz 0 EIxx 0

0 0 0 0

2

666664

3

777775

Sa

1 0 0

0 1 0

0 0 1

za 0 xa

2

666664

3

777775

K12

0 0 0 0

EAx+ 0 EAz+ 0

0 0 0 0

2

666664

3

777775

K21

0 EAx+ 0 0

0 0 0 0

0 EAz+ 0 0

0 0 0 0

2

666664

3

777775

Then (321) becomes

r u

y$ v

w

8gtlt

gt

9gt=

gt$ E

xlyy

z

8gtlt

gt

9gt=

gt (326)

ARTICLE IN PRESS

x

z

yu

w

vO

Orsquoz

xz

xu

w amptamp

ampamp

amp +ˆ

Z L

0duT Zu _u uampdy

)X

e

Z Le

X

e

Z Le

0NeT ZNeueN

Muamp u$ Cu

amp _u$ Ku

amp u Qu

amp (3210)

ARTICLE IN PRESS

0k

rGk

OG O

xG x

zG z

yG

y y

z

x

O

x

y

z

O

ARTICLE IN PRESS

y

O

q8 q9 q10 q11q12 q13

yG y

z

x

O

x

y

z

OG O

xG x

zG z

q1 q2 q3 q4q5 q6

q7

q14

py L sin fD cos f

(413)

m1

amp 2=NampPN

y2tamp $ ~y2tamp

p[183184]

ARTICLE IN PRESS

3540

4550

tim

e (s

ec)

-35

E+0

6

-3E

+06

-25

E+0

6

-2E

+06

-15

E+0

6

-1E

+06

12

34

freq

uenc

y (H

z)

0

5E+

08

1E+

08

15E

+08

2E+

08

25E

+08

3E+

08

-t

data

19

190

5

191

5

192

5

193

5

194

00

51

15

22

53

t (se

c)

- 1

n

Fig21Schem

atic

dam

pingan

alysis

ARTICLE IN PRESS

-15e+06

-1e+06

-500000

0

500000

1e+06

15e+06

2 4 6 8 10 12 14

tow

er to

p til

t ben

ding

mom

ent (

KN

m)

time (s)

1000

800

600

400

200

0

-2000 50 100 150 200 250 300

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

1000900800700600500400300200100

0-100

200 210 220 230 240 250 260

ARTICLE IN PRESS

wake operation

complex inflow

ARTICLE IN PRESS

2

18

16

14

12C1

Cd

1

08

06

040 5 10 15 20 25 30

r=14r=28r=42r=56r=69r=89r=94r=97

35 40$ $

05

045

04

035

03

025

02

015

01

005

00

5 10 15 20 25 30 35 40

ARTICLE IN PRESS

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

ARTICLE IN PRESS

700800900

10001100120013001400150016001700

6 8 10 12 14 16 18 20 22 24 26

Low

Spe

ed S

haft

Tor

que

[Nm

]

Wind Speed [m s]

MeasuredComputed

1000

1500

2000

2500

3000

3500

4000

4500

5000

6 8 10 12 14 16 18 20 22 24 26

Roo

t Fla

p M

omen

t [N

m]

Wind Speed [ms]

MeasuredComputed

ARTICLE IN PRESS

0

05

1

15

2

25

3

02 03 04 05 06 07 08 09 1

CN

rR

MeasuredComputed

-008

-006

-004

-002

0

002

004

02 03 04 05 06 07 08 09 1

CT

rR

MeasuredComputed

-15

-1

-05

0

05

1

15

2

25

3

0 02 04 06 08 1

-Cp

X Chord

rR=063

M x$ C _x$ Kx Fg (311)

ARTICLE IN PRESS

-2

0

2

4

6

8

10

12

0 50 100 150 200 250 300 350

CN

Azimuth Angle [deg]

S1500600 r R=030

-04

-02

0

02

04

06

08

1

0 50 100 150 200 250 300 350

CT

Azimuth Angle [deg]

S1500600 rR=030

Fgidxi Z

S

p uidS (312)

and

r n0 $ S0 u$ S1 yu

S0 1 0 0 z

0 1 0 0

0 0 1 x

2

64

3

75

S1 0 0 0 0

x 0 z 0

0 0 0 0

2

64

3

75

(321)

My GItampqyy

322amp

ARTICLE IN PRESS

y

x

z

A

u

w

u+du

M + dMz

z

M + dMyy

M + dMxx

zF + dF

yyF + dF

xxF + dF

Mx

Fx

Mz

FzMy

Fy dr

ra

)P

x

z

y

A(a) (b)

a

b

Z

A

rdm

$ dy dF$

Z

A

g dm

$ dy$ dpdy

(323)

Z

A

r0 rdm

$Z

A

r0 gdm

fx$ zy dy z xygT

A

S0ampT r dm qyZ

A

$ q2yy-K21 qyu( $Z

A

S0ampT g dm$ Sa dP

325amp

K11

Fy 0 0 0

0 EA 0 0

0 0 Fy 0

0 0 0 GIt

2

666664

3

777775

K22

EIzz 0 EIxz 0

0 0 0 0

EIxz 0 EIxx 0

0 0 0 0

2

666664

3

777775

Sa

1 0 0

0 1 0

0 0 1

za 0 xa

2

666664

3

777775

K12

0 0 0 0

EAx+ 0 EAz+ 0

0 0 0 0

2

666664

3

777775

K21

0 EAx+ 0 0

0 0 0 0

0 EAz+ 0 0

0 0 0 0

2

666664

3

777775

Then (321) becomes

r u

y$ v

w

8gtlt

gt

9gt=

gt$ E

xlyy

z

8gtlt

gt

9gt=

gt (326)

ARTICLE IN PRESS

x

z

yu

w

vO

Orsquoz

xz

xu

w amptamp

ampamp

amp +ˆ

Z L

0duT Zu _u uampdy

)X

e

Z Le

X

e

Z Le

0NeT ZNeueN

Muamp u$ Cu

amp _u$ Ku

amp u Qu

amp (3210)

ARTICLE IN PRESS

0k

rGk

OG O

xG x

zG z

yG

y y

z

x

O

x

y

z

O

ARTICLE IN PRESS

y

O

q8 q9 q10 q11q12 q13

yG y

z

x

O

x

y

z

OG O

xG x

zG z

q1 q2 q3 q4q5 q6

q7

q14

py L sin fD cos f

(413)

m1

amp 2=NampPN

y2tamp $ ~y2tamp

p[183184]

ARTICLE IN PRESS

3540

4550

tim

e (s

ec)

-35

E+0

6

-3E

+06

-25

E+0

6

-2E

+06

-15

E+0

6

-1E

+06

12

34

freq

uenc

y (H

z)

0

5E+

08

1E+

08

15E

+08

2E+

08

25E

+08

3E+

08

-t

data

19

190

5

191

5

192

5

193

5

194

00

51

15

22

53

t (se

c)

- 1

n

Fig21Schem

atic

dam

pingan

alysis

ARTICLE IN PRESS

-15e+06

-1e+06

-500000

0

500000

1e+06

15e+06

2 4 6 8 10 12 14

tow

er to

p til

t ben

ding

mom

ent (

KN

m)

time (s)

1000

800

600

400

200

0

-2000 50 100 150 200 250 300

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

1000900800700600500400300200100

0-100

200 210 220 230 240 250 260

ARTICLE IN PRESS

wake operation

complex inflow

ARTICLE IN PRESS

2

18

16

14

12C1

Cd

1

08

06

040 5 10 15 20 25 30

r=14r=28r=42r=56r=69r=89r=94r=97

35 40$ $

05

045

04

035

03

025

02

015

01

005

00

5 10 15 20 25 30 35 40

ARTICLE IN PRESS

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

ARTICLE IN PRESS

700800900

10001100120013001400150016001700

6 8 10 12 14 16 18 20 22 24 26

Low

Spe

ed S

haft

Tor

que

[Nm

]

Wind Speed [m s]

MeasuredComputed

1000

1500

2000

2500

3000

3500

4000

4500

5000

6 8 10 12 14 16 18 20 22 24 26

Roo

t Fla

p M

omen

t [N

m]

Wind Speed [ms]

MeasuredComputed

ARTICLE IN PRESS

0

05

1

15

2

25

3

02 03 04 05 06 07 08 09 1

CN

rR

MeasuredComputed

-008

-006

-004

-002

0

002

004

02 03 04 05 06 07 08 09 1

CT

rR

MeasuredComputed

-15

-1

-05

0

05

1

15

2

25

3

0 02 04 06 08 1

-Cp

X Chord

rR=063

M x$ C _x$ Kx Fg (311)

ARTICLE IN PRESS

-2

0

2

4

6

8

10

12

0 50 100 150 200 250 300 350

CN

Azimuth Angle [deg]

S1500600 r R=030

-04

-02

0

02

04

06

08

1

0 50 100 150 200 250 300 350

CT

Azimuth Angle [deg]

S1500600 rR=030

Fgidxi Z

S

p uidS (312)

and

r n0 $ S0 u$ S1 yu

S0 1 0 0 z

0 1 0 0

0 0 1 x

2

64

3

75

S1 0 0 0 0

x 0 z 0

0 0 0 0

2

64

3

75

(321)

My GItampqyy

322amp

ARTICLE IN PRESS

y

x

z

A

u

w

u+du

M + dMz

z

M + dMyy

M + dMxx

zF + dF

yyF + dF

xxF + dF

Mx

Fx

Mz

FzMy

Fy dr

ra

)P

x

z

y

A(a) (b)

a

b

Z

A

rdm

$ dy dF$

Z

A

g dm

$ dy$ dpdy

(323)

Z

A

r0 rdm

$Z

A

r0 gdm

fx$ zy dy z xygT

A

S0ampT r dm qyZ

A

$ q2yy-K21 qyu( $Z

A

S0ampT g dm$ Sa dP

325amp

K11

Fy 0 0 0

0 EA 0 0

0 0 Fy 0

0 0 0 GIt

2

666664

3

777775

K22

EIzz 0 EIxz 0

0 0 0 0

EIxz 0 EIxx 0

0 0 0 0

2

666664

3

777775

Sa

1 0 0

0 1 0

0 0 1

za 0 xa

2

666664

3

777775

K12

0 0 0 0

EAx+ 0 EAz+ 0

0 0 0 0

2

666664

3

777775

K21

0 EAx+ 0 0

0 0 0 0

0 EAz+ 0 0

0 0 0 0

2

666664

3

777775

Then (321) becomes

r u

y$ v

w

8gtlt

gt

9gt=

gt$ E

xlyy

z

8gtlt

gt

9gt=

gt (326)

ARTICLE IN PRESS

x

z

yu

w

vO

Orsquoz

xz

xu

w amptamp

ampamp

amp +ˆ

Z L

0duT Zu _u uampdy

)X

e

Z Le

X

e

Z Le

0NeT ZNeueN

Muamp u$ Cu

amp _u$ Ku

amp u Qu

amp (3210)

ARTICLE IN PRESS

0k

rGk

OG O

xG x

zG z

yG

y y

z

x

O

x

y

z

O

ARTICLE IN PRESS

y

O

q8 q9 q10 q11q12 q13

yG y

z

x

O

x

y

z

OG O

xG x

zG z

q1 q2 q3 q4q5 q6

q7

q14

py L sin fD cos f

(413)

m1

amp 2=NampPN

y2tamp $ ~y2tamp

p[183184]

ARTICLE IN PRESS

3540

4550

tim

e (s

ec)

-35

E+0

6

-3E

+06

-25

E+0

6

-2E

+06

-15

E+0

6

-1E

+06

12

34

freq

uenc

y (H

z)

0

5E+

08

1E+

08

15E

+08

2E+

08

25E

+08

3E+

08

-t

data

19

190

5

191

5

192

5

193

5

194

00

51

15

22

53

t (se

c)

- 1

n

Fig21Schem

atic

dam

pingan

alysis

ARTICLE IN PRESS

-15e+06

-1e+06

-500000

0

500000

1e+06

15e+06

2 4 6 8 10 12 14

tow

er to

p til

t ben

ding

mom

ent (

KN

m)

time (s)

1000

800

600

400

200

0

-2000 50 100 150 200 250 300

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

1000900800700600500400300200100

0-100

200 210 220 230 240 250 260

ARTICLE IN PRESS

wake operation

complex inflow

ARTICLE IN PRESS

2

18

16

14

12C1

Cd

1

08

06

040 5 10 15 20 25 30

r=14r=28r=42r=56r=69r=89r=94r=97

35 40$ $

05

045

04

035

03

025

02

015

01

005

00

5 10 15 20 25 30 35 40

ARTICLE IN PRESS

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

ARTICLE IN PRESS

700800900

10001100120013001400150016001700

6 8 10 12 14 16 18 20 22 24 26

Low

Spe

ed S

haft

Tor

que

[Nm

]

Wind Speed [m s]

MeasuredComputed

1000

1500

2000

2500

3000

3500

4000

4500

5000

6 8 10 12 14 16 18 20 22 24 26

Roo

t Fla

p M

omen

t [N

m]

Wind Speed [ms]

MeasuredComputed

ARTICLE IN PRESS

0

05

1

15

2

25

3

02 03 04 05 06 07 08 09 1

CN

rR

MeasuredComputed

-008

-006

-004

-002

0

002

004

02 03 04 05 06 07 08 09 1

CT

rR

MeasuredComputed

-15

-1

-05

0

05

1

15

2

25

3

0 02 04 06 08 1

-Cp

X Chord

rR=063

M x$ C _x$ Kx Fg (311)

ARTICLE IN PRESS

-2

0

2

4

6

8

10

12

0 50 100 150 200 250 300 350

CN

Azimuth Angle [deg]

S1500600 r R=030

-04

-02

0

02

04

06

08

1

0 50 100 150 200 250 300 350

CT

Azimuth Angle [deg]

S1500600 rR=030

Fgidxi Z

S

p uidS (312)

and

r n0 $ S0 u$ S1 yu

S0 1 0 0 z

0 1 0 0

0 0 1 x

2

64

3

75

S1 0 0 0 0

x 0 z 0

0 0 0 0

2

64

3

75

(321)

My GItampqyy

322amp

ARTICLE IN PRESS

y

x

z

A

u

w

u+du

M + dMz

z

M + dMyy

M + dMxx

zF + dF

yyF + dF

xxF + dF

Mx

Fx

Mz

FzMy

Fy dr

ra

)P

x

z

y

A(a) (b)

a

b

Z

A

rdm

$ dy dF$

Z

A

g dm

$ dy$ dpdy

(323)

Z

A

r0 rdm

$Z

A

r0 gdm

fx$ zy dy z xygT

A

S0ampT r dm qyZ

A

$ q2yy-K21 qyu( $Z

A

S0ampT g dm$ Sa dP

325amp

K11

Fy 0 0 0

0 EA 0 0

0 0 Fy 0

0 0 0 GIt

2

666664

3

777775

K22

EIzz 0 EIxz 0

0 0 0 0

EIxz 0 EIxx 0

0 0 0 0

2

666664

3

777775

Sa

1 0 0

0 1 0

0 0 1

za 0 xa

2

666664

3

777775

K12

0 0 0 0

EAx+ 0 EAz+ 0

0 0 0 0

2

666664

3

777775

K21

0 EAx+ 0 0

0 0 0 0

0 EAz+ 0 0

0 0 0 0

2

666664

3

777775

Then (321) becomes

r u

y$ v

w

8gtlt

gt

9gt=

gt$ E

xlyy

z

8gtlt

gt

9gt=

gt (326)

ARTICLE IN PRESS

x

z

yu

w

vO

Orsquoz

xz

xu

w amptamp

ampamp

amp +ˆ

Z L

0duT Zu _u uampdy

)X

e

Z Le

X

e

Z Le

0NeT ZNeueN

Muamp u$ Cu

amp _u$ Ku

amp u Qu

amp (3210)

ARTICLE IN PRESS

0k

rGk

OG O

xG x

zG z

yG

y y

z

x

O

x

y

z

O

ARTICLE IN PRESS

y

O

q8 q9 q10 q11q12 q13

yG y

z

x

O

x

y

z

OG O

xG x

zG z

q1 q2 q3 q4q5 q6

q7

q14

py L sin fD cos f

(413)

m1

amp 2=NampPN

y2tamp $ ~y2tamp

p[183184]

ARTICLE IN PRESS

3540

4550

tim

e (s

ec)

-35

E+0

6

-3E

+06

-25

E+0

6

-2E

+06

-15

E+0

6

-1E

+06

12

34

freq

uenc

y (H

z)

0

5E+

08

1E+

08

15E

+08

2E+

08

25E

+08

3E+

08

-t

data

19

190

5

191

5

192

5

193

5

194

00

51

15

22

53

t (se

c)

- 1

n

Fig21Schem

atic

dam

pingan

alysis

ARTICLE IN PRESS

-15e+06

-1e+06

-500000

0

500000

1e+06

15e+06

2 4 6 8 10 12 14

tow

er to

p til

t ben

ding

mom

ent (

KN

m)

time (s)

1000

800

600

400

200

0

-2000 50 100 150 200 250 300

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

1000900800700600500400300200100

0-100

200 210 220 230 240 250 260

ARTICLE IN PRESS

wake operation

complex inflow

ARTICLE IN PRESS

2

18

16

14

12C1

Cd

1

08

06

040 5 10 15 20 25 30

r=14r=28r=42r=56r=69r=89r=94r=97

35 40$ $

05

045

04

035

03

025

02

015

01

005

00

5 10 15 20 25 30 35 40

ARTICLE IN PRESS

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

ARTICLE IN PRESS

0

05

1

15

2

25

3

02 03 04 05 06 07 08 09 1

CN

rR

MeasuredComputed

-008

-006

-004

-002

0

002

004

02 03 04 05 06 07 08 09 1

CT

rR

MeasuredComputed

-15

-1

-05

0

05

1

15

2

25

3

0 02 04 06 08 1

-Cp

X Chord

rR=063

M x$ C _x$ Kx Fg (311)

ARTICLE IN PRESS

-2

0

2

4

6

8

10

12

0 50 100 150 200 250 300 350

CN

Azimuth Angle [deg]

S1500600 r R=030

-04

-02

0

02

04

06

08

1

0 50 100 150 200 250 300 350

CT

Azimuth Angle [deg]

S1500600 rR=030

Fgidxi Z

S

p uidS (312)

and

r n0 $ S0 u$ S1 yu

S0 1 0 0 z

0 1 0 0

0 0 1 x

2

64

3

75

S1 0 0 0 0

x 0 z 0

0 0 0 0

2

64

3

75

(321)

My GItampqyy

322amp

ARTICLE IN PRESS

y

x

z

A

u

w

u+du

M + dMz

z

M + dMyy

M + dMxx

zF + dF

yyF + dF

xxF + dF

Mx

Fx

Mz

FzMy

Fy dr

ra

)P

x

z

y

A(a) (b)

a

b

Z

A

rdm

$ dy dF$

Z

A

g dm

$ dy$ dpdy

(323)

Z

A

r0 rdm

$Z

A

r0 gdm

fx$ zy dy z xygT

A

S0ampT r dm qyZ

A

$ q2yy-K21 qyu( $Z

A

S0ampT g dm$ Sa dP

325amp

K11

Fy 0 0 0

0 EA 0 0

0 0 Fy 0

0 0 0 GIt

2

666664

3

777775

K22

EIzz 0 EIxz 0

0 0 0 0

EIxz 0 EIxx 0

0 0 0 0

2

666664

3

777775

Sa

1 0 0

0 1 0

0 0 1

za 0 xa

2

666664

3

777775

K12

0 0 0 0

EAx+ 0 EAz+ 0

0 0 0 0

2

666664

3

777775

K21

0 EAx+ 0 0

0 0 0 0

0 EAz+ 0 0

0 0 0 0

2

666664

3

777775

Then (321) becomes

r u

y$ v

w

8gtlt

gt

9gt=

gt$ E

xlyy

z

8gtlt

gt

9gt=

gt (326)

ARTICLE IN PRESS

x

z

yu

w

vO

Orsquoz

xz

xu

w amptamp

ampamp

amp +ˆ

Z L

0duT Zu _u uampdy

)X

e

Z Le

X

e

Z Le

0NeT ZNeueN

Muamp u$ Cu

amp _u$ Ku

amp u Qu

amp (3210)

ARTICLE IN PRESS

0k

rGk

OG O

xG x

zG z

yG

y y

z

x

O

x

y

z

O

ARTICLE IN PRESS

y

O

q8 q9 q10 q11q12 q13

yG y

z

x

O

x

y

z

OG O

xG x

zG z

q1 q2 q3 q4q5 q6

q7

q14

py L sin fD cos f

(413)

m1

amp 2=NampPN

y2tamp $ ~y2tamp

p[183184]

ARTICLE IN PRESS

3540

4550

tim

e (s

ec)

-35

E+0

6

-3E

+06

-25

E+0

6

-2E

+06

-15

E+0

6

-1E

+06

12

34

freq

uenc

y (H

z)

0

5E+

08

1E+

08

15E

+08

2E+

08

25E

+08

3E+

08

-t

data

19

190

5

191

5

192

5

193

5

194

00

51

15

22

53

t (se

c)

- 1

n

Fig21Schem

atic

dam

pingan

alysis

ARTICLE IN PRESS

-15e+06

-1e+06

-500000

0

500000

1e+06

15e+06

2 4 6 8 10 12 14

tow

er to

p til

t ben

ding

mom

ent (

KN

m)

time (s)

1000

800

600

400

200

0

-2000 50 100 150 200 250 300

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

1000900800700600500400300200100

0-100

200 210 220 230 240 250 260

ARTICLE IN PRESS

wake operation

complex inflow

ARTICLE IN PRESS

2

18

16

14

12C1

Cd

1

08

06

040 5 10 15 20 25 30

r=14r=28r=42r=56r=69r=89r=94r=97

35 40$ $

05

045

04

035

03

025

02

015

01

005

00

5 10 15 20 25 30 35 40

ARTICLE IN PRESS

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

M x$ C _x$ Kx Fg (311)

ARTICLE IN PRESS

-2

0

2

4

6

8

10

12

0 50 100 150 200 250 300 350

CN

Azimuth Angle [deg]

S1500600 r R=030

-04

-02

0

02

04

06

08

1

0 50 100 150 200 250 300 350

CT

Azimuth Angle [deg]

S1500600 rR=030

Fgidxi Z

S

p uidS (312)

and

r n0 $ S0 u$ S1 yu

S0 1 0 0 z

0 1 0 0

0 0 1 x

2

64

3

75

S1 0 0 0 0

x 0 z 0

0 0 0 0

2

64

3

75

(321)

My GItampqyy

322amp

ARTICLE IN PRESS

y

x

z

A

u

w

u+du

M + dMz

z

M + dMyy

M + dMxx

zF + dF

yyF + dF

xxF + dF

Mx

Fx

Mz

FzMy

Fy dr

ra

)P

x

z

y

A(a) (b)

a

b

Z

A

rdm

$ dy dF$

Z

A

g dm

$ dy$ dpdy

(323)

Z

A

r0 rdm

$Z

A

r0 gdm

fx$ zy dy z xygT

A

S0ampT r dm qyZ

A

$ q2yy-K21 qyu( $Z

A

S0ampT g dm$ Sa dP

325amp

K11

Fy 0 0 0

0 EA 0 0

0 0 Fy 0

0 0 0 GIt

2

666664

3

777775

K22

EIzz 0 EIxz 0

0 0 0 0

EIxz 0 EIxx 0

0 0 0 0

2

666664

3

777775

Sa

1 0 0

0 1 0

0 0 1

za 0 xa

2

666664

3

777775

K12

0 0 0 0

EAx+ 0 EAz+ 0

0 0 0 0

2

666664

3

777775

K21

0 EAx+ 0 0

0 0 0 0

0 EAz+ 0 0

0 0 0 0

2

666664

3

777775

Then (321) becomes

r u

y$ v

w

8gtlt

gt

9gt=

gt$ E

xlyy

z

8gtlt

gt

9gt=

gt (326)

ARTICLE IN PRESS

x

z

yu

w

vO

Orsquoz

xz

xu

w amptamp

ampamp

amp +ˆ

Z L

0duT Zu _u uampdy

)X

e

Z Le

X

e

Z Le

0NeT ZNeueN

Muamp u$ Cu

amp _u$ Ku

amp u Qu

amp (3210)

ARTICLE IN PRESS

0k

rGk

OG O

xG x

zG z

yG

y y

z

x

O

x

y

z

O

ARTICLE IN PRESS

y

O

q8 q9 q10 q11q12 q13

yG y

z

x

O

x

y

z

OG O

xG x

zG z

q1 q2 q3 q4q5 q6

q7

q14

py L sin fD cos f

(413)

m1

amp 2=NampPN

y2tamp $ ~y2tamp

p[183184]

ARTICLE IN PRESS

3540

4550

tim

e (s

ec)

-35

E+0

6

-3E

+06

-25

E+0

6

-2E

+06

-15

E+0

6

-1E

+06

12

34

freq

uenc

y (H

z)

0

5E+

08

1E+

08

15E

+08

2E+

08

25E

+08

3E+

08

-t

data

19

190

5

191

5

192

5

193

5

194

00

51

15

22

53

t (se

c)

- 1

n

Fig21Schem

atic

dam

pingan

alysis

ARTICLE IN PRESS

-15e+06

-1e+06

-500000

0

500000

1e+06

15e+06

2 4 6 8 10 12 14

tow

er to

p til

t ben

ding

mom

ent (

KN

m)

time (s)

1000

800

600

400

200

0

-2000 50 100 150 200 250 300

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

1000900800700600500400300200100

0-100

200 210 220 230 240 250 260

ARTICLE IN PRESS

wake operation

complex inflow

ARTICLE IN PRESS

2

18

16

14

12C1

Cd

1

08

06

040 5 10 15 20 25 30

r=14r=28r=42r=56r=69r=89r=94r=97

35 40$ $

05

045

04

035

03

025

02

015

01

005

00

5 10 15 20 25 30 35 40

ARTICLE IN PRESS

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

Page 19: State of the art in wind turbine aerodynamics and ...phyexp/uploads/EolienneFlexible/Eoli… · ARTICLE IN PRESS Progress in Aerospace Sciences 42 (2006) 285–330 State of the art

Fgidxi Z

S

p uidS (312)

and

r n0 $ S0 u$ S1 yu

S0 1 0 0 z

0 1 0 0

0 0 1 x

2

64

3

75

S1 0 0 0 0

x 0 z 0

0 0 0 0

2

64

3

75

(321)

My GItampqyy

322amp

ARTICLE IN PRESS

y

x

z

A

u

w

u+du

M + dMz

z

M + dMyy

M + dMxx

zF + dF

yyF + dF

xxF + dF

Mx

Fx

Mz

FzMy

Fy dr

ra

)P

x

z

y

A(a) (b)

a

b

Z

A

rdm

$ dy dF$

Z

A

g dm

$ dy$ dpdy

(323)

Z

A

r0 rdm

$Z

A

r0 gdm

fx$ zy dy z xygT

A

S0ampT r dm qyZ

A

$ q2yy-K21 qyu( $Z

A

S0ampT g dm$ Sa dP

325amp

K11

Fy 0 0 0

0 EA 0 0

0 0 Fy 0

0 0 0 GIt

2

666664

3

777775

K22

EIzz 0 EIxz 0

0 0 0 0

EIxz 0 EIxx 0

0 0 0 0

2

666664

3

777775

Sa

1 0 0

0 1 0

0 0 1

za 0 xa

2

666664

3

777775

K12

0 0 0 0

EAx+ 0 EAz+ 0

0 0 0 0

2

666664

3

777775

K21

0 EAx+ 0 0

0 0 0 0

0 EAz+ 0 0

0 0 0 0

2

666664

3

777775

Then (321) becomes

r u

y$ v

w

8gtlt

gt

9gt=

gt$ E

xlyy

z

8gtlt

gt

9gt=

gt (326)

ARTICLE IN PRESS

x

z

yu

w

vO

Orsquoz

xz

xu

w amptamp

ampamp

amp +ˆ

Z L

0duT Zu _u uampdy

)X

e

Z Le

X

e

Z Le

0NeT ZNeueN

Muamp u$ Cu

amp _u$ Ku

amp u Qu

amp (3210)

ARTICLE IN PRESS

0k

rGk

OG O

xG x

zG z

yG

y y

z

x

O

x

y

z

O

ARTICLE IN PRESS

y

O

q8 q9 q10 q11q12 q13

yG y

z

x

O

x

y

z

OG O

xG x

zG z

q1 q2 q3 q4q5 q6

q7

q14

py L sin fD cos f

(413)

m1

amp 2=NampPN

y2tamp $ ~y2tamp

p[183184]

ARTICLE IN PRESS

3540

4550

tim

e (s

ec)

-35

E+0

6

-3E

+06

-25

E+0

6

-2E

+06

-15

E+0

6

-1E

+06

12

34

freq

uenc

y (H

z)

0

5E+

08

1E+

08

15E

+08

2E+

08

25E

+08

3E+

08

-t

data

19

190

5

191

5

192

5

193

5

194

00

51

15

22

53

t (se

c)

- 1

n

Fig21Schem

atic

dam

pingan

alysis

ARTICLE IN PRESS

-15e+06

-1e+06

-500000

0

500000

1e+06

15e+06

2 4 6 8 10 12 14

tow

er to

p til

t ben

ding

mom

ent (

KN

m)

time (s)

1000

800

600

400

200

0

-2000 50 100 150 200 250 300

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

1000900800700600500400300200100

0-100

200 210 220 230 240 250 260

ARTICLE IN PRESS

wake operation

complex inflow

ARTICLE IN PRESS

2

18

16

14

12C1

Cd

1

08

06

040 5 10 15 20 25 30

r=14r=28r=42r=56r=69r=89r=94r=97

35 40$ $

05

045

04

035

03

025

02

015

01

005

00

5 10 15 20 25 30 35 40

ARTICLE IN PRESS

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

Page 20: State of the art in wind turbine aerodynamics and ...phyexp/uploads/EolienneFlexible/Eoli… · ARTICLE IN PRESS Progress in Aerospace Sciences 42 (2006) 285–330 State of the art

r n0 $ S0 u$ S1 yu

S0 1 0 0 z

0 1 0 0

0 0 1 x

2

64

3

75

S1 0 0 0 0

x 0 z 0

0 0 0 0

2

64

3

75

(321)

My GItampqyy

322amp

ARTICLE IN PRESS

y

x

z

A

u

w

u+du

M + dMz

z

M + dMyy

M + dMxx

zF + dF

yyF + dF

xxF + dF

Mx

Fx

Mz

FzMy

Fy dr

ra

)P

x

z

y

A(a) (b)

a

b

Z

A

rdm

$ dy dF$

Z

A

g dm

$ dy$ dpdy

(323)

Z

A

r0 rdm

$Z

A

r0 gdm

fx$ zy dy z xygT

A

S0ampT r dm qyZ

A

$ q2yy-K21 qyu( $Z

A

S0ampT g dm$ Sa dP

325amp

K11

Fy 0 0 0

0 EA 0 0

0 0 Fy 0

0 0 0 GIt

2

666664

3

777775

K22

EIzz 0 EIxz 0

0 0 0 0

EIxz 0 EIxx 0

0 0 0 0

2

666664

3

777775

Sa

1 0 0

0 1 0

0 0 1

za 0 xa

2

666664

3

777775

K12

0 0 0 0

EAx+ 0 EAz+ 0

0 0 0 0

2

666664

3

777775

K21

0 EAx+ 0 0

0 0 0 0

0 EAz+ 0 0

0 0 0 0

2

666664

3

777775

Then (321) becomes

r u

y$ v

w

8gtlt

gt

9gt=

gt$ E

xlyy

z

8gtlt

gt

9gt=

gt (326)

ARTICLE IN PRESS

x

z

yu

w

vO

Orsquoz

xz

xu

w amptamp

ampamp

amp +ˆ

Z L

0duT Zu _u uampdy

)X

e

Z Le

X

e

Z Le

0NeT ZNeueN

Muamp u$ Cu

amp _u$ Ku

amp u Qu

amp (3210)

ARTICLE IN PRESS

0k

rGk

OG O

xG x

zG z

yG

y y

z

x

O

x

y

z

O

ARTICLE IN PRESS

y

O

q8 q9 q10 q11q12 q13

yG y

z

x

O

x

y

z

OG O

xG x

zG z

q1 q2 q3 q4q5 q6

q7

q14

py L sin fD cos f

(413)

m1

amp 2=NampPN

y2tamp $ ~y2tamp

p[183184]

ARTICLE IN PRESS

3540

4550

tim

e (s

ec)

-35

E+0

6

-3E

+06

-25

E+0

6

-2E

+06

-15

E+0

6

-1E

+06

12

34

freq

uenc

y (H

z)

0

5E+

08

1E+

08

15E

+08

2E+

08

25E

+08

3E+

08

-t

data

19

190

5

191

5

192

5

193

5

194

00

51

15

22

53

t (se

c)

- 1

n

Fig21Schem

atic

dam

pingan

alysis

ARTICLE IN PRESS

-15e+06

-1e+06

-500000

0

500000

1e+06

15e+06

2 4 6 8 10 12 14

tow

er to

p til

t ben

ding

mom

ent (

KN

m)

time (s)

1000

800

600

400

200

0

-2000 50 100 150 200 250 300

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

1000900800700600500400300200100

0-100

200 210 220 230 240 250 260

ARTICLE IN PRESS

wake operation

complex inflow

ARTICLE IN PRESS

2

18

16

14

12C1

Cd

1

08

06

040 5 10 15 20 25 30

r=14r=28r=42r=56r=69r=89r=94r=97

35 40$ $

05

045

04

035

03

025

02

015

01

005

00

5 10 15 20 25 30 35 40

ARTICLE IN PRESS

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

Page 21: State of the art in wind turbine aerodynamics and ...phyexp/uploads/EolienneFlexible/Eoli… · ARTICLE IN PRESS Progress in Aerospace Sciences 42 (2006) 285–330 State of the art

Z

A

rdm

$ dy dF$

Z

A

g dm

$ dy$ dpdy

(323)

Z

A

r0 rdm

$Z

A

r0 gdm

fx$ zy dy z xygT

A

S0ampT r dm qyZ

A

$ q2yy-K21 qyu( $Z

A

S0ampT g dm$ Sa dP

325amp

K11

Fy 0 0 0

0 EA 0 0

0 0 Fy 0

0 0 0 GIt

2

666664

3

777775

K22

EIzz 0 EIxz 0

0 0 0 0

EIxz 0 EIxx 0

0 0 0 0

2

666664

3

777775

Sa

1 0 0

0 1 0

0 0 1

za 0 xa

2

666664

3

777775

K12

0 0 0 0

EAx+ 0 EAz+ 0

0 0 0 0

2

666664

3

777775

K21

0 EAx+ 0 0

0 0 0 0

0 EAz+ 0 0

0 0 0 0

2

666664

3

777775

Then (321) becomes

r u

y$ v

w

8gtlt

gt

9gt=

gt$ E

xlyy

z

8gtlt

gt

9gt=

gt (326)

ARTICLE IN PRESS

x

z

yu

w

vO

Orsquoz

xz

xu

w amptamp

ampamp

amp +ˆ

Z L

0duT Zu _u uampdy

)X

e

Z Le

X

e

Z Le

0NeT ZNeueN

Muamp u$ Cu

amp _u$ Ku

amp u Qu

amp (3210)

ARTICLE IN PRESS

0k

rGk

OG O

xG x

zG z

yG

y y

z

x

O

x

y

z

O

ARTICLE IN PRESS

y

O

q8 q9 q10 q11q12 q13

yG y

z

x

O

x

y

z

OG O

xG x

zG z

q1 q2 q3 q4q5 q6

q7

q14

py L sin fD cos f

(413)

m1

amp 2=NampPN

y2tamp $ ~y2tamp

p[183184]

ARTICLE IN PRESS

3540

4550

tim

e (s

ec)

-35

E+0

6

-3E

+06

-25

E+0

6

-2E

+06

-15

E+0

6

-1E

+06

12

34

freq

uenc

y (H

z)

0

5E+

08

1E+

08

15E

+08

2E+

08

25E

+08

3E+

08

-t

data

19

190

5

191

5

192

5

193

5

194

00

51

15

22

53

t (se

c)

- 1

n

Fig21Schem

atic

dam

pingan

alysis

ARTICLE IN PRESS

-15e+06

-1e+06

-500000

0

500000

1e+06

15e+06

2 4 6 8 10 12 14

tow

er to

p til

t ben

ding

mom

ent (

KN

m)

time (s)

1000

800

600

400

200

0

-2000 50 100 150 200 250 300

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

1000900800700600500400300200100

0-100

200 210 220 230 240 250 260

ARTICLE IN PRESS

wake operation

complex inflow

ARTICLE IN PRESS

2

18

16

14

12C1

Cd

1

08

06

040 5 10 15 20 25 30

r=14r=28r=42r=56r=69r=89r=94r=97

35 40$ $

05

045

04

035

03

025

02

015

01

005

00

5 10 15 20 25 30 35 40

ARTICLE IN PRESS

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

Page 22: State of the art in wind turbine aerodynamics and ...phyexp/uploads/EolienneFlexible/Eoli… · ARTICLE IN PRESS Progress in Aerospace Sciences 42 (2006) 285–330 State of the art

Z L

0duT Zu _u uampdy

)X

e

Z Le

X

e

Z Le

0NeT ZNeueN

Muamp u$ Cu

amp _u$ Ku

amp u Qu

amp (3210)

ARTICLE IN PRESS

0k

rGk

OG O

xG x

zG z

yG

y y

z

x

O

x

y

z

O

ARTICLE IN PRESS

y

O

q8 q9 q10 q11q12 q13

yG y

z

x

O

x

y

z

OG O

xG x

zG z

q1 q2 q3 q4q5 q6

q7

q14

py L sin fD cos f

(413)

m1

amp 2=NampPN

y2tamp $ ~y2tamp

p[183184]

ARTICLE IN PRESS

3540

4550

tim

e (s

ec)

-35

E+0

6

-3E

+06

-25

E+0

6

-2E

+06

-15

E+0

6

-1E

+06

12

34

freq

uenc

y (H

z)

0

5E+

08

1E+

08

15E

+08

2E+

08

25E

+08

3E+

08

-t

data

19

190

5

191

5

192

5

193

5

194

00

51

15

22

53

t (se

c)

- 1

n

Fig21Schem

atic

dam

pingan

alysis

ARTICLE IN PRESS

-15e+06

-1e+06

-500000

0

500000

1e+06

15e+06

2 4 6 8 10 12 14

tow

er to

p til

t ben

ding

mom

ent (

KN

m)

time (s)

1000

800

600

400

200

0

-2000 50 100 150 200 250 300

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

1000900800700600500400300200100

0-100

200 210 220 230 240 250 260

ARTICLE IN PRESS

wake operation

complex inflow

ARTICLE IN PRESS

2

18

16

14

12C1

Cd

1

08

06

040 5 10 15 20 25 30

r=14r=28r=42r=56r=69r=89r=94r=97

35 40$ $

05

045

04

035

03

025

02

015

01

005

00

5 10 15 20 25 30 35 40

ARTICLE IN PRESS

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

ARTICLE IN PRESS

y

O

q8 q9 q10 q11q12 q13

yG y

z

x

O

x

y

z

OG O

xG x

zG z

q1 q2 q3 q4q5 q6

q7

q14

py L sin fD cos f

(413)

m1

amp 2=NampPN

y2tamp $ ~y2tamp

p[183184]

ARTICLE IN PRESS

3540

4550

tim

e (s

ec)

-35

E+0

6

-3E

+06

-25

E+0

6

-2E

+06

-15

E+0

6

-1E

+06

12

34

freq

uenc

y (H

z)

0

5E+

08

1E+

08

15E

+08

2E+

08

25E

+08

3E+

08

-t

data

19

190

5

191

5

192

5

193

5

194

00

51

15

22

53

t (se

c)

- 1

n

Fig21Schem

atic

dam

pingan

alysis

ARTICLE IN PRESS

-15e+06

-1e+06

-500000

0

500000

1e+06

15e+06

2 4 6 8 10 12 14

tow

er to

p til

t ben

ding

mom

ent (

KN

m)

time (s)

1000

800

600

400

200

0

-2000 50 100 150 200 250 300

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

1000900800700600500400300200100

0-100

200 210 220 230 240 250 260

ARTICLE IN PRESS

wake operation

complex inflow

ARTICLE IN PRESS

2

18

16

14

12C1

Cd

1

08

06

040 5 10 15 20 25 30

r=14r=28r=42r=56r=69r=89r=94r=97

35 40$ $

05

045

04

035

03

025

02

015

01

005

00

5 10 15 20 25 30 35 40

ARTICLE IN PRESS

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

Page 24: State of the art in wind turbine aerodynamics and ...phyexp/uploads/EolienneFlexible/Eoli… · ARTICLE IN PRESS Progress in Aerospace Sciences 42 (2006) 285–330 State of the art

py L sin fD cos f

(413)

m1

amp 2=NampPN

y2tamp $ ~y2tamp

p[183184]

ARTICLE IN PRESS

3540

4550

tim

e (s

ec)

-35

E+0

6

-3E

+06

-25

E+0

6

-2E

+06

-15

E+0

6

-1E

+06

12

34

freq

uenc

y (H

z)

0

5E+

08

1E+

08

15E

+08

2E+

08

25E

+08

3E+

08

-t

data

19

190

5

191

5

192

5

193

5

194

00

51

15

22

53

t (se

c)

- 1

n

Fig21Schem

atic

dam

pingan

alysis

ARTICLE IN PRESS

-15e+06

-1e+06

-500000

0

500000

1e+06

15e+06

2 4 6 8 10 12 14

tow

er to

p til

t ben

ding

mom

ent (

KN

m)

time (s)

1000

800

600

400

200

0

-2000 50 100 150 200 250 300

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

1000900800700600500400300200100

0-100

200 210 220 230 240 250 260

ARTICLE IN PRESS

wake operation

complex inflow

ARTICLE IN PRESS

2

18

16

14

12C1

Cd

1

08

06

040 5 10 15 20 25 30

r=14r=28r=42r=56r=69r=89r=94r=97

35 40$ $

05

045

04

035

03

025

02

015

01

005

00

5 10 15 20 25 30 35 40

ARTICLE IN PRESS

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

Page 25: State of the art in wind turbine aerodynamics and ...phyexp/uploads/EolienneFlexible/Eoli… · ARTICLE IN PRESS Progress in Aerospace Sciences 42 (2006) 285–330 State of the art

(413)

m1

amp 2=NampPN

y2tamp $ ~y2tamp

p[183184]

ARTICLE IN PRESS

3540

4550

tim

e (s

ec)

-35

E+0

6

-3E

+06

-25

E+0

6

-2E

+06

-15

E+0

6

-1E

+06

12

34

freq

uenc

y (H

z)

0

5E+

08

1E+

08

15E

+08

2E+

08

25E

+08

3E+

08

-t

data

19

190

5

191

5

192

5

193

5

194

00

51

15

22

53

t (se

c)

- 1

n

Fig21Schem

atic

dam

pingan

alysis

ARTICLE IN PRESS

-15e+06

-1e+06

-500000

0

500000

1e+06

15e+06

2 4 6 8 10 12 14

tow

er to

p til

t ben

ding

mom

ent (

KN

m)

time (s)

1000

800

600

400

200

0

-2000 50 100 150 200 250 300

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

1000900800700600500400300200100

0-100

200 210 220 230 240 250 260

ARTICLE IN PRESS

wake operation

complex inflow

ARTICLE IN PRESS

2

18

16

14

12C1

Cd

1

08

06

040 5 10 15 20 25 30

r=14r=28r=42r=56r=69r=89r=94r=97

35 40$ $

05

045

04

035

03

025

02

015

01

005

00

5 10 15 20 25 30 35 40

ARTICLE IN PRESS

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

Page 26: State of the art in wind turbine aerodynamics and ...phyexp/uploads/EolienneFlexible/Eoli… · ARTICLE IN PRESS Progress in Aerospace Sciences 42 (2006) 285–330 State of the art

y2tamp $ ~y2tamp

p[183184]

ARTICLE IN PRESS

3540

4550

tim

e (s

ec)

-35

E+0

6

-3E

+06

-25

E+0

6

-2E

+06

-15

E+0

6

-1E

+06

12

34

freq

uenc

y (H

z)

0

5E+

08

1E+

08

15E

+08

2E+

08

25E

+08

3E+

08

-t

data

19

190

5

191

5

192

5

193

5

194

00

51

15

22

53

t (se

c)

- 1

n

Fig21Schem

atic

dam

pingan

alysis

ARTICLE IN PRESS

-15e+06

-1e+06

-500000

0

500000

1e+06

15e+06

2 4 6 8 10 12 14

tow

er to

p til

t ben

ding

mom

ent (

KN

m)

time (s)

1000

800

600

400

200

0

-2000 50 100 150 200 250 300

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

1000900800700600500400300200100

0-100

200 210 220 230 240 250 260

ARTICLE IN PRESS

wake operation

complex inflow

ARTICLE IN PRESS

2

18

16

14

12C1

Cd

1

08

06

040 5 10 15 20 25 30

r=14r=28r=42r=56r=69r=89r=94r=97

35 40$ $

05

045

04

035

03

025

02

015

01

005

00

5 10 15 20 25 30 35 40

ARTICLE IN PRESS

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

ARTICLE IN PRESS

3540

4550

tim

e (s

ec)

-35

E+0

6

-3E

+06

-25

E+0

6

-2E

+06

-15

E+0

6

-1E

+06

12

34

freq

uenc

y (H

z)

0

5E+

08

1E+

08

15E

+08

2E+

08

25E

+08

3E+

08

-t

data

19

190

5

191

5

192

5

193

5

194

00

51

15

22

53

t (se

c)

- 1

n

Fig21Schem

atic

dam

pingan

alysis

ARTICLE IN PRESS

-15e+06

-1e+06

-500000

0

500000

1e+06

15e+06

2 4 6 8 10 12 14

tow

er to

p til

t ben

ding

mom

ent (

KN

m)

time (s)

1000

800

600

400

200

0

-2000 50 100 150 200 250 300

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

1000900800700600500400300200100

0-100

200 210 220 230 240 250 260

ARTICLE IN PRESS

wake operation

complex inflow

ARTICLE IN PRESS

2

18

16

14

12C1

Cd

1

08

06

040 5 10 15 20 25 30

r=14r=28r=42r=56r=69r=89r=94r=97

35 40$ $

05

045

04

035

03

025

02

015

01

005

00

5 10 15 20 25 30 35 40

ARTICLE IN PRESS

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

ARTICLE IN PRESS

-15e+06

-1e+06

-500000

0

500000

1e+06

15e+06

2 4 6 8 10 12 14

tow

er to

p til

t ben

ding

mom

ent (

KN

m)

time (s)

1000

800

600

400

200

0

-2000 50 100 150 200 250 300

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

ED

GE

WIS

E B

LA

DE

RO

OT

MO

ME

NT

[UN

CA

LIB

RA

TE

D]

1000900800700600500400300200100

0-100

200 210 220 230 240 250 260

ARTICLE IN PRESS

wake operation

complex inflow

ARTICLE IN PRESS

2

18

16

14

12C1

Cd

1

08

06

040 5 10 15 20 25 30

r=14r=28r=42r=56r=69r=89r=94r=97

35 40$ $

05

045

04

035

03

025

02

015

01

005

00

5 10 15 20 25 30 35 40

ARTICLE IN PRESS

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

ARTICLE IN PRESS

wake operation

complex inflow

ARTICLE IN PRESS

2

18

16

14

12C1

Cd

1

08

06

040 5 10 15 20 25 30

r=14r=28r=42r=56r=69r=89r=94r=97

35 40$ $

05

045

04

035

03

025

02

015

01

005

00

5 10 15 20 25 30 35 40

ARTICLE IN PRESS

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

ARTICLE IN PRESS

wake operation

complex inflow

ARTICLE IN PRESS

2

18

16

14

12C1

Cd

1

08

06

040 5 10 15 20 25 30

r=14r=28r=42r=56r=69r=89r=94r=97

35 40$ $

05

045

04

035

03

025

02

015

01

005

00

5 10 15 20 25 30 35 40

ARTICLE IN PRESS

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

ARTICLE IN PRESS

wake operation

complex inflow

ARTICLE IN PRESS

2

18

16

14

12C1

Cd

1

08

06

040 5 10 15 20 25 30

r=14r=28r=42r=56r=69r=89r=94r=97

35 40$ $

05

045

04

035

03

025

02

015

01

005

00

5 10 15 20 25 30 35 40

ARTICLE IN PRESS

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

ARTICLE IN PRESS

wake operation

complex inflow

ARTICLE IN PRESS

2

18

16

14

12C1

Cd

1

08

06

040 5 10 15 20 25 30

r=14r=28r=42r=56r=69r=89r=94r=97

35 40$ $

05

045

04

035

03

025

02

015

01

005

00

5 10 15 20 25 30 35 40

ARTICLE IN PRESS

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

ARTICLE IN PRESS

wake operation

complex inflow

ARTICLE IN PRESS

2

18

16

14

12C1

Cd

1

08

06

040 5 10 15 20 25 30

r=14r=28r=42r=56r=69r=89r=94r=97

35 40$ $

05

045

04

035

03

025

02

015

01

005

00

5 10 15 20 25 30 35 40

ARTICLE IN PRESS

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

ARTICLE IN PRESS

wake operation

complex inflow

ARTICLE IN PRESS

2

18

16

14

12C1

Cd

1

08

06

040 5 10 15 20 25 30

r=14r=28r=42r=56r=69r=89r=94r=97

35 40$ $

05

045

04

035

03

025

02

015

01

005

00

5 10 15 20 25 30 35 40

ARTICLE IN PRESS

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

Page 35: State of the art in wind turbine aerodynamics and ...phyexp/uploads/EolienneFlexible/Eoli… · ARTICLE IN PRESS Progress in Aerospace Sciences 42 (2006) 285–330 State of the art

wake operation

complex inflow

ARTICLE IN PRESS

2

18

16

14

12C1

Cd

1

08

06

040 5 10 15 20 25 30

r=14r=28r=42r=56r=69r=89r=94r=97

35 40$ $

05

045

04

035

03

025

02

015

01

005

00

5 10 15 20 25 30 35 40

ARTICLE IN PRESS

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

ARTICLE IN PRESS

2

18

16

14

12C1

Cd

1

08

06

040 5 10 15 20 25 30

r=14r=28r=42r=56r=69r=89r=94r=97

35 40$ $

05

045

04

035

03

025

02

015

01

005

00

5 10 15 20 25 30 35 40

ARTICLE IN PRESS

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

ARTICLE IN PRESS

2

18

16

14

12C1

Cd

1

08

06

040 5 10 15 20 25 30

r=14r=28r=42r=56r=69r=89r=94r=97

35 40$ $

05

045

04

035

03

025

02

015

01

005

00

5 10 15 20 25 30 35 40

ARTICLE IN PRESS

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

ARTICLE IN PRESS

2

18

16

14

12C1

Cd

1

08

06

040 5 10 15 20 25 30

r=14r=28r=42r=56r=69r=89r=94r=97

35 40$ $

05

045

04

035

03

025

02

015

01

005

00

5 10 15 20 25 30 35 40

ARTICLE IN PRESS

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

ARTICLE IN PRESS

2

18

16

14

12C1

Cd

1

08

06

040 5 10 15 20 25 30

r=14r=28r=42r=56r=69r=89r=94r=97

35 40$ $

05

045

04

035

03

025

02

015

01

005

00

5 10 15 20 25 30 35 40

ARTICLE IN PRESS

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

ARTICLE IN PRESS

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

Page 41: State of the art in wind turbine aerodynamics and ...phyexp/uploads/EolienneFlexible/Eoli… · ARTICLE IN PRESS Progress in Aerospace Sciences 42 (2006) 285–330 State of the art

6 Discussion

References

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

Page 42: State of the art in wind turbine aerodynamics and ...phyexp/uploads/EolienneFlexible/Eoli… · ARTICLE IN PRESS Progress in Aerospace Sciences 42 (2006) 285–330 State of the art

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

Page 43: State of the art in wind turbine aerodynamics and ...phyexp/uploads/EolienneFlexible/Eoli… · ARTICLE IN PRESS Progress in Aerospace Sciences 42 (2006) 285–330 State of the art

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

Page 44: State of the art in wind turbine aerodynamics and ...phyexp/uploads/EolienneFlexible/Eoli… · ARTICLE IN PRESS Progress in Aerospace Sciences 42 (2006) 285–330 State of the art

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

Page 45: State of the art in wind turbine aerodynamics and ...phyexp/uploads/EolienneFlexible/Eoli… · ARTICLE IN PRESS Progress in Aerospace Sciences 42 (2006) 285–330 State of the art

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

Page 46: State of the art in wind turbine aerodynamics and ...phyexp/uploads/EolienneFlexible/Eoli… · ARTICLE IN PRESS Progress in Aerospace Sciences 42 (2006) 285–330 State of the art

- Introduction
    - Yawtilt model
    - Dynamic stall
    - Airfoil data
    - Wind simulation
      - Vortex methods
        
        Panel methods
        
        
        
        
        Approaches
        
        
        
        Numerical issues
        
        
        Future
        
        
        
        
        
        
        
        
        
        
        Flutter instability
        
        
        
        
        
        
        
        
        
        Wake operation
        
        
        Complex inflow
        
        
        
        Discussion
        
        References

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