Enhancement of Free Vortex Filament Method for ...publications.lib.chalmers.se/records/fulltext/249148/...for Aerodynamic Loads on Rotor Blades The aerodynamics of a wind turbine is

Hamidreza AbediDivision of Fluid Dynamics

Department of Applied Mechanics

Chalmers University of Technology

Geurooteborg SE-412 96 Sweden

e-mail abedihchalmersse

Lars DavidsonDivision of Fluid Dynamics

Department of Applied Mechanics

Chalmers University of Technology

Geurooteborg SE-412 96 Sweden

e-mail ladachalmersse

Spyros VoutsinasFluid Section

School of Mechanical Engineering

National Technical University of Athens

Athens 15780 Greece

e-mail spyrosfluidmechntuagr

Enhancement of FreeVortex Filament Methodfor Aerodynamic Loadson Rotor BladesThe aerodynamics of a wind turbine is governed by the flow around the rotor where theprediction of air loads on rotor blades in different operational conditions and its relationto rotor structural dynamics is one of the most important challenges in wind turbine rotorblade design Because of the unsteady flow field around wind turbine blades predictionof aerodynamic loads with high level of accuracy is difficult and increases the uncertaintyof load calculations An in-house vortex lattice free wake (VLFW) code based on theinviscid incompressible and irrotational flow (potential flow) was developed to studythe aerodynamic loads Since it is based on the potential flow it cannot be used to predictviscous phenomena such as drag and boundary layer separation Therefore it must becoupled to tabulated airfoil data to take the viscosity effects into account Additionally adynamic approach must be introduced to modify the aerodynamic coefficients forunsteady operating conditions This approach which is called dynamic stall adjusts thelift the drag and the moment coefficients for each blade element on the basis of the two-dimensional (2D) static airfoil data together with the correction for separated flow Twodifferent turbines NREL and MEXICO are used in the simulations Predicted normaland tangential forces using the VLFW method are compared with the blade elementmomentum (BEM) method the GENUVP code and the MEXICO wind tunnel measure-ments The results show that coupling to the 2D static airfoil data improves the load andpower predictions while employing the dynamic stall model to take the time-varying oper-ating conditions into consideration is crucial [DOI 10111514035887]

1 Introduction

The methods for predicting wind turbine performance aresimilar to propeller and helicopter theories There are differentmethods for modeling the aerodynamics of a wind turbine withdifferent levels of complexity and accuracy such as the BEMtheory and solving the NavierndashStokes equations using computa-tional fluid dynamics (CFD) Today the BEM method is usedextensively by wind turbine manufacturers to analyze the aerody-namic performance of a wind turbine Although it is computation-ally fast and is easily implemented it is acceptable only for acertain range of flow conditions [1] A number of empirical andsemi-empirical correction factors have been added to the BEMmethod in order to increase its application range such as yaw mis-alignment dynamic inflow finite number of blades and bladecone angle [2] but they are not relevant to all operating conditionsand are often incorrect at high tip speed ratios where wake distor-tion is significant [3] Moreover because of the axisymmetricinflow assumption for the BEM method it is no longer valid topredict the aerodynamic loads on rotor blades when the wind tur-bine operates under the yaw condition (because of nonuniformblade loading)

The vortex theory which is based on the potential inviscidand irrotational flow can also be used to predict the aerodynamicperformance of wind turbines It has been widely used for aerody-namic analysis of airfoils and aircrafts Although the standardmethod cannot be used to predict viscous phenomena such as drag

and boundary layer separation its combination with tabulated air-foil data makes it a powerful tool for the prediction of fluid flowCompared with the BEM method the vortex method is able toprovide more physical solutions for attached flow conditions usingboundary layer corrections and it is also valid over a wider rangeof turbine operating conditions Although it is computationallymore expensive than the BEM method it is still feasible as anengineering method

In vortex methods the trailing and shed vortices are modeledby either vortex particles (characterized by a position a volumeand a strength) [4ndash6] or vortex filaments [78] moving eitherfreely known as free wake [9ndash12] or restrictedly by imposingthe wake geometry known as prescribed wake [1314] The pre-scribed wake requires less computational effort than the freewake but it requires experimental data to be valid for a broadrange of operating conditions The free wake model which is themost computationally expensive vortex method is able to predictthe wake geometry and loads more accurately than the prescribedwake because of less restrictive assumptions Therefore it can beused for load calculations especially for unsteady flow environ-ment However its application is limited to attached flow and itmust be linked to tabulated airfoil data to predict the air loads inthe presence of the drag and the flow separation

Additionally wind turbines always operate in unsteady flowconditions The unsteadiness sources are classified according tothe atmospheric conditions eg wind shear and turbulent inflowtogether with the turbine structure such as yaw misalignmentrotor tilt and blade elastic deformation [15] which are consideredas perturbations of the local angle of attack and the velocity fieldParticularly unsteady inflow dynamically affects the local angleof attack along the blade due to three-dimensional cross flows andseparation Since the variation in frequency of these sources may

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Journal of Solar Energy Engineering JUNE 2017 Vol 139 031007-1Copyright VC 2017 by ASME

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be high (rapid transient loads) the quasi-static aerodynamic is nolonger valid [1617] As a consequence a dynamic approach mustbe introduced to modify the aerodynamic coefficients for unsteadyoperating conditions This approach which is called dynamicstall adjusts the lift the drag and the moment coefficients foreach blade element on the basis of the 2D static airfoil datatogether with the correction for separated flow Furthermorebecause of the dynamic stall the predicted aerodynamic coeffi-cients may result in noticeable differences [16] in comparisonwith the static ones A dynamic stall model for the unsteady aero-dynamic loads prediction is therefore crucial for the wind turbinetechnology development

In this paper an in-house time-marching VLFW code is usedfor the simulation where its potential solution is coupled to thetabulated airfoil data for the wind turbine load calculation Inaddition a semi-empirical model called extended ONERAmodel is added to account for the dynamic stall effects Theresults of three different load calculation methods namely stand-ard potential method (potential solution) 2D static airfoil datamodel (viscous solution) and the dynamic stall model are com-pared with the BEM method [1] the GENUVP code1 and theMEXICO [18] wind tunnel measurements2

2 Theory

Vortex flow theory is based on assuming incompressible(r V frac14 0) and irrotational (r V frac14 0) flow at every pointexcept at the origin of the vortex where the velocity is infinite[19] A region containing a concentrated amount of vorticity iscalled a vortex where a vortex line is defined as a line whose tan-gent is parallel to the local vorticity vector everywhere Vortexlines surrounded by a given closed curve make a vortex tube witha strength equal to the circulation C A vortex filament with astrength of C is represented as a vortex tube of an infinitesimalcross section with strength C

According to the Helmholtz theorem an irrotational motion ofan inviscid fluid which started from rest remains irrotationalAlso a vortex line cannot end in the fluid It must form a closedpath end at a solid boundary or go to infinity this implies thatvorticity can only be generated at solid boundaries Therefore asolid surface may be considered as a source of vorticity Hencethe solid surface in contact with fluid is replaced by a distributionof vorticity

For an irrotational flow a velocity potential U can be definedas V frac14 rU where in order to find the velocity field the Laplacersquosequation r2Ufrac14 0 is solved using a proper boundary conditionfor the velocity on the body and at infinity In addition in vortextheory the vortical structure of a wake can be modeled by eithervortex filaments or vortex particles where a vortex filament ismodeled as concentrated vortices along an axis with a singularityat the center

The velocity induced by a straight vortex filament can be deter-mined by the BiotndashSavart law as

Vind frac14C4p

jrj3(1)

which can also be written as

Vind frac14C4p

r1 thorn r2eth THORN r1 r2eth THORNr1r2eth THORN r1r2 thorn r1 r2eth THORN

where C denotes the strength of the vortex filament r1 and r2 arethe distance vectors from the beginning A and end B of a vortexsegment to an arbitrary point C respectively (see Fig 1)

The BiotndashSavart law has a singularity when the point of evalua-tion (C) of induced velocity is located on the vortex filament axis(L) Also when the evaluation point is very near to the vortex fila-ment there is an unphysically large induced velocity at that pointThe remedy is either to use a cut-off radius d [20] or to use a vis-cous vortex model with a finite core size by multiplying a factorto remove the singularity [21]

The BiotndashSavart law correction based on the viscous vortexmodel can be made by introducing a finite core size rc for a vor-tex filament [22] In this paper for simplicity a constant viscouscore size model which is one of the general approaches usingdesingularized algebraic profile is employed for the inducedvelocity calculations A general form of a desingularized algebraicswirl-velocity profile for stationary vortices is proposed byVasitas et al [23] as

Vh reth THORN frac14 C2pr

r2nc thorn r2n

where r and n are the distance of a vortex segment to an arbitrarypoint and an integer number respectively

Bagai and Leishman [24] suggested the velocity profile basedon Eq (3) for nfrac14 2 for rotor tip vortices Therefore in order totake into account the effect of the viscous vortex core a factor ofKv must be added to the BiotndashSavart law as [24]

Vind frac14 KvC4p

Kv frac14h2

r2nc thorn h2n

1=n(5)

and h is defined as the perpendicular distance of the evaluationpoint (see Fig 1) Factor Kv desingularizes the BiotndashSavart equa-tion when the evaluation point distance tends to zero and preventsa high induced velocity in the vicinity region of the vortex coreradius

3 Model

31 Assumptions In this study as a first effort to evaluatethe load calculation methods (see Sec 33) the upstream flow isset to be uniform both in time and space For the nonyawed flowit is perpendicular to the rotor plane whereas for the yawed flow it

Fig 1 Schematic for the BiotndashSavart law

1GENUVP is an unsteady flow solver based on vortex blob approximationsdeveloped for rotor systems by National Technical University of Athens

2The MEXICO wind turbine measurements were carried out in 2006 in the LargeScale Low Speed Facility (LLF) of the German Dutch Wind Tunnel (DNW)

031007-2 Vol 139 JUNE 2017 Transactions of the ASME

is not (it deviates from the rotating axis) It should be noted that ayaw misalignment makes the inflow unsteady even if the upstreamflow does not change in time and space (steady state)

However the VLFW code can handle both uniform or nonuni-form flow (varying both in time and space) The blades areassumed to be rigid ie the elastic effects of the blades areneglected

32 Vortex Lattice Free Wake The vortex lattice method(Fig 3) is based on the thin lifting surface theory of vortex ringelements [25] in which the blade surface is replaced by vortexpanels that are constructed based on the airfoil camber line ofeach blade section (see Fig 2) The solution of Laplacersquos equationwith a proper boundary condition gives the flow around the bladeresulting in an aerodynamic load calculation generating powerand thrust of the wind turbine To take the blade surface curvatureinto account the lifting surface is divided into a number of panelsboth in the chordwise and spanwise directions where each panelcontains a vortex ring with strength Cij in which i and j indicatepanel indices in the chordwise and spanwise directions respec-tively The strength of each blade bound vortex ring element Cijis assumed to be constant and the positive circulation is definedon the basis of the right-hand rotation rule

In order to fulfill the 2D Kutta condition (which can beexpressed as cTEfrac14 0 in terms of the strength of the vortex sheet

where the TE subscripts denotes the trailing edge) the leadingsegment of a vortex ring is located at 14 of the panel length (seeFig 4) The control point of each panel is located at 34 of thepanel length meaning that the control point is placed at the centerof the panelrsquos vortex ring

The wake elements which induce a velocity field around therotor blades are modeled as vortex ring elements and they aretrailed and shed from the trailing edge based on a time-marchingmethod To satisfy the 3D trailing edge condition for each span-wise section the strength of the trailing vortex wake rings must beequal to the last vortex ring row in the chordwise direction(CTEfrac14CWake) This mechanism allows that the blade bound vor-ticity is transformed into free wake vortices

To find the blade bound vorticesrsquo strength at each time step theflow tangency condition at each bladersquos control point must bespecified by establishing a system of equations The velocity com-ponents at each blade control point include the free stream ethV1THORNrotational ethXrTHORN blade vortex rings self-induced ethVindboundTHORN andwake induced ethVindwakeTHORN velocities The blade-induced componentis known as influence coefficient aij and is defined as the inducedvelocity of a jth blade vortex ring with a strength equal to one onthe ith blade control point given by

aij frac14 ethVindboundTHORNij ni (6)

If the blade is assumed to be rigid then the influencecoefficients are constant at each time step which means that theleft-hand side of the equation system is computed only onceHowever if the blade is modeled as a flexible blade they must becalculated at each time step Since the wind and rotational veloc-ities are known during the wind turbine operation they are trans-ferred to the right-hand side of the equation system In addition ateach time step the strength of the wake vortex panels is knownfrom the previous time step so the induced velocity contributionby the wake panels is also transferred to the right-hand sideTherefore the system of equations can be expressed as

a11 a12 a1m

a21 a22 a2m

am1 am2 amm

1CCA frac14

1CCA (7)

where m is defined as mfrac14MN for a blade with M spanwise and Nchordwise panels and the right-hand side is computed as

RHSk frac14 ethV1 thorn Xrthorn VindwakeTHORNk nk (8)

The blade bound vortex strength (Cij) is calculated by solvingEq (7) at each time step At the first time step (see Figs 5 and 6)

Fig 2 Lifting surface and vortex panels construction

Fig 3 Schematic of vortex lattice free wake

Fig 4 Numbering procedure

Journal of Solar Energy Engineering JUNE 2017 Vol 139 031007-3

there are no free wake elements At the second time step (seeFigs 5 and 7) when the blade is rotating the first wake panels areshed Their strength is equal to the bound vortex circulation of thelast row of the blade vortex ring elements (Kutta condition)located at the trailing edge at the previous time step (see Fig 7)which means that CWt2

frac14 CTEt1 where the W and TE subscriptsrepresent the wake and the trailing edge respectively At thesecond time step the strength of the blade bound vortex rings iscalculated by specifying the flow tangency boundary conditionwhere in addition to the blade vortex ring elements the contribu-tion of the first row of the wake panels is considered

This methodology is repeated and vortex wake elements aretrailed and shed at each time step where their strengths remainconstant (Kelvin theorem) In addition the corner points of vortexwake elements are moved based on the governing equation for thewake geometry given by

dtfrac14 Vtot reth THORN r t frac14 0eth THORN frac14 r0 (9)

where r Vtot and t denote the position vector of a Lagrangianmarker the total velocity field and time respectively The totalvelocity field expressed in the rotating reference frame ieVrot frac14 0 can be written as

Vtot frac14 V1 thorn Vindblade thorn Vindwake (10)

including the wind velocity and the induced velocity by all bladesand wake vortex rings

Different numerical schemes may be used for Eq (9) such asthe explicit Euler method the implicit method AdamsndashBashforthmethod and the PredictorndashCorrector method The numericalintegration scheme must be considered in terms of the accuracystability and computational efficiency Here the first-order Eulerexplicit method is used as

rtthorn1 frac14 rt thorn VtotethrtTHORNDt (11)

where Vtot is taken at the old time step

33 Load Calculation In the vortex flow the only force act-ing on the rotor blades is the lift force which can be calculatedeither by the KuttandashJukowski theory or by the Bernoulli equationwhere the viscous effects such as the skin friction and the flowseparation are not included Therefore in order to take intoaccount the viscous effects and the flow separation the inviscidlift force must be combined with the aerodynamic coefficientsthrough the tabulated airfoil data along with the dynamic stallmodel to take the unsteady effects into account

The currently developed model (VLFW) is based on the thinlifting surface theory of vortex ring elements where the body ispart of the flow domain Therefore the effective angle of attack iscalculated by projecting the lift force acting on rotor blades intothe normal and tangential directions with respect to the rotorplane In general the predicted angle of attack computed on thebasis of the potential flow solution (ie the lifting surface theory)is always greater than that calculated by the viscous flow There-fore it cannot be directly used as entry to look up the tabulatedairfoil data to provide the aerodynamic coefficients This leads usto modify the predicted angle of attack by introducing a methodcalled the 2D static airfoil data method (viscous solution)

In the 2D static airfoil data method the new angle of attack iscalculated by using the tabulated airfoil data where it is directlyconnected to both the tabulated airfoil data and the potential solu-tion parameter (C) This angle of attack is used as the entry tolook-up the airfoil table and then we are able to calculate the liftdrag and moment coefficients giving the lift and drag forces foreach blade element It is worth noting that both the standardpotential method and 2D static airfoil data method are based onthe quasi-static assumption

In the fully unsteady condition since the lift drag and momentcoefficients are not following the tabulated airfoil data curve (seeAppendix) they should be corrected and this is done by employ-ing a dynamic stall model Generally the aim of the dynamic stallmodel is to correct the aerodynamic coefficients under the differ-ent time-dependent events which were described in the introduc-tion Hence in case of uniform steady inflow condition and in the

Fig 5 Schematic of generation and moving of wake panels ateach time step

Fig 6 Schematic of wake evolution at the first time step

Fig 7 Schematic of wake evolution at the second time step

absence of the yaw misalignment the rotor tilt and the bladeaeroelastic motion it is not necessary to use the dynamic stallmodel

331 The Standard Potential Method In the VLFW methodwhen the positions of all Lagrangian markers are calculated ateach time step we are able to compute the velocity field aroundthe rotor blade where as a consequence the lift force can be cal-culated according to the KuttandashJukowski theorem The differentialsteady-state form of the KuttandashJukowski theorem reads

dL frac14 qVtot Cdl (12)

where q Vtot C and dl denote the air density total velocityvector vortex filament strength and vortex filament length vectorrespectively

The KuttandashJukowski theorem is applied at the midpoint of thefront edge of each blade vortex ring and gives the potential liftforce where the lift force of each spanwise blade section is calcu-lated by summing up the lift force of all panels along the chordThe lift force for each blade panel is computed using the generalform of the KuttandashJukowski theorem given by

Lij frac14 qVtotij Cij Ci1jeth THORNDyij thorn qAijDCij

where Dyij Aij DCij=Dt and nij denote the width vector of ablade vortex panel in the chordwise direction blade vortex panelarea time-gradient of circulation and unit vector normal to thevortex panel in which i and j indicate panel indices in the chord-wise and spanwise directions respectively Moreover Vtotij iscomputed as

Vtotij frac14 V1ij thorn Xrj thorn Vindwakeij thorn Vindboundij (14)

The second term in Eq (13) is the unsteady term which may beneglected for the steady-state computations For the blade panelsadjacent to the leading edge Eq (13) can be written as

L1j frac14 qVtot1j C1jDy1j thorn qA1jDC1j

The total lift of each blade section in the spanwise direction isobtained as

Lj frac14XN

ifrac141

Lij (16)

where N denotes the number of chordwise sections Decomposi-tion of the lift force for each blade spanwise section into the nor-mal and tangential directions with respect to the rotor plane (seeFig 8) gives the effective potential angle of attack for eachsection

a frac14 tan1ethFt=FnTHORN ht hp (17)

where a Ft Fn ht and hp represent the angle of attack tangentialforce normal force blade section twist and blade pitchrespectively

332 Two-Dimensional Static Airfoil Data Method In potentialflow the lift coefficient is expressed by the thin airfoil theorywhich is a linear function of angle of attack with constant slopeequal to 2p This means that for the thick airfoil commonly usedin wind turbine blades the thin airfoil theory is no longer valid Inaddition because of this linear relation of the lift coefficient andthe angle of attack this assumption gives the higher lift the higherthe angle of attack Hence considerable lift reduction due to flowseparation at higher angles of attack cannot be predicted In other

words the application of thin airfoil theory is limited to attachedflow and it must be linked to tabulated airfoil data to predict airloads in the presence of drag and flow separation Coupling thethin airfoil theory (standard potential method) to the tabulated air-foil data for wind turbine load calculation is done by employingthe 2D static airfoil data method as described here

According to the KuttandashJukowski theory the magnitude of thelift force per unit spanwise length L0 is proportional to the circu-lation C and it is given by

L0 frac14 qVtotC (18)

where q Vtot denote the air density and the total velocity magni-tude respectively The circulation for each spanwise section isequal to the bound vortex circulation of the last row vortex ringelement located at the trailing edge In addition in the linear air-foil theory the lift coefficient is expressed by

CL frac14 metha a0THORN (19)

where mfrac14 2p a and a0 indicate the slope the angle of attack andthe zero-lift angle of attack respectively The lift coefficient isgenerally defined as

CL frac14L0

05qV2totc

where c denotes the airfoil chord length Combination of Eqs(18)ndash(20) gives the modified angle of attack as

a frac14 2CmVtotc

thorn a0 (21)

For an arbitrary airfoil both m and a0 are determined according tothe CL versus a curve where the constant lift coefficient slope mis computed over the linear region (attached flow) The modifiedangle of attack based on the Eq (21) is used as entry to calculatethe lift the drag and the moment coefficients through the tabu-lated airfoil data As a result the lift and drag forces are computedfor each blade element in the spanwise direction which conse-quently gives the tangential and normal forces acting on the rotorblade (see Fig 9)

333 Dynamic Stall Method The semi-empirical dynamicstall model called the extended ONERA is used to predict theunsteady lift drag and moment coefficients for each blade span-wise section based on 2D static airfoil data In this model theunsteady airfoil coefficients are described by a set of differentialequations including the excitation and the response variables

Fig 8 Potential load decomposition

where they are applied separately for both the attached and sepa-rated flows

In the extended ONERA model the lift (L0) and the drag (D0)forces per unit spanwise length are written as

L0 frac14 qc

2Vtot C1L thorn C2Leth THORN thorn SLc

2_W 0 thorn

D0 frac14 qc

totCDLin thornrDc

2_W 0 thorn VtotC2D

where the symbol ethTHORN q c Vtot C1L C2L W0 W1 C2D and CDLin

denote the derivation with respect to time air density blade ele-ment chord length total velocity linear circulation related to theattached flow lift nonlinear circulation related to the separatedflow lift total velocity component perpendicular to the sectional

chord rotational velocity of the blade section due to the pitchingoscillation nonlinear circulation related to the separated flowdrag and linear drag coefficient (see Fig 25) respectively Itshould be noted that different circulation terms in Eqs (22) and(23) are the circulation divided by half chord length For detaileddescription of other coefficients in Eqs (22) and (23) seeAppendix

4 Results

To validate different load calculation methods implemented inthe VLFW code the NREL 5-MW [26] and MEXICO [18]turbines are used in the simulations For the NREL 5-MWmachine in addition to the power and thrust curves the angle ofattack and tangential force along the rotor blade are studied andthey are compared with the BEM method and the GENUVP codeFor the MEXICO turbine two different steady inflow conditions(with and without yaw misalignment) are employed in the VLFWsimulations The tangential and normal forces acting on the rotorblades are compared with the existing experimental data

41 NREL-5 MW Turbine Table 1 shows the operatingconditions in which the simulations have been done for the NREL5-MW reference wind turbine Among these operating conditionsthree cases based on the low middle and high freestream veloc-ities were chosen for further studies In the vortex method simula-tions made with VLFW code the blade is discretized with Mfrac14 24spanwise sections (see Fig 10) and Nfrac14 8 equally spaced chord-wise sections Because of the large circulation gradients (dCdr)near the tip of the rotor blade the cosine rule for the blade span-wise segmentation [2] is used where the blade elements are dis-tributed at equi-angle increments in the radial direction resultingin a fine tip resolution Increments of 10 deg are employed for thewake segmentation and the wake length is truncated after fourrotor diameters [27] The free stream is assumed to be uniformsteady and perpendicular to the rotor plane The vortex core sizeis one of the most important parameters in the free vortex wakemodels which affects the vortex roll-up and wake developmentDifferent parameters may affect the vortex core radius such asoperating condition and blade radius Choosing a large vortexcore size delays the vortex roll-up On the other hand a small vor-tex core size does not significantly affect the tip vortex roll-upbut it makes the trailing wake vortices to deflect earlier which

Fig 9 Viscous load decomposition

Table 1 NREL 5-MW turbine operating conditions

Case No 1 2 3 4 5 6 7 8 9 10 11

V1 (ms) 5 6 7 8 9 10 11 12 13 14 15X (rads) 0627 0753 0878 1003 1129 1255 1267 1267 1267 1267 1267hp (deg) 00 00 00 00 00 00 00 40 665 870 1046

Fig 10 Radial distribution of blade elements for NREL-5 MW turbine

increases the wake instability For the NREL-5 MW turbine it isassumed that the wake vortex filament core radius is constant andis equal to 10 m

Figure 11 shows the angle of attack along the blade As can beseen the potential angle of attack is greater than the viscous onefor rRgt 05 which is consistent with the higher power produc-tion predicted by the potential solution The blade of theNREL-5 MW machine is constructed by different airfoil profiles[26] Computing the lift coefficient slope in the linear region(attached flow) for each airfoil profile shows that this slope islarger than the slope for the thin airfoil theory (mfrac14 2p) Equation(21) implies that the larger the lift coefficient slope (m) the lowerthe angle of attack Therefore the modification of the potentialangle of attack by coupling to the 2D airfoil data influences aero-dynamic loads and generated power

Figure 12 shows the tangential force along the blade withrespect to the rotor plane The predicted tangential force by thepotential solution is significantly larger near the blade tip makingmore power in comparison with the viscous solution The tangen-tial force calculated by the viscous solution gives larger valuesthan the potential solution near the blade root region This differ-ence between the potential and viscous solutions for the tangential

force close to the blade root increases even more for higher windvelocity where the turbine is pitch regulated to prevent the turbineoperating above the rated power

Figure 13 shows the normal force along the blade with respectto the rotor plane Contrary to the tangential force the viscoussolution predicts larger normal forces than the potential solutionThe reason for this difference is the drag force which is onlytaken into account the viscous solution Furthermore for higherwind velocity where the turbine is pitch regulated the normalforce along the blade decreases which reduces the turbinersquos thrustcoefficient (see Fig 14 (right))

The power and thrust curves are obtained by integrating the tan-gential and normal forces along the blade respectively Thereforethe comparison between the different methods was only per-formed for the power and thrust curves As seen in Fig 14 (left)there is a good quantitative agreement among the different VLFWload calculation methods For the wind velocity less than 11 mswhere the turbine is not pitch regulated the difference betweenthe potential and viscous solutions is negligible But this differ-ence increases for wind velocity higher than 11 ms (pitch regu-lated zone) showing the necessity to modify the standardpotential method using the tabulated airfoil data Moreover

Fig 11 Distribution of angle of attack along the blade for the NREL turbine (a) case 2 (b) case6 and (c) case 10 mdash potential and ndash ndash viscous

Fig 12 Distribution of tangential force along the blade for the NREL turbine (a) case 2(b) case 6 and (c) case 10 mdash potential and ndash ndash viscous

Fig 13 Distribution of normal force along the blade for the NREL turbine (a) case 2 (b) case 6and (c) case 10 mdash potential and ndash ndash viscous

Fig 14 (right) displays the thrust curve for the 5-MW NREL tur-bine By increasing the upstream flow the wind turbine thrust lin-early increases and it suddenly drops at freestream velocity of11 ms when the blade pitch angle is increased As observed thedifferent load calculation methods approximately provide similarresults but there are distinctive differences between thembecause of the unsteady environment which will be discussed inSec 42

42 MEXICO Turbine The 3-bladed MEXICO wind turbine[18] is used in the simulations The radius of the MEXICOturbinersquos blades is 205 m The simulations are performed for bothnonyawed and yawed inflow conditions Table 2 shows the operat-ing conditions at which the MEXICO measurements have beendone V1 X q hp and v denote the free stream velocityrotational velocity air density blade pitch angle and yaw mis-alignment respectively The MEXICO turbinersquos rotational direc-tion is clockwise and the positive yaw angle is defined asclockwise with respect to the radial (y) axis Moreover in theVLFW simulation the turbinersquos nacelle is not modeled

In the VLFW code the blade is discretized with Mfrac14 24 span-wise sections (see Fig 15) and Nfrac14 10 equally spaced chordwisesections The same methodology is applied for the blade wakesegmentation and vortex filament core radius as described in Sec

41 for the NREL 5-MW turbine It is assumed that the wake vor-tex filament core radius is constant and is equal to 01 m More-over for the nonyawed flow the free stream is assumed to beuniform steady and perpendicular to the rotor plane and for theyawed flow it is assumed to be uniform and steady devia-tedthorn 30 deg with respect to the rotor axis (z) For the yawed flowsimulation all coefficients in the dynamic stall model are takenaccording to the flat plat and the mean profile values (seeAppendix)

421 Nonyawed Flow For the nonyawed inflow the normaland axial forces along the rotor for three different operatingconditions (see Table 2) are compared against the MEXICOexperiment As observed in Fig 16 both the standard potentialmethod (potential solution) and the 2D static airfoil data method(viscous solution) overpredict the normal force with respect to theexperiments However there is a rather good agreement betweenthe simulation (especially for the viscous solution) and the mea-surement The difference between the potential and viscous solu-tions increases by increasing the freestream velocity This meansthat for the higher freestream velocities the viscous phenomenasuch as flow separation and stall condition increase the flow com-plexity around the blade making the potential assumption lessaccurate The discrepancy between the potential and viscous solu-tions indicates that the flow separation occurs for the blade rootregion (inboard positions) even for the lower wind velocities andit moves toward the blade tip region (outboard positions) forhigher wind velocities

Figure 17 displays the tangential force along the blade com-pared to the measurements where a qualitative agreement betweenthe simulation and measurement is seen Like the normal forcethe potential solution shows the same trend as the experimentalresults the higher the wind velocity the larger the forces actingon the rotor However including the viscosity effects using thetabulated airfoil data in the viscous solution gives a better

Fig 14 (a) power and (b) thrust curves for the NREL turbine mdash VLFW potentialndash ndash VLFW viscous ndash ndash BEM ndash 8 ndash GENUVP potential and ndash 3 ndash GENUVP viscous

Table 2 MEXICO turbine operating conditions

Item Case V1 (ms) X (rads) q (kgm3) hp (8) v (8)

No yaw 1 1001 4445 1245 23 02 1493 4445 1246 23 03 2396 4445 1236 23 0

Yaw 1 1499 4445 1237 23 30

Fig 15 Radial distribution of blade elements for MEXICO turbine

prediction compared with the potential solution As explained inRef [16] the centrifugal and Coriolis forces acting on the bound-ary layer increases the maximum lift coefficient (especially for theblade root region) and delays the flow separation and stall as wellTherefore the larger tangential force given by the experiments forthe inboard positions (compared to the viscous solution) may bedue to the centrifugal and Coriolis forces which are not consideredin the tabulated airfoil data

422 Yawed Flow For the yawed flow the aerodynamicforces normal and parallel to the local chord as a function ofazimuthal position for the five radial stations along blade 1 arecompared against measurement data These five stations arelocated at 025R 035R 060R 082R and 092R respectivelywhere R denotes the blade radius According to the MEXICOexperiment the zero rotor azimuthal angle is defined as the 12orsquoclock position (pointing upward) for blade 1 Moreover theupwind and downwind sides of the rotor plane are the 9 orsquoclockand 3 orsquoclock positions respectively

Figures 18 and 19 display azimuthal variation of the tangentialand normal forces at 025R and 035R of blade 1 The potentialsolution and the measurement data show almost the same trendalthough the VLFW method overpredicts the tangential forceabout 20 and 75 at 025R and 035R respectively about 50and 35 at 025R and 035R respectively for the normal forcerespectively The viscous solution makes a slight improvement interms of magnitude of the tangential and normal forces howeverit cannot predict the extrema especially for the tangential forceThis implies that the viscous effects (separated flow and stall con-dition) are not well captured by the viscous solution in the bladeroot region Moreover it is expected that the dynamic stall solu-tion makes adjustment between the potential and viscous solutionsin terms of the force magnitude and phasing But it seems that itmakes a slight improvement only for the blade root region(inboard positions) especially for the tangential forces

Azimuthal variation of the tangential and normal forces at 06Rof the blade 1 is presented in Fig 20 Despite a rather good trendbetween different methods and the experiment there is an

overprediction of approximately 40 and 30 for the tangentialand normal forces respectively The same tendency between thecurves implies that the flow is attached to the blade surface in themidboard region (no separation) Nevertheless the large offsetbetween the simulation and the measurement may be explained

Fig 16 Distribution of normal force along the MEXICO turbinersquos blade nonyawed flow (a)case 1 (b) case 2 and (c) case 3 mdash potential ndash ndash viscous and 8 experiment

Fig 17 Distribution of tangential force along the MEXICO turbinersquos blade nonyawed flow (a)case 1 (b) case 2 and (c) case 3 mdash potential ndash ndash viscous and 8 experiment

Fig 18 Azimuthal variation of (a) tangential and (b) normalforces at 025R of radial position MEXICO turbine yawed flowcase 1 mdash potential ndash ndash viscous ndash - ndash dynamic stall and 8experiment

due to the poor quality of the interpolated airfoil profile at 06Rlocated at the transition region between the Risoslash and NACA air-foils (see Fig 15)

Figures 21 and 22 show the tangential and normal forces at082R and 092R of the blade 1 as a function of azimuthal angleApart from a qualitative agreement between the simulations andmeasurements as discussed in the previous paragraph the sametendency between the different methods and experiments revealsthat the flow over the outboard region of the blade 1 is not sepa-rated Moreover a significant phase shift between the simulationand measurement is observed where the simulations predict thelater maximum peak than the measurement Finally the adjust-ment role of the dynamic stall model between the potential andviscous solutions is noticeable for the tangential force

5 Conclusions

A time-marching vortex lattice free wake is used for the predic-tion of aerodynamic loads on rotor blades It is based on the

inviscid incompressible and irrotational flow (potential flow)where its potential solution is coupled to the tabulated airfoil dataand a semi-empirical model to take into account the viscosity andthe dynamic stall effects respectively Three different methodscalled the standard potential method the 2D static airfoil datamethod and the dynamic stall method are introduced and they arecompared with the BEM method and the GENUVP code

The results show that for more accurate load and powerprediction coupling to the 2D static airfoil data is necessaryeven though some complex conditions such as separatedflow stall condition and centrifugal forces cannot be wellpredicted

Furthermore the predicted forces using the dynamic stall solu-tion do not make considerable improvements with respect to theother load calculation methods This may be because of the sev-eral airfoil-dependent coefficients used in the extended ONERAmodel while in case of no wind tunnel measurements they aretaken from the flat plate and mean airfoil

The predicted power production by different methods forthe NREL 5-MW turbine shows that the potential inviscid andirrotational assumptions of the vortex flow are relevant to a broadrange of operating conditions The VLFW method predicts highernormal and tangential forces compared with the MEXICO experi-ment For the normal forces the maximum peak occurs downwindof the rotor plane whereas it occurs upwind of the rotor planewhen moving from inboard sections toward the outboard sectionsThe difference between the maximum peak positions along therotor blade with respect to the rotor plane induces an additionalmoment on the rotor due to the yaw misalignment Furthermorefor almost all spanwise sections the simulation presents a phaseshift against the experiments for both the normal and tangentialforces nevertheless it predicts the azimuthal load variation ratherwell

A considerable discrepancy between the simulations and mea-surement data for the MEXICO turbine close to the blade root(inboard sections) may be physically explained due to the thickairfoil profiles which consequently results in the flow separationand stall condition even if at lower wind velocities This is alsocertified for the NREL 5-MW machine

According to Refs [2829] a deficiency in the streamwisevelocity (around 1 ms) due to open type wind tunnel employed inthe experimental investigation has been reported It is expectedthat the predicted forces by the VLFW simulation are slightlyimproved by taking the tunnel effect into account

Acknowledgment

The technical support of National Technical University ofAthens (NTUA) to use the GENUVP is gratefully acknowledged(GENUVP is an unsteady flow solver based on vortex blobapproximations developed for rotor systems by National Techni-cal University of Athens)

The data used have been supplied by the consortium which car-ried out the EU FP5 project MEXICO ldquoModel rotor EXperimentsin COntrolled conditionsrdquo

This work was financed through the Swedish Wind PowerTechnology Centre (SWPTC) SWPTC is a research center for thedesign and production of wind turbines The purpose of the centeris to support Swedish industry with knowledge of design techni-ques and maintenance in the field of wind power The work is car-ried out in six theme groups and is funded by the Swedish EnergyAgency and by academic and industrial partners

Appendix Extended ONERA Model

The extended ONERA model is used to predict the unsteadylift drag and moment coefficients based on 2D static airfoil dataIn the initial version of the ONERA model the excitation variable

is the angle of attack with respect to the chord line whereas inthe extended version the excitation variables are W0 and W1 thevelocity component perpendicular to the sectional chord and theblade element angular velocity for the pitching oscillation respec-tively Furthermore compared with the initial version of theONERA model in the extended model instead of the lift coeffi-cient (CL) the circulation (C) which is responsible for producinglift is the response variable Also the variation of the wind veloc-ity is included in the extended model which does not exist in theearly version [17]

In steady flow when the angle of attack for some blade regionsexceeds from the critical angle of attack (astall) which is equiva-lent to the maximum lift coefficient (CLmax) the flow is separatedThis phenomenon is called static stall This phenomenon for anairfoil in an unsteady flow is associated with so-called dynamicstall where its major effect is stall delay and an excessive force(see Fig 23) In other words when an airfoil or a lifting surface isexposed to time-varying pitching plunging and incident velocitythe stall condition happens at an angle of attack higher than thestatic stall angle which means that the flow separates at a higherangle of attack than in steady flow When stall occurs there is asudden decrease in lift By decreasing the angle of attack the flowreattaches again (stall recovery) but at a lower angle than thestatic stall angle [30] This scenario which is called dynamic stalloccurs around the stall angle and the result is hysteresis loops anda sudden decrease of the lift coefficient Hence the dynamic stalldescribes a series of event resulting in dynamic delay of stall toangles above the static stall angle and it provides the unsteadyevolution of lift drag and moment coefficients along the rotorblade

In Eqs (22) and (23) SL KL and rD are airfoil dependent coef-ficients However in case of no wind tunnel measurement datathe flat plate values are applied as SLfrac14 p and KLfrac14p2 for smallMach number The term rD is expressed by

rD frac14 r0Dathorn r1DjDCLj (A1)

where for the flat plate r0Dfrac14 0 and r1Dfrac14 0 MoreoverDCLfrac14CLLinCLStat where the Lin and Stat subscripts representthe linear region and the static condition respectively(see Figs 24 and 25) The linear circulation concerning theattached flow lift (C1L) is calculated by the first-order differentialequation as

_C1L thorn kL2Vp

frac14 kL2Vp

W0 Vpa0eth THORN thorn kL2Vp

thorn aLdCL

thorn dL

_W 0 thorn aLrL

_W 1 (A2)

where Vp dCLda and a0 are the total velocity component parallelto the airfoil chord slope of the CL versus a curve in the linearregion and the zero-lift angle of attack of each blade elementrespectively

The nonlinear circulation concerning the stall correctionof lift (C2L) is calculated by the second-order differentialequation as

Fig 23 Hysteresis loop around the stall angle

Fig 24 Definition of the lift coefficient parameters in theONERA model

Fig 25 Definition of the drag coefficient parameters in theONERA model

euroC2L thorn aL2Vp

c_C2L thorn rL

frac14 rL2Vp

VpDCLeL2Vp

c_W 0 (A3)

Furthermore the nonlinear circulation concerning the stallcorrection of drag (C2L) is given by the second-order differentialequation as

euroC2D thorn aD2V

c_C2D thorn rD

frac14 rD2V

VDCDeD2V

c_W 0 (A4)

In Eqs (22) (A2) (A3) and (A4) the symbol ethTHORN denotes the deri-vation with respect to time

In the above equations kL rL and aL depend on the specificairfoil type and they must be determined from experimental meas-urements If experimental data for a particular airfoil are not avail-able these coefficients take the flat plate values as kLfrac14 017rLfrac14 2p aLfrac14 053 dL in Eq (A2) and the coefficients inEqs (A3) and (A4) are functions of DCL due to the flow separa-tion and they are defined as

dL frac14 r1LjDCLjaL frac14 a0L thorn a2LethDCLTHORN2

aD frac14 a0D thorn a2DethDCLTHORN2ffiffiffiffirLp frac14 r0L thorn r2LethDCLTHORN2ffiffiffiffiffi

rDp frac14 r0D thorn r2DethDCLTHORN2

eL frac14 e2LethDCLTHORN2

eD frac14 e2DethDCLTHORN2

The coefficients in Eq (A5) are airfoil dependent In case of nowind tunnel measurements the values for a mean airfoil may betaken and the flat plate values cannot be used For the mean air-foil r1Lfrac14 00 a0Lfrac14 01 a2Lfrac14 00 r0Lfrac14 01 r2Lfrac14 00 e2Lfrac14 00a0Dfrac14 00 a2Dfrac14 00 r0Dfrac14 01 r2Dfrac14 00 and e2Dfrac14 00

References[1] Hansen M O 2008 Aerodynamics of Wind Turbines 2nd ed EarthScan

London[2] van Garrel A 2003 ldquoDevelopment of a Wind Turbine Aerodynamics Simula-

tion Modulerdquo Energy Research Centre of the Netherlands (ECN) Petten TheNetherlands Report No ECN-Cndash03-079

[3] Vermeer L Soslashrensen J and Crespo A 2003 ldquoWind Turbine Wake Aero-dynamicsrdquo Prog Aerosp Sci 39(6ndash7) pp 467ndash510

[4] Opoku D G Triantos D G Nitzsche F and Voutsinas S G 2002ldquoRotorcraft Aerodynamic and Aeroacoustic Modeling Using Vortex ParticleMethodsrdquo 23rd International Congress of the Aeronautical Sciences (ICAS)Toronto ON Canada Sept 8ndash13 Paper No ICAS 2002-213

[5] Voutsinas S G Beleiss M A and Rados K G 1995 ldquoInvestigation of theYawed Operation of Wind Turbines by Means of a Vortex Particle MethodrdquoAGARD Conference Proceedings Vol 552 pp 111ndash11

[6] Zhao J and He C 2010 ldquoA Viscous Vortex Particle Model for Rotor Wakeand Interference Analysisrdquo J Am Helicopter Soc 55(1) p 12007

[7] Egolf T A 1988 ldquoHelicopter Free Wake Prediction of Complex Wake Struc-tures Under Blade-Vortex Interaction Operating Conditionsrdquo 44th AnnualForum of the American Helicopter Society Washington DC June 16ndash18 pp819ndash832

[8] Rosen A and Graber A 1988 ldquoFree Wake Model of Hovering RotorsHaving Straight or Curved Bladesrdquo J Am Helicopter Soc 33(3) pp 11ndash21

[9] Bagai A 1995 ldquoContribution to the Mathematical Modeling of Rotor Flow-Fields Using a Pseudo-Implicit Free Wake Analysisrdquo PhD thesis Universityof Maryland College Park MD

[10] Gupta S 2006 ldquoDevelopment of a Time-Accurate Viscous Lagrangian VortexWake Model for Wind Turbine Applicationsrdquo PhD thesis University of Mary-land College Park MD

[11] Pesmajoglou S and Graham J 2000 ldquoPrediction of Aerodynamic Forces onHorizontal Axis Wind Turbines in Free Yaw and Turbulencerdquo J Wind EngInd Aerodyn 86(1) pp 1ndash14

[12] Voutsinas S 2006 ldquoVortex Methods in Aeronautics How To Make ThingsWorkrdquo Int J Comput Fluid Dyn 20(1) pp 3ndash18

[13] Chattot J 2007 ldquoHelicoidal Vortex Model For Wind Turbine Aeroelastic Sim-ulationrdquo Comput Struct 85(11ndash14) pp 1072ndash1079

[14] Chattot J 2003 ldquoOptimization of Wind Turbines Using Helicoidal VortexModelrdquo ASME J Sol Energy Eng 125(4) pp 418ndash424

[15] Holierhoek J de Vaal J van Zuijlen A and Bijl H 2013 ldquoComparing Dif-ferent Dynamic Stall Modelsrdquo J Wind Energy 16(1) pp 139ndash158

[16] Leishman J 2002 ldquoChallenges in Modeling the Unsteady Aerodynamics ofWind Turbinesrdquo AIAA Paper No 2002-0037

[17] Bierbooms W 1992 ldquoA Comparison Between Unsteady Aerodynamic Mod-elsrdquo J Wind Eng Ind Aerodyn 39(1ndash3) pp 23ndash33

[18] Schepers J and Boorsma K 2006 ldquoDescription of Experimental SetupMEXICO Measurementsrdquo Energy Research Centre of the Netherlands (ECN)Petten The Netherlands Report No ECN-Xndash09-0XX

[19] Anderson J 2001 Fundamentals of Aerodynamics 3rd ed McGraw-HillNew York

[20] van Garrel A 2001 ldquoRequirements for a Wind Turbine Aerodynamics Simu-lation Modulerdquo 1st ed Energy Research Centre of the Netherlands (ECN) Pet-ten The Netherlands Report No ECN-Cndash01-099

[21] Leishman J and Bagai M J 2002 ldquoFree Vortex Filament Methods for theAnalysis of Helicopter Rotor Wakesrdquo J Aircr 39(5) pp 759ndash775

[22] Bhagwat M and Leishman J 2002 ldquoGeneralized Viscous Vortex Model forApplication to Free-Vortex Wake and Aeroacoustic Calculationsrdquo 58th AnnualForum and Technology Display of the American Helicopter Society Interna-tional Montreal Canada June 11ndash13 pp 2042ndash2057

[23] Vasitas G Kozel V and Mih W 1991 ldquoA Simpler Model for ConcentratedVorticesrdquo Exp Fluids 11(1) pp 73ndash76

[24] Bagai A and Leishman J 1993 ldquoFlow Visualization of Compressible VortexStructures Using Density Gradient Techniquesrdquo Exp Fluids 15(6)pp 431ndash442

[25] Katz J and Plotkin A 2001 Low-Speed Aerodynamics 2nd ed CambridgeUniversity Press New York

[26] Jonkman J Butterfield S Musial W and Scott G 2009 ldquoDefinition of a5-MW Reference Wind Turbine for Offshore System Developmentrdquo NationalRenewable Energy Laboratory Golden CO Report No NRELTP-500-38060

[27] Abedi H 2013 ldquoDevelopment of Vortex Filament Method for AerodynamicLoads on Rotor Bladesrdquo Licentiate thesis Chalmers University of TechnologyGothenburg Sweden

[28] Chasapogiannis P and Voutsinas S 2014 ldquoAerodynamic Simulations of theFlow Around a Horizontal Axis Wind Turbine Using the GAST SoftwarerdquoNational Technical University of Athens Athens Greece Ref No 681110(in Greek)

[29] Abedi H Davidson L and Voutsinas S 2015 ldquoNumerical Studies of theUpstream Flow Field Around a Horizontal Axis Wind Turbinerdquo AIAA PaperNo 2015-0495

[30] Reddy T S R and Kaza K R V 1989 ldquoAnalysis of an Unswept PropfanBlade With a Semi Empirical Dynamic Stall Modelrdquo NASA Lewis ResearchCenter Cleveland OH Report No NASA-TM-4083

be high (rapid transient loads) the quasi-static aerodynamic is nolonger valid [1617] As a consequence a dynamic approach mustbe introduced to modify the aerodynamic coefficients for unsteadyoperating conditions This approach which is called dynamicstall adjusts the lift the drag and the moment coefficients foreach blade element on the basis of the 2D static airfoil datatogether with the correction for separated flow Furthermorebecause of the dynamic stall the predicted aerodynamic coeffi-cients may result in noticeable differences [16] in comparisonwith the static ones A dynamic stall model for the unsteady aero-dynamic loads prediction is therefore crucial for the wind turbinetechnology development

In this paper an in-house time-marching VLFW code is usedfor the simulation where its potential solution is coupled to thetabulated airfoil data for the wind turbine load calculation Inaddition a semi-empirical model called extended ONERAmodel is added to account for the dynamic stall effects Theresults of three different load calculation methods namely stand-ard potential method (potential solution) 2D static airfoil datamodel (viscous solution) and the dynamic stall model are com-pared with the BEM method [1] the GENUVP code1 and theMEXICO [18] wind tunnel measurements2

2 Theory

Vortex flow theory is based on assuming incompressible(r V frac14 0) and irrotational (r V frac14 0) flow at every pointexcept at the origin of the vortex where the velocity is infinite[19] A region containing a concentrated amount of vorticity iscalled a vortex where a vortex line is defined as a line whose tan-gent is parallel to the local vorticity vector everywhere Vortexlines surrounded by a given closed curve make a vortex tube witha strength equal to the circulation C A vortex filament with astrength of C is represented as a vortex tube of an infinitesimalcross section with strength C

According to the Helmholtz theorem an irrotational motion ofan inviscid fluid which started from rest remains irrotationalAlso a vortex line cannot end in the fluid It must form a closedpath end at a solid boundary or go to infinity this implies thatvorticity can only be generated at solid boundaries Therefore asolid surface may be considered as a source of vorticity Hencethe solid surface in contact with fluid is replaced by a distributionof vorticity

For an irrotational flow a velocity potential U can be definedas V frac14 rU where in order to find the velocity field the Laplacersquosequation r2Ufrac14 0 is solved using a proper boundary conditionfor the velocity on the body and at infinity In addition in vortextheory the vortical structure of a wake can be modeled by eithervortex filaments or vortex particles where a vortex filament ismodeled as concentrated vortices along an axis with a singularityat the center

The velocity induced by a straight vortex filament can be deter-mined by the BiotndashSavart law as

Vind frac14C4p

jrj3(1)

which can also be written as

Vind frac14C4p

where C denotes the strength of the vortex filament r1 and r2 arethe distance vectors from the beginning A and end B of a vortexsegment to an arbitrary point C respectively (see Fig 1)

The BiotndashSavart law has a singularity when the point of evalua-tion (C) of induced velocity is located on the vortex filament axis(L) Also when the evaluation point is very near to the vortex fila-ment there is an unphysically large induced velocity at that pointThe remedy is either to use a cut-off radius d [20] or to use a vis-cous vortex model with a finite core size by multiplying a factorto remove the singularity [21]

The BiotndashSavart law correction based on the viscous vortexmodel can be made by introducing a finite core size rc for a vor-tex filament [22] In this paper for simplicity a constant viscouscore size model which is one of the general approaches usingdesingularized algebraic profile is employed for the inducedvelocity calculations A general form of a desingularized algebraicswirl-velocity profile for stationary vortices is proposed byVasitas et al [23] as

Vh reth THORN frac14 C2pr

r2nc thorn r2n

where r and n are the distance of a vortex segment to an arbitrarypoint and an integer number respectively

Bagai and Leishman [24] suggested the velocity profile basedon Eq (3) for nfrac14 2 for rotor tip vortices Therefore in order totake into account the effect of the viscous vortex core a factor ofKv must be added to the BiotndashSavart law as [24]

Vind frac14 KvC4p

Kv frac14h2

r2nc thorn h2n

1=n(5)

and h is defined as the perpendicular distance of the evaluationpoint (see Fig 1) Factor Kv desingularizes the BiotndashSavart equa-tion when the evaluation point distance tends to zero and preventsa high induced velocity in the vicinity region of the vortex coreradius

3 Model

31 Assumptions In this study as a first effort to evaluatethe load calculation methods (see Sec 33) the upstream flow isset to be uniform both in time and space For the nonyawed flowit is perpendicular to the rotor plane whereas for the yawed flow it

Fig 1 Schematic for the BiotndashSavart law

1GENUVP is an unsteady flow solver based on vortex blob approximationsdeveloped for rotor systems by National Technical University of Athens

2The MEXICO wind turbine measurements were carried out in 2006 in the LargeScale Low Speed Facility (LLF) of the German Dutch Wind Tunnel (DNW)

a11 a12 a1m

a21 a22 a2m

am1 am2 amm

1CCA frac14

1CCA (7)

Lj frac14XN

ifrac141

Lij (16)

CL frac14L0

05qV2totc

a frac14 2CmVtotc

thorn a0 (21)

L0 frac14 qc

2_W 0 thorn

D0 frac14 qc

totCDLin thornrDc

2_W 0 thorn VtotC2D

4 Results

Case No 1 2 3 4 5 6 7 8 9 10 11

V1 (ms) 5 6 7 8 9 10 11 12 13 14 15X (rads) 0627 0753 0878 1003 1129 1255 1267 1267 1267 1267 1267hp (deg) 00 00 00 00 00 00 00 40 665 870 1046

No yaw 1 1001 4445 1245 23 02 1493 4445 1246 23 03 2396 4445 1236 23 0

Yaw 1 1499 4445 1237 23 30

5 Conclusions

Acknowledgment

_C1L thorn kL2Vp

frac14 kL2Vp

thorn aLdCL

thorn dL

_W 0 thorn aLrL

_W 1 (A2)

euroC2L thorn aL2Vp

c_C2L thorn rL

frac14 rL2Vp

VpDCLeL2Vp

c_W 0 (A3)

euroC2D thorn aD2V

c_C2D thorn rD

frac14 rD2V

VDCDeD2V

c_W 0 (A4)

a11 a12 a1m

a21 a22 a2m

am1 am2 amm

1CCA frac14

1CCA (7)

Lj frac14XN

ifrac141

Lij (16)

CL frac14L0

05qV2totc

a frac14 2CmVtotc

thorn a0 (21)

L0 frac14 qc

2_W 0 thorn

D0 frac14 qc

totCDLin thornrDc

2_W 0 thorn VtotC2D

4 Results

Case No 1 2 3 4 5 6 7 8 9 10 11

V1 (ms) 5 6 7 8 9 10 11 12 13 14 15X (rads) 0627 0753 0878 1003 1129 1255 1267 1267 1267 1267 1267hp (deg) 00 00 00 00 00 00 00 40 665 870 1046

No yaw 1 1001 4445 1245 23 02 1493 4445 1246 23 03 2396 4445 1236 23 0

Yaw 1 1499 4445 1237 23 30

5 Conclusions

Acknowledgment

_C1L thorn kL2Vp

frac14 kL2Vp

thorn aLdCL

thorn dL

_W 0 thorn aLrL

_W 1 (A2)

euroC2L thorn aL2Vp

c_C2L thorn rL

frac14 rL2Vp

VpDCLeL2Vp

c_W 0 (A3)

euroC2D thorn aD2V

c_C2D thorn rD

frac14 rD2V

VDCDeD2V

c_W 0 (A4)

Lj frac14XN

ifrac141

Lij (16)

CL frac14L0

05qV2totc

a frac14 2CmVtotc

thorn a0 (21)

L0 frac14 qc

2_W 0 thorn

D0 frac14 qc

totCDLin thornrDc

2_W 0 thorn VtotC2D

4 Results

Case No 1 2 3 4 5 6 7 8 9 10 11

V1 (ms) 5 6 7 8 9 10 11 12 13 14 15X (rads) 0627 0753 0878 1003 1129 1255 1267 1267 1267 1267 1267hp (deg) 00 00 00 00 00 00 00 40 665 870 1046

No yaw 1 1001 4445 1245 23 02 1493 4445 1246 23 03 2396 4445 1236 23 0

Yaw 1 1499 4445 1237 23 30

5 Conclusions

Acknowledgment

_C1L thorn kL2Vp

frac14 kL2Vp

thorn aLdCL

thorn dL

_W 0 thorn aLrL

_W 1 (A2)

euroC2L thorn aL2Vp

c_C2L thorn rL

frac14 rL2Vp

VpDCLeL2Vp

c_W 0 (A3)

euroC2D thorn aD2V

c_C2D thorn rD

frac14 rD2V

VDCDeD2V

c_W 0 (A4)

Lj frac14XN

ifrac141

Lij (16)

CL frac14L0

05qV2totc

a frac14 2CmVtotc

thorn a0 (21)

L0 frac14 qc

2_W 0 thorn

D0 frac14 qc

totCDLin thornrDc

2_W 0 thorn VtotC2D

4 Results

Case No 1 2 3 4 5 6 7 8 9 10 11

V1 (ms) 5 6 7 8 9 10 11 12 13 14 15X (rads) 0627 0753 0878 1003 1129 1255 1267 1267 1267 1267 1267hp (deg) 00 00 00 00 00 00 00 40 665 870 1046

No yaw 1 1001 4445 1245 23 02 1493 4445 1246 23 03 2396 4445 1236 23 0

Yaw 1 1499 4445 1237 23 30

5 Conclusions

Acknowledgment

_C1L thorn kL2Vp

frac14 kL2Vp

thorn aLdCL

thorn dL

_W 0 thorn aLrL

_W 1 (A2)

euroC2L thorn aL2Vp

c_C2L thorn rL

frac14 rL2Vp

VpDCLeL2Vp

c_W 0 (A3)

euroC2D thorn aD2V

c_C2D thorn rD

frac14 rD2V

VDCDeD2V

c_W 0 (A4)

L0 frac14 qc

2_W 0 thorn

D0 frac14 qc

totCDLin thornrDc

2_W 0 thorn VtotC2D

4 Results

Case No 1 2 3 4 5 6 7 8 9 10 11

V1 (ms) 5 6 7 8 9 10 11 12 13 14 15X (rads) 0627 0753 0878 1003 1129 1255 1267 1267 1267 1267 1267hp (deg) 00 00 00 00 00 00 00 40 665 870 1046

No yaw 1 1001 4445 1245 23 02 1493 4445 1246 23 03 2396 4445 1236 23 0

Yaw 1 1499 4445 1237 23 30

5 Conclusions

Acknowledgment

_C1L thorn kL2Vp

frac14 kL2Vp

thorn aLdCL

thorn dL

_W 0 thorn aLrL

_W 1 (A2)

euroC2L thorn aL2Vp

c_C2L thorn rL

frac14 rL2Vp

VpDCLeL2Vp

c_W 0 (A3)

euroC2D thorn aD2V

c_C2D thorn rD

frac14 rD2V

VDCDeD2V

c_W 0 (A4)

No yaw 1 1001 4445 1245 23 02 1493 4445 1246 23 03 2396 4445 1236 23 0

Yaw 1 1499 4445 1237 23 30

5 Conclusions

Acknowledgment

_C1L thorn kL2Vp

frac14 kL2Vp

thorn aLdCL

thorn dL

_W 0 thorn aLrL

_W 1 (A2)

euroC2L thorn aL2Vp

c_C2L thorn rL

frac14 rL2Vp

VpDCLeL2Vp

c_W 0 (A3)

euroC2D thorn aD2V

c_C2D thorn rD

frac14 rD2V

VDCDeD2V

c_W 0 (A4)

No yaw 1 1001 4445 1245 23 02 1493 4445 1246 23 03 2396 4445 1236 23 0

Yaw 1 1499 4445 1237 23 30

5 Conclusions

Acknowledgment

_C1L thorn kL2Vp

frac14 kL2Vp

thorn aLdCL

thorn dL

_W 0 thorn aLrL

_W 1 (A2)

euroC2L thorn aL2Vp

c_C2L thorn rL

frac14 rL2Vp

VpDCLeL2Vp

c_W 0 (A3)

euroC2D thorn aD2V

c_C2D thorn rD

frac14 rD2V

VDCDeD2V

c_W 0 (A4)

5 Conclusions

Acknowledgment

_C1L thorn kL2Vp

frac14 kL2Vp

thorn aLdCL

thorn dL

_W 0 thorn aLrL

_W 1 (A2)

euroC2L thorn aL2Vp

c_C2L thorn rL

frac14 rL2Vp

VpDCLeL2Vp

c_W 0 (A3)

euroC2D thorn aD2V

c_C2D thorn rD

frac14 rD2V

VDCDeD2V

c_W 0 (A4)

5 Conclusions

Acknowledgment

_C1L thorn kL2Vp

frac14 kL2Vp

thorn aLdCL

thorn dL

_W 0 thorn aLrL

_W 1 (A2)

euroC2L thorn aL2Vp

c_C2L thorn rL

frac14 rL2Vp

VpDCLeL2Vp

c_W 0 (A3)

euroC2D thorn aD2V

c_C2D thorn rD

frac14 rD2V

VDCDeD2V

c_W 0 (A4)

_C1L thorn kL2Vp

frac14 kL2Vp

thorn aLdCL

thorn dL

_W 0 thorn aLrL

_W 1 (A2)

euroC2L thorn aL2Vp

c_C2L thorn rL

frac14 rL2Vp

VpDCLeL2Vp

c_W 0 (A3)

euroC2D thorn aD2V

c_C2D thorn rD

frac14 rD2V

VDCDeD2V

c_W 0 (A4)

euroC2L thorn aL2Vp

c_C2L thorn rL

frac14 rL2Vp

VpDCLeL2Vp

c_W 0 (A3)

euroC2D thorn aD2V

c_C2D thorn rD

frac14 rD2V

VDCDeD2V

c_W 0 (A4)

Documents

Enhancement of Free Vortex Filament Method for ...publications.lib.chalmers.se/records/fulltext/249148/...for Aerodynamic Loads on Rotor Blades The aerodynamics of a wind turbine is