Gaunaa, Mac

Guntur, Srinivas

Troldborg, Niels

A simple analytical rotor model based on vortex theory is briefly presented and used to investigate the main mechanisms for wind turbine rotors operating at yaw misalignment. The overall findings of the model is verified by comparing with an existing model as well as with results obtained using the actuator disc technique combined with full Navier- Stokes computations

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Downloaded from orbit.dtu.dk on: Dec 20, 2017Modelling the influence of yaw using a simple vortex rotor modelTroldborg, Niels; Gaunaa, Mac; Guntur, SrinivasPublication date:2012Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Troldborg, N., Gaunaa, M., & Guntur, S. (2012). Modelling the influence of yaw using a simple vortex rotormodel. Poster session presented at EWEA 2012 - European Wind Energy Conference & Exhibition,Copenhagen, Denmark.1. Glauert, H. The elements of aerofoil and airscrew theory, second edition. Cambridge University Press, Cambridge, England, 1947. 2. Øye, S. A simple vortex model. In Proc. of the third IEA Symposium on the Aerodynamics of Wind Turbines, ETSU, Harwell, 1990, p. 4.1-4.15. 3. Mikkelsen, R. Actuator Disc Methods Applied to Wind Turbines, MEK-FM-PHD 2003-02, Technical University of Denmark, 2003.  The objective of the present work is to develop a simple model for including the effect of yaw misalignment on induced velocities and power production of a wind turbine  and to verify it by comparing it to the model by Glauert [1] as well as to Navier-Stokes actuator disc computations.  A simple analytical rotor model based on vortex theory is presented and used to investigate the main mechanisms for wind turbine rotors operating at yaw misalignment. The overall findings of the model is verified by comparing with an existing model as well as with results obtained using the actuator disc technique combined with full Navier-Stokes computations. The proposed vortex model is shown to give predictions of all induction components and power performance, which are in close agreement with actuator disc simulations at low to medium loads and over a wide range of yaw angles. However, for higher loads and yaw angles the vortex model is less accurate. In these cases the model is outperformed by the model of Glauert, which is shown to be accurate for all load conditions in terms of integral values. However, unlike the proposed vortex model, the model of Glauert does not provide the inductions in the radial and tangential directions and cannot predict the nonlinear behavior of the induced velocity on the outer part of the disc. Abstract Modelling the influence of yaw using a simple vortex rotor model Niels Troldborg, Mac Gaunaa, Srinivas Guntur Department of Wind Energy, DTU Wind Energy, Frederiksborgvej 399, P.O. Box 49, 4000 Roskilde  PO. ID 243 Results Objectives Conclusions Methods References                EWEA 2012, Copenhagen, Denmark: Europe’s Premier Wind Energy Event   A simple vortex model for yawed rotors: The proposed vortex model is a generalization of the model originally developed by Øye [2]. The basic assumptions and governing equations behind the model is outlined below. Basic assumptions: • Potential flow (i.e. incompressible, irrotational and inviscid) • Steady and uniform inflow • Rotational effects neglected (infinite rotational speed) • Wake expansion neglected • Transport velocity of vorticity sheet is (see Figure 1): Vsheet=½(V1+V2)      Governing equations: The velocity field is everywhere governed by Biot Savart’s law while the local forces per unit span on the blades are determined using Joukowski’s relation:  where dΓ is the local bound circulation, which is aligned with the blade, Vrel is the local velocity relative to the blade and ρ denotes density. Using these relations together with the above assumptions the following  can be shown:     where Fn signifies the local force per unit area in the normal direction and V∞ denotes the free stream velocity. Γ signifies the total bound circulation on the rotor, ψ is the yaw angle and r and θ is the radial and tangential position where the forces are evaluated. Wn denotes the induced velocity in the normal direction at the rotor disc.   Navier Stokes actuator disc model: The used actuator disc model combines the Navier-Stokes flow solver EllipSys3D with an actuator disc technique where body forces are distributed on a disc representing a wind turbine rotor [3]. The applied disc forces are distributed smoothly on several mesh point to avoid singular behavior. The steady state flow field is solved in a Cartesian computational domain with 18.9·106 number of grid points. ΓdVdF rel ×= ρ2221,∞∞ΓΩ==VVFC nlocT πρ+=∞VrWCC nlocTlocP),(cos,,θψFigure 2 compares actuator disc (AD) and vortex model (VM) predictions of CP,loc and an = cos(ψ) −Wn(r, θ)/V∞, respectively for CT = 0.64 and ψ = 30°. Figure 3 shows a comparison of the predicted inductions in the radial and tangential directions in the same case as in Figure 2. Figure 2: Local power coefficient (left) and normal induction in the rotor plane (right) predicted using the simple vortex model and actuator disc model, respectively. The thrust coefficient and yaw angle is CT = 0.64 and ψ = 30°. Figure 3: Radial (left) and tangential (right) induction in the rotor plane predicted using the simple vortex model and actuator disc model, respectively. The thrust coefficient and yaw angle is CT = 0.64 and ψ = 30°. The color and symbol code in the left plot is as in Figure 2. Figure 4 shows the global power coefficient as a function of yaw angle predicted by the vortex model and Glauert’s model, respectively in comparison with actuator disc results. Figure 4: Power coefficient as a function of yaw angle for various thrust coefficients. Full lines are AD predictions and symbols are predictions by the vortex model (left) and Glauert’s model (right).  Figure 1: Sketch of vorticity sheet of strength γ=V2-V1 

Modelling the influence of yaw using a simple vortex rotor model

                       General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.  • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal   If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.     Downloaded from orbit.dtu.dk on: Dec 20, 2017Modelling the influence of yaw using a simple vortex rotor modelTroldborg, Niels; Gaunaa, Mac; Guntur, SrinivasPublished in:Proceedings of EWEA 2012 - European Wind Energy Conference &amp; ExhibitionPublication date:2012Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Troldborg, N., Gaunaa, M., & Guntur, S. (2012). Modelling the influence of yaw using a simple vortex rotormodel. In Proceedings of EWEA 2012 - European Wind Energy Conference & Exhibition European Wind EnergyAssociation (EWEA).Modelling the influence of yaw using a simple vortex rotormodelNiels Troldborg∗ Mac Gaunaa∗ and Srinivas Guntur ∗∗DTU Wind Energy, Department of Wind Energy, Technical University of Denmark, Risø Campus, DK-4000Roskilde, Denmarkniet@risoe.dtu.dkAbstractA simple analytical rotor model based on vortextheory is brieﬂy presented and used to investigate themain mechanisms for wind turbine rotors operatingat yaw misalignment. The overall ﬁndings of themodel is veriﬁed by comparing with an existingmodel as well as with results obtained using theactuator disc technique combined with full Navier-Stokes computations.Keywords: Yawed rotor, vortex model, actuator disc1 IntroductionWind turbines operate in an unsteady environmentwhere the wind direction changes frequently andtherefore they may operate in yaw for longer peri-ods of time. A yaw misaligned wind turbine experi-ences cyclic loads which reduces its fatigue life timeand therefore the inﬂuence of yaw on turbine loadsneeds to be understood and modelled properly. Forthis reason the eﬀects of yaw misalignment on rotorloads have been studied extensively, both experimen-tally [1] and numerically [2, 6]. However, in a recentstudy [7] it was revealed that state-of-the-art engi-neering models attempting to include the eﬀect of yawmisalignment on wind turbine performance and loadsstill show important discrepancies when compared tomeasurements.In this paper we employ a simple analytical rotormodel based on vortex theory to investigate and ex-plain the main mechanisms for yawed wind turbinerotors. The overall ﬁndings of the model is veriﬁedby comparing both with a model by Glauert [3, 4, 5]and with results obtained using the actuator disctechnique combined with full Navier-Stokes compu-tations.2 Glauert’s model for yawed ro-torsFor a rotor operating at an angle ψ relative to asteady uniform inﬂow Glauert [3, 4, 5] showed thefollowing relation between the thrust coeﬃcient CTand the normal induction factor a = −Wn/V0:CT = 4a√sinψ2 + (cosψ − a)2 (1)Here Wn is the induced velocity in the direction nor-mal to disc, V0 is the free-stream velocity andCT =T0.5ρAV 20(2)where T is the trust on the disc and A is the disc area.In equation 1 T and a are averages over the disc. Inorder to describe the induced velocities over a yawmisaligned rotor disc Glauert proposed the followingmodel [3]an(r, θ) = a (1 + (r/R) tan (χ/2) cos (θ − θ0)) (3)where an(r, θ) is the local induction factor, a is themean induced induction factor found from equation 1,χ is the wake skew angle deﬁned as the angle betweenthe velocity in the wake and the rotational axis of therotor. θ is the azimuthal position and θ0 is the anglewhere the blade is deepest into the wake. The wakeskew angle is assumed to be constant with the radiusand is in the present case computed ascosχ =cosψ − a√(1− a cosψ)2 + a2 sin2 ψ(4)Having computed an(r, θ) the local power coeﬃcientfor a constantly loaded rotor is determined as:CP,loc = CT (cos(ψ)− an(r, θ)) (5)3 A simple vortex model foryawed rotorsThe proposed vortex model is a generalization of themodel originally developed by Øye [8, 9] where theidealized wind turbine is modelled with an inﬁnitenumber of blades of constant circulation. In orderto further simplify the analysis we assume inﬁnitelyhigh tip speed ratio, such that rotational eﬀects inthe wake are avoided, as in classical 1D momentumtheory [10]. Moreover, we assume steady uniformﬂow and neglect the expansion of the wake. Besidesthese assumptions the present model uses the usualpotential-ﬂow assumptions, which are an incompress-ible, irrotational and inviscid ﬂow corresponding tovery high Reynolds numbers and low mach-numbers.Eﬀects of drag on rotor performance are not includedin the model, so that the local forces per unit spanon the blades can be determined using Joukowski’srelation.dF = ρV rel × dΓ (6)where dΓ is the local bound circulation, which isaligned with the blade, V rel is the local velocity rel-ative to the blade and ρ denotes density. Using thisrelation together with the above assumptions the fol-lowing can be shown:CT,loc =F˜n0.5ρV 20=ΓΩπV 20(7)CP,loc = CT,loc (cos(ψ) + Wn(r, θ)/V0) (8)where F˜n signify the local force per unit area in theand normal direction and V0 denotes the free streamvelocity. Γ signiﬁes the total bound circulation onthe rotor, ψ is the yaw angle and r and θ is the radialand tangential position where the forces are evalu-ated. Wn is the induced velocity in the normal direc-tion at the rotor disc.The fact that the local thrust coeﬃcient is constant(equation 7) is a consequence of the assumption of avery high tip speed ratio. From this assumption itnamely follows, that the free-stream velocity compo-nent relative to the spinning rotor is negligible, yield-ing that a rotor with a constant circulation corre-sponds to a rotor with constant local thrust coeﬃ-cient.4 The Navier-Stokes actuatordisc modelThe actuator disc model used in the present workis described in [12, 16] and was originally developedby Mikkelsen as an extension of the model presentedin [11]. The model combines the Navier-Stokes ﬂowsolver EllipSys3D [13, 14, 15], with an actuator disctechnique where body forces are distributed on a discrepresenting a wind turbine rotor. As described in[16], the applied disc forces are distributed smoothlyon several mesh point to avoid singular behavior.The steady state ﬂow ﬁeld is solved in acartesian computational domain with dimensions(Lx, Ly, Lz) = (30R, 30R, 100R), where Lx, Ly andLz is respectively the width, height and length of thedomain and R is the radius of the disc. The disc islocated 15R downstream of the inlet. In the regionaround the rotor disc grid points were distributedequidistantly with a resolution of R/30 in order to re-solve the strong gradients here. From 1.3R to 20.2Rdownstream of the disc the grid was stretched slightlyusing a relative stretch of 0.27% in the ﬂow directionin order to limit the number of grid point while stillkeeping a good resolution of the wake. The total num-ber of grid points in the used grid is 18.9 · 106. Thereason for having a high resolution of the ﬁrst 20R ofthe wake is that the present computations originallywas designed to study wake deﬂection as a functionof yaw angle [17]. The boundary conditions were asfollows: Uniform inﬂow at the inlet, Neumann condi-tions at the outlet, symmetry conditions on the topand bottom boundaries and periodic conditions onthe sides.5 Preliminary resultsThis section presents some results of the simple vor-tex model and compare them to Glauert’s modelas well as to predictions from actuator disc simu-lations. We consider a uniformly loaded disc, (i.e.CT,loc = CT ) in a uniform inﬂow at diﬀerent yawangles. Assuming uniform loading as well as uni-form and steady inﬂow is of course very crude in re-lation to wind turbine rotors but will still highlightthe main mechanism of rotors in yaw. In the fol-lowing, results are presented for thrust coeﬃcients ofCT = 0.36, 0.64, 0.80 and 0.89, respectively, and yawangles of ψ = 0o, 5o, 10o, 20o, 30o, 45o and 60o, respec-tively.Figures 1-2 compares actuator disc and vortex modelpredictions of CP,loc and an = cos(ψ) −Wn(r, θ)/V0,respectively for CT = 0.64 and ψ = 30o. Azimuthθ = 0o is horizontal in the direction pointing up-stream and θ = 90o is vertically upwards. The cor-responding comparisons of actuator disc predictionswith Glauert’s model is shown in Figures 3-4. Notethat the predictions at 90o and 270o (the two verticaldirections) are coinciding.0 0.2 0.4 0.6 0.8 100.10.20.30.40.5r/RCPloc  AD (0o)AD (90o)AD (180o)AD (270o)VM (0o)VM (90o)VM (180o)VM (270o)Figure 1: Local power coeﬃcient predicted using thesimple vortex model and actuator disc model, re-spectively. The thrust coeﬃcient and yaw angle isCT = 0.64 and ψ = 30o.0 0.2 0.4 0.6 0.8 1−0.100.10.20.30.4r/Ra nFigure 2: Normal induction in the rotor plane pre-dicted using the simple vortex model and actuator discmodel, respectively. The thrust coeﬃcient and yawangle is CT = 0.64 and ψ = 30o. The legend is as inFigure 1As seen the predictions of both the proposedvortex model and Glauert’s model are in goodagreement with the actuator disc computations,which suggests that the two models captures themost important physics of rotors in yaw. It appearsthat Glauerts model is slightly more accurate in theinner part of the disc, whereas the vortex model isslightly more accurate on the outer part where itcaptures the nonlinear behavior of both CP,loc and an.0 0.2 0.4 0.6 0.8 100.10.20.30.40.5r/RC P,loc  AD (0o)AD (90o)AD (180o)AD (270o)Glauert (0o)Glauert (90o)Glauert (180o)Glauert (270o)Figure 3: Comparison of local power coeﬃcient pre-dicted by Glauert’s model and the actuator disc model,respectively. The thrust coeﬃcient and yaw angle isCT = 0.64 and ψ = 30o.0 0.2 0.4 0.6 0.8 1−0.100.10.20.30.4r/Ra nFigure 4: Normal induction in the rotor plane pre-dicted using Glauert’s model and the actuator discmodel, respectively. The thrust coeﬃcient and yawangle is CT = 0.64 and ψ = 30o. The legend is as inFigure 3One of the strengths of the proposed vortex modelis its ability to also predicting the induced velocitiesin the radial and tangential directions. The vortexmodel predictions of these induced velocity compo-nents are compared with actuator disc results in Fig-ures 5 and 6 for the same case as in Figures 1-2. Asseen the resemblance between the predictions is good,which further veriﬁes the the proposed vortex model.Finally, Figures 7-8 shows the global power coeﬃ-cient as a function of yaw angle predicted by the vor-tex model and Glauert’s model, respectively in com-parison with actuator disc results. Generally, bothmodels show good agreement with actuator disc pre-dictions, however, the proposed vortex model showsigniﬁcant discrepancies with increasing yaw angleand thrust coeﬃcient. The reason for this is that thevortex model, in contrast to the model of Glauert,0 0.2 0.4 0.6 0.8 1−0.1−0.0500.050.10.150.20.25r/Ra rFigure 5: Spanwise distribution of radial induction,ar = Wr/V0 (The legend is as in Figure 1); Thethrust coeﬃcient and yaw angle is CT = 0.64 andψ = 30o.0 90 180 270 360−0.1−0.0500.050.1θ [o]a θ  AD (0.25R)AD (0.5R)AD (0.75R)VM (0.25R)VM (0.5R)VM (0.75R)Figure 6: Azimuthal variation of tangential induc-tion, aθ = Wθ/V0, at r/R = 0.25, 0.50 and 0.75. Thethrust coeﬃcient and yaw angle is CT = 0.64 andψ = 30o.assumes the average induction in the rotor plane tobe constant with yaw angle. It is seen that the modelof Glauert performs very well on an integral basis fornearly all yaw angles and loads.Future work will be directed towards improving thevortex model prediction for high loading and yaw an-gles as well as to formulate in a form which is suitablefor implementation in an aero-elastic code.6 ConclusionsAn analytical model based on classical vortex the-ory is proposed for predicting performance of uni-formly loaded rotors operating in yaw. The capabil-ity of the model has been shown by comparing it toNavier-Stokes actuator disc simulations and with pre-dictions of the model by Glauert. The vortex model is0 10 20 30 40 50 600.10.20.30.40.50.60.70.8ψ [o]C pCT=8/9CT=0.80CT=0.64CT=0.36Figure 7: Global power coeﬃcient as a function ofyaw angle at diﬀerent thrust coeﬃcients. Lines: Ac-tuator disc predictions; Symbols; Vortex model pre-dictions.0 10 20 30 40 50 600.10.20.30.40.50.60.70.8ψ [o]C pCT=8/9CT=0.80CT=0.64CT=0.36Figure 8: Global power coeﬃcient as a function ofyaw angle at diﬀerent thrust coeﬃcients. Lines: Ac-tuator disc predictions; Symbols; Glauert’s model pre-dictions.proven to give predictions of all induction componentsand power performance, which are in close agreementwith the actuator disc simulations at low to mediumloads and over a wide range of yaw angles. However,for higher loads and yaw angles the proposed vor-tex model is less accurate. In these cases the modelis outperformed by the model of Glauert, which isshown to be accurate for all load conditions in termsof integral values. However, unlike the proposed vor-tex model, the model of Glauert does not provide theinductions in the radial and tangential directions andcannot predict the nonlinear behavior of the inducedvelocity on the outer part of the disc.References[1] Haans, W., van Kuik, G. and van Bussel, GJW.Experimentally observed eﬀects of yaw misalign-ment on the inﬂow in the rotor plane. Journal ofPhysics: Conference Series. The Science of Mak-ing Torque from Wind. Technical University ofDenmark, Lyngby, August 2007.[2] Sant, T., van Kuik, G. and van Bussel, GJW. Es-timating the Angle of Attack from Blade Pres-sure Measurements on the National RenewableEnergy Laboratory Phase VI Rotor Using a FreeWake Vortex Model: Yawed Conditions. WindEnergy 2009, 12:1-32.[3] Glauert, H. A general theory for the autogyro.Technical Report ARC R & M 786, ARC, 1926.[4] Glauert, H. The elements of aerofoil and airscrewtheory, second edition. Cambridge UniversityPress, Cambridge, England, 1947.[5] Glauert, H. Airplane propellers. In W.F. Du-rand, editor, Aerodynamic Theory, chapter Divi-sion L, pages 169-360. Dover Publications, NewYork, NY., USA, 1963.[6] Shen, WZ. Actuator Line / Navier Stokes Com-putations for Flows past the Yawed MEXICORotor. Wake Conference, held at Gotland Uni-versity 2011.[7] Schepers, JG et al. Final report of IEA Task 29,Mexnext (Phase 1): Analysis of Mexico windtunnel measurements. To be published 2012.[8] Øye, S. A simple vortex model. In Proc. of thethird IEA Symposium on the Aerodynamics ofWind Turbines, ETSU, Harwell, 1990, p. 4.1-4.15.[9] Madsen, HA, Mikkelsen, R, Johansen, J,Bak,C, Øye, S & Sørensen, NN. Inboard rotor/bladeaerodynamics and its inﬂuence on blade design.Chapter in Risøtechnical report: Research inaeroelasticity EFP-2005. Ed. Bak, C. Risø-R-1559(EN), Roskilde, Denmark, May 2006. p.19-39.[10] Hansen, MOL. Aerodynamics of Wind Turbines.Second edition. Earthscan, 2008.[11] Mikkelsen, R., Actuator Disc Methods Ap-plied to Wind Turbines, MEK-FM-PHD 2003-02, Technical University of Denmark, 2003.[12] Ivanell, SSA. Numerical Computations of WindTurbine Wakes, PhD Thesis, Linne´ Flow Centre,KTH Mechanics, Stockholm, Sweden, 2009.[13] Michelsen, JA. Basis3D - a Platform for Devel-opment of Multiblock PDE Solvers. Report AFM92-05, Dept. of Fluid Mechanics, Technical Uni-versity of Denmark, DTU, 1994.[14] Michelsen, JA., Block Structured Multigrid So-lution of 2D and 3D elliptic PDE’s, Report AFM94-06, Dept. of Fluid Mechanics, Technical Uni-versity of Denmark, DTU, 1994.[15] Sørensen, NN., General Purpose Flow Solver Ap-plied to Flow over Hills, PhD Dissertation, Risø-R-827(EN), RisøNational Laboratory, Roskilde,Denmark, 1995.[16] Troldborg, N., Gaunaa, M. and Mikkelsen, R.Actuator Disc Simulations of Inﬂuence of WindShear on Power Production of Wind Turbines.Torque conference, 2010.[17] Guntur, S., Troldborg, N. and Gaunaa, M. Ap-plication of engineering models to predict wakedeﬂection due to a tilted wind turbine. EWEA,Denmark, Copenhagen, 2012.

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Modelling the influence of yaw using a simple vortex rotor model

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