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Faculty of Engineering & TechnologyImproving the prediction of pressure gradient field inwind farm numerical simulation method, WAKEFARMGopala Krishnan Sankara SubramanianSustainable energy technologyM Sc. ThesisOctober 2018EFD-293Supervisors:dr. ir. Arne van Garreldr. Hüseyin Özdemir (Mentor ECN)ir. Edwin Bot (Mentor ECN)Faculty of Engineering & Technology,Engineering Fluid DynamicsUniversity of TwenteP.O. Box 2177500 AE EnschedeThe Netherlands

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Assessment committeeSupervisordr.ir.Arne van GarrelEngineering fluid dynamics,University of TwenteChairmanProf.dr.ir.C.H.VennerEngineering fluid dynamics,University of TwenteSupervisor from companyir. Edwin BotResearcher, wind energyEnergy research centre of the netherlandsExternal memberdr. Maarten.J. ArentsenFaculty of Behavioural, Management and Social SciencesUniversity of Twenteiii

IVA SSESSMENT COMMITTEE

AbstractThe wind turbines in a wind farm interact aerodynamically through their wakes. Thewakes are characterized by reduced wind speeds and increased turbulence. Thesewake effects influence the overall power production of the wind farm and cause additional fatigue loading. The numerical modelling of wind farms is vital, as it helpsus to understand the wind turbine wake interactions and to predict the total poweroutput of the wind farm better. The current wake model at the Energy researchCentre of the Netherlands is implemented into a Fortran code named WAKEFARM.It simulates the wake properties of a single turbine or a row of turbines. WAKEFARM solves Reynolds averaged Navier Stokes equations in perturbation form. TheReynolds averaged Navier Stokes equations used in WAKEFARM are parabolizedin the streamwise direction. The two momentum equations in the transverse directions are elliptic and are solved iteratively. The axial pressure gradient in the axialmomentum equation is neglected in the far wake and prescribed along with the bodyforce, in the near wake. This axial pressure gradient is calculated using an inviscidvortex models. The induced velocity vectors calculated from the vortex model aregiven as initial guesses to the perturbation variables in the three momentum equations. In this thesis work, two vortex methods that give an improved prediction ofthe pressure gradient field are developed: model of a wind turbine rotor with morethan three blades with span varying circulation and constant axial induction alongthe span and a model of a real wind turbine having three blades with span varyingcirculation and axial induction distributed along the span. Both the models trail ahelical wake. The root vortex is included in both the models. The axial pressure gradient is calculated from inviscid, incompressible Navier Stokes equations. The trendin the calculated axial pressure gradient is in good agreement with the momentumtheory. The two new near wake models (vortex models) are implemented in WAKEFARM. The horizontal velocity profiles in the cross-flow direction at hub height arevalidated with field measurements in ECNs wind turbines test site in wieringemeer.The constant axial induction model correlates well with the experiments comparedto the existing vortex models. The implementation of the root vortex was successful.The new method shows a flattened velocity profile near the centre of the wake, dueto the influence of root vortex. The centerline velocity profiles are validated with windv

VIA BSTRACTtunnel measurements in Marchwood laboratories. The centerline velocity profile at 5rotor diameters downstream of the turbine, predicted by the constant axial inductionmodel has the best correlation with the Marchwood experiments. The new constantaxial induction model shows an improvement in comparisons with experiments.

ContentsAssessment committeeiiiAbstractvList of SymbolsxxAbbreviationsxxiii1 Introduction1.1 Scope of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.2 Research methodology . . . . . . . . . . . . . . . . . . . . . . . . . .1.3 Report organization . . . . . . . . . . . . . . . . . . . . . . . . . . . .12342 WAKEFARM model description2.1 Coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.2 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . .2.2.1 Perturbed Navier Stokes equations . . . . . . . . . . . . . . .2.3 Rotor model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.3.1 Induction factor and thrust coefficient . . . . . . . . . . . . . .2.4 Undisturbed flow model - Atmospheric boundary layer stability model2.4.1 Panofsky Dutton ABL model . . . . . . . . . . . . . . . . . .2.4.2 Gryning ABL model . . . . . . . . . . . . . . . . . . . . . . .2.5 Grid generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.6 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . .2.6.1 ADI method . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.6.2 The SIMPLE method . . . . . . . . . . . . . . . . . . . . . .2.6.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . .2.7 Example WAKEFARM result . . . . . . . . . . . . . . . . . . . . . .56771112131315171719202121.3 Inviscid near wake model improvement233.1 Existing inviscid vortex model . . . . . . . . . . . . . . . . . . . . . . . 233.1.1 Vortex tube model . . . . . . . . . . . . . . . . . . . . . . . . . 23vii

C 464 Varying axial induction along the blade4.1 Calculating the bound circulation . . . . . . . . . . . . . . . . . . . . .4.1.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.1.2 Input requirements . . . . . . . . . . . . . . . . . . . . . . . . .4.2 Results from the model . . . . . . . . . . . . . . . . . . . . . . . . . .4.2.1 Effect of Nr . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.2.2 Comparison between constant axial induction rotor model andvarying axial induction rotor model . . . . . . . . . . . . . . . .4.2.3 Axial pressure gradient . . . . . . . . . . . . . . . . . . . . . .4949495154565 WAKEFARM Results5.1 Validation with EWTW measurements . . . . . . . . . . . . . . . . . .5.1.1 Comparison of different near wake models . . . . . . . . . . .5.1.2 Effect of Number of blades, Nb . . . . . . . . . . . . . . . . . .5.1.3 Comparison between constant and varying axial induction model5.2 Sensitivity to atmospheric boundary layer model . . . . . . . . . . . .5.3 Far wake validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .616162676869746 Conclusions793.23.33.43.53.63.1.2 Oye’s vortex ring model . . . . . . . . . . . . . . . . . . . .Purpose of improving the vortex models . . . . . . . . . . . . . . .Development of varying circulation, constant axial induction modelSpan varying bound circulation . . . . . . . . . . . . . . . . . . . .3.4.1 Discretization of the wake and blade . . . . . . . . . . . . .3.4.2 Coordinate system . . . . . . . . . . . . . . . . . . . . . . .3.4.3 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . .3.4.4 Constructing a vortex system . . . . . . . . . . . . . . . . .3.4.5 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . .3.4.6 Vortex line induced velocity . . . . . . . . . . . . . . . . . .3.4.7 Influence coefficients . . . . . . . . . . . . . . . . . . . . .3.4.8 Wake expansion effects: prescribed wake model . . . . . .Results from the vortex model . . . . . . . . . . . . . . . . . . . . .3.5.1 Evolution of wake . . . . . . . . . . . . . . . . . . . . . . . .3.5.2 Effect of Number of blades . . . . . . . . . . . . . . . . . .3.5.3 Effect of number of span-wise vortex filaments . . . . . . .3.5.4 Cosine spacing . . . . . . . . . . . . . . . . . . . . . . . . .3.5.5 Calculating pressure gradients . . . . . . . . . . . . . . . .Root vortex inclusion . . . . . . . . . . . . . . . . . . . . . . . . . .3.6.1 Contour plots . . . . . . . . . . . . . . . . . . . . . . . . . .5759

C ONTENTSReferencesIX81AppendicesA Velocity induced by Vortex ring85B Wind turbine Geometrical and Aerodynamic data87C Stability classification and Monin Obukhov length93C.0.1 Monin Obukhov Length (L) . . . . . . . . . . . . . . . . . . . . 93C.0.2 Stability correction functions . . . . . . . . . . . . . . . . . . . . 94D Components of vorticity in a wind turbine blade97

XC ONTENTS

List of Figures2.1 Different regions in the wake of a wind turbine . . . . . . . . . . . . . .62.2 Coordinate system used in WAKEFARM . . . . . . . . . . . . . . . . .72.3 Pressure and Velocity distributions in actuator disk model . . . . . . . 122.4 Profiles of the length scale for neutral conditions with boundary layerheight of 1000m and roughness length z0 0.05m The dashed-dottedline correlates to the surface layer scaling. The dashed line includesthe effect of first two terms and the full line includes the effect of allthree terms in the formulation of the length scale [1]. . . . . . . . . . . 162.5 Computational domain in y z plane in WAKEFARM [2] . . . . . . . . 182.6 First step in ADI method . . . . . . . . . . . . . . . . . . . . . . . . . . 192.7 Staggered grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.8 SIMPLE method [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.9 Comparison of horizontal velocity profile in the cross flow directionat hub height and at downstream distance 2.5D with EWTW measurements. The axial induction factor at rotor a 0.245 and the freestream velocity at hub height is 11m/s . . . . . . . . . . . . . . . . . . 222.10 Comparison of centerline velocity profile at downstream distance ofx 5D with Marchwood experimental data. The thrust coefficient atthe rotor Ct 0.62 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.1 Wake represented as a vortex tube [4] . . . . . . . . . . . . . . . . . . 243.2 Model of wind turbine wake with discrete ring vortices . . . . . . . . . 253.3 Resolving the vortex density on the surface of a single tip vortex intoits axial and tangential components[5]. . . . . . . . . . . . . . . . . . 253.4 Side view of trailing helices, for different tip speed ratios.λ 6(left),λ 9(right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.5 Helical wake structure behind a wind turbine. . . . . . . . . . . . . . 303.6 Geometrical parameters of a helical filament. . . . . . . . . . . . . . 303.7 Vortex system in a single bladed wind turbine. . . . . . . . . . . . . . . 323.8 Representation of helix by straight vortex segments, the helix has aradius r0 and length 7.5D . . . . . . . . . . . . . . . . . . . . . . . . . 33xi

XIILIST OF FIGURES3.9 Schematic for Biot Savart law [6]. . . . . . . . . . . . . . . . . . . . . . 343.10 Non-dimensional wake radius calculated from Oye’s vortex ring model[5], as a function of distance from the turbine for an axial inductionfactor, a 0.33 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.11 Trailed vorticity distribution along the blade span . . . . . . . . . . . . 373.12 Bound circulation distribution along the lifting line corresponding to anaxial induction factor, a 0.33 . . . . . . . . . . . . . . . . . . . . . . . 383.13 The horizontal velocity profile in the cross flow direction at hub heightfor a free stream velocity of 11m/s at a distance of 2.5D downstreamof the rotor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.14 Normalized horizontal velocity profiles in the cross flow direction athub height for an axial induction factor a of 0.33 at various locationsdownstream of the wind turbine rotor. The parameters used are Nb 6, Nr 25, Nh 318 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.15 The horizontal velocity profile in the cross flow direction at hub heightfor an axial induction factor, a of 0.33 at a distance of 2.5D downstreamof the rotor, for Nb 3, 6, 12, 24, 48, 96 with Nr 25 and Nh 320 . . . 403.16 The horizontal velocity profile in the cross flow direction at hub heightfor an axial induction factor, a of 0.33 at a distance of 2.5D downstreamof the rotor. The number of span-wise vortex filaments are Nr 15, 25, 45, 75 with Nh 320 and Nb 6 . . . . . . . . . . . . . . . . . . 403.17 The horizontal velocity profile in the cross flow direction at hub heightat a distance of 2.5D downstream of the rotor obtained using cosineand uniform spacing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.18 Bound circulation distribution along the lifting line obtained using cosine and uniform spacing . . . . . . . . . . . . . . . . . . . . . . . . . 413.19 Axial pressure gradient at hub-height and centre of nacelle as a function of distance downstream of the turbine . . . . . . . . . . . . . . . . 433.20 Bound circulation distribution along the lifting line corresponding to anaxial induction factor, a 0.33 and Root radius, Rr 6.2m (left),Rr 2.5m (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.21 Trailed vorticity distribution along lifting line corresponding to an axialinduction factor, a 0.33 and Rr 6.2m (left),Rr 2.5m (right). . . . . 443.22 The horizontal velocity profile in the cross flow direction at hub heightfor an axial induction factor a 0.33 and root radius, Rr 6.2m (left),Rr 2.5m (right) at a distance of 2.5D downstream of the rotor. Ahundred points were used along the scan line. . . . . . . . . . . . . . 45

LIST OF FIGURESXIII3.23 Normalized horizontal velocity profiles in the cross flow direction athub height for an axial induction factor a of 0.333 at various locationsdownstream of the wind turbine rotor. A hundred points were usedalong the scan line. The root radius is taken as 6.2m . . . . . . . . . . 453.24 The hub designs in Enercon (left) and GE (right) blades [7]. . . . . . . 463.25 Axial induced velocities in the X Z plane and Y 0 for an uniformfree stream velocity of 11m/s and axial induction factor a 0.245. Theroot radius is selected as 6.2m . . . . . . . . . . . . . . . . . . . . . . .15Velocity triangle in a wind turbine blade . . . . . . . . . . . . . . . . .normal vectors in a wind turbine blade . . . . . . . . . . . . . . . . . .Flow chart of the lifting line algorithm . . . . . . . . . . . . . . . . . . .Distribution of lift coefficient, Cl along the lifting line . . . . . . . . . . .Distribution of bound circulation, Γb along the lifting line . . . . . . . .Distribution of local induction factor, a along the lifting line . . . . . . .Distribution of angle of attack, α along the lifting line . . . . . . . . . .Horizontal velocity profile in the cross flow direction at hub height atrotor plane. Number of points along the scan line is 100 . . . . . . . .Horizontal velocity profile in the cross flow direction at hub height andat x 2.5D. Number of points along the scan line is 100 . . . . . . . .Distribution of local axial induction factor, a, along the lifting line fortwo different values of Nr . . . . . . . . . . . . . . . . . . . . . . . . . .Distribution of lift coefficient, Cl , along the lifting line for two differentvalues of Nr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Horizontal velocity profile in the cross flow direction at hub height andat x 2.5D for two different values of Nr 18, 36. Number of pointsalong the scan line is 100 . . . . . . . . . . . . . . . . . . . . . . . . .Horizontal velocity profile in the cross flow direction at hub height andat x 2.5D predicted by two different rotor models, constant axialinduction model, and radially varying axial induction model. Hundredpoints were used along the scan line . . . . . . . . . . . . . . . . . . .Horizontal velocity profile in the cross flow direction at hub height androtor plane predicted by two different rotor models, the constant axialinduction model, and radially varying axial induction model. Hundredpoints were used along the scan line . . . . . . . . . . . . . . . . . . .Axial pressure gradient at hub-height and at y 0.5R as a function ofdistance downstream of the turbine . . . . . . . . . . . . . . . . . . . .5051535454555555565657575858595.1 Location of wind turbines and meteorological measurement mast atEWTW [8]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

XIVLIST OF FIGURES5.2 Top view of EWTW experimental setup. In yawed conditions, the metmast shifts along the dashed line with respect to the rotor. . . . . . . . 625.3 Horizontal velocity profile in a single wake at x-2.5D measured atEWTW site for 3 ambient wind speed classes [8]. . . . . . . . . . . . . 635.4 Horizontal velocity profile at x-3.5D behind the rotor, measured atEWTW site for 3 ambient wind speed classes [8]. . . . . . . . . . . . . 635.5 Comparison of horizontal wind speed profiles predicted by three different near wake models at hub height and x 2.5D, for a wind speedof 11m/s. The root radius is taken as 2.5m . . . . . . . . . . . . . . . . 655.6 Comparison of horizontal wind speed profiles predicted by three different near wake models at hub height and x 3.5D, for a wind speedof 11m/s. The root radius is taken as 2.5m . . . . . . . . . . . . . . . . 655.7 Comparison of horizontal wind speed profiles predicted by three different near wake models at hub height and x 2.5D, for a wind speedof 11m/s. The root radius is taken as 6.2m . . . . . . . . . . . . . . . . 665.8 Comparison of horizontal wind speed profiles predicted by three different near wake models at z hub height and x 3.5D, for a windspeed of 11m/s. The root radius is taken as 6.2m . . . . . . . . . . . . 665.9 Horizontal wind speed profiles at hub height, predicted by helicalmodel at various x 2.5D downstream of the turbine for 3 differentvalues of Nb 3, 6, 12 and for Nr 25, Nh 320. The wind speed athub height is 11m/s. The root radius is taken as 2.5m. . . . . . . . . . 675.10 Horizontal wind speed profiles at hub height, predicted by helicalmodel at various x-locations downstream of the turbine. The windspeed at hub height is 11m/s. The root radius is taken as 6.2m . . . . 685.11 Local axial induction distribution along the blade. . . . . . . . . . . . . 695.12 Comparison of horizontal velocity profiles at x 2.5D predicted byconstant axial induction model and radially varying axial induction withEWTW experiments. The parameters used in the both the helicalmodels Nh 320, Nb 3, Nr 36. . . . . . . . . . . . . . . . . . . . . 705.13 Comparison of horizontal velocity profiles at x 3.5D predicted byconstant axial induction model and varying axial induction with EWTWexperiments. The parameters used in the both the helical modelsNh 320, Nb 3, Nr 36 . . . . . . . . . . . . . . . . . . . . . . . . . 705.14 Free-stream velocity u0 profiles obtained from Panofsky-Dutton andGryning models. The free stream velocity at hub height is 11m/s. . . . 715.15 Horizontal velocity profiles along z-axis at x 5D, y 0 obtainedfrom the WAKEFARM method using Panofsky-Dutton and Gryningmodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

LIST OF FIGURES5.16 Comparison of horizontal wind speed profiles at z hub height and x 2.5D, predicted by the WAKEFARM method using Panofsky-Duttonand Gryning models for a wind speed of 11m/s. The root radius istaken as 2.5m and Nb 12. . . . . . . . . . . . . . . . . . . . . . . . .5.17 Comparison of horizontal wind speed profiles at z hub height and x 3.5D, predicted by the WAKEFARM method using Panofsky-Duttonand Gryning models for a wind speed of 11m/s. The root radius istaken as 2.5m and Nb 12. . . . . . . . . . . . . . . . . . . . . . . . .5.18 Comparison of horizontal wind speed profiles at z hub height and x 2.5D, predicted by the WAKEFARM method using Panofsky-Duttonand Gryning models for a wind speed of 11m/s. The root radius istaken as 6.2m and Nb 12

The wind turbines in a wind farm interact aerodynamically through their wakes. The wakes are characterized by reduced wind speeds and increased turbulence. These wake effects influence the overall power production of the wind farm and cause ad-ditional fatigue loading. The numerical modelling of wind farms is vital, as it helps