aerodynamics of vertical axis wind turbines

ACTAUNIVERSITATIS

UPSALIENSISUPPSALA

2015

Digital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 1274

Aerodynamics of Vertical AxisWind Turbines

Development of Simulation Tools and Experiments

EDUARD DYACHUK

ISSN 1651-6214ISBN 978-91-554-9307-3urn:nbn:se:uu:diva-260573

Dissertation presented at Uppsala University to be publicly examined in Polhemssalen,Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, Friday, 9 October 2015 at 09:00 forthe degree of Doctor of Philosophy. The examination will be conducted in English. Facultyexaminer: Mac Gaunaa (Technical University of Denmark, Department of Wind Energy).

AbstractDyachuk, E. 2015. Aerodynamics of Vertical Axis Wind Turbines. Development ofSimulation Tools and Experiments. Digital Comprehensive Summaries of UppsalaDissertations from the Faculty of Science and Technology 1274. 86 pp. Uppsala: ActaUniversitatis Upsaliensis. ISBN 978-91-554-9307-3.

This thesis combines measurements with the development of simulation tools for vertical axiswind turbines (VAWT). Numerical models of aerodynamic blade forces are developed andvalidated against experiments. The studies were made on VAWTs which were operated at opensites. Significant progress within the modeling of aerodynamics of VAWTs has been achievedby the development of new simulation tools and by conducting experimental studies.

An existing dynamic stall model was investigated and further modified for the conditions ofthe VAWT operation. This model was coupled with a streamtube model and assessed againstblade force measurements from a VAWT with curved blades, operated by Sandia NationalLaboratories. The comparison has shown that the accuracy of the streamtube model has beenimproved compared to its previous versions. The dynamic stall model was further modified bycoupling it with a free vortex model. The new model has become less dependent on empiricalconstants and has shown an improved accuracy.

Unique blade force measurements on a 12 kW VAWT were conducted. The turbine wasoperated north of Uppsala. Load cells were used to measure the forces on the turbine. Acomprehensive analysis of the measurement accuracy has been performed and the major errorsources have been identified.

The measured aerodynamic normal force has been presented and analyzed for a wide rangeof operational conditions including dynamic stall, nominal operation and the region of highflow expansion. The improved vortex model has been validated against the data from the newmeasurements. The model agrees quite well with the experiments for the regions of nominaloperation and high flow expansion. Although it does not reproduce all measurements in greatdetail, it is suggested that the presented vortex model can be used for preliminary estimationsof blade forces due to its high computational speed and reasonable accuracy.

Keywords: wind power, vertical axis turbine, H-rotor, simulations, streamtube model, vortexmodel, dynamic stall, measurements, blade, force

Eduard Dyachuk, Department of Engineering Sciences, Electricity, Box 534, UppsalaUniversity, SE-75121 Uppsala, Sweden.

© Eduard Dyachuk 2015

ISSN 1651-6214ISBN 978-91-554-9307-3urn:nbn:se:uu:diva-260573 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-260573)

To my mama & papa, Lidochka,babushkam & dedushkam

List of papers

This thesis is based on the following papers, which are referred to in the textby their Roman numerals.

I Dyachuk, E., Goude A., and Bernhoff H., “Dynamic Stall Modelingfor the Conditions of Vertical Axis Wind Turbines”, AIAA journal,vol. 52, iss. 1, pp. 72 – 81, 2014.

II Dyachuk, E., and Goude A., “Simulating Dynamic Stall Effects forVertical Axis Wind Turbines Applying a Double Multiple StreamtubeModel”, Energies, vol. 8, iss. 2, pp. 1353 – 1372, 2015.

III Dyachuk, E., Goude A., and Bernhoff H., “Simulating Pitching BladeWith Free Vortex Model Coupled With Dynamic Stall Model forConditions of Straight Bladed Vertical Axis Turbine”, Journal of SolarEnergy Engineering, vol. 137, iss. 4, pp. 041008, 2015.

IV Rossander M., Dyachuk, E., Apelfröjd S., Trolin K., Goude A.,Bernhoff H., and Eriksson S. “Evaluation of a Blade ForceMeasurement System for a Vertical Axis Wind Turbine Using LoadCells”, Energies, vol. 8, iss. 6, pp. 5973 – 5996, 2015.

V Dyachuk, E., Rossander M., Goude A., and Bernhoff H.,“Measurements of the Aerodynamic Normal Forces on a 12-kWStraight-Bladed Vertical Axis Wind Turbine”, Energies, vol. 8, iss. 8,pp. 8482 – 8496, 2015.

VI Dyachuk, E., and Goude A., “Numerical Validation of a Vortex ModelAgainst Experimental Data on a Straight-Bladed Vertical Axis WindTurbine”, Submitted to Energies, August 2015.

Reprints were made with permission from the publishers.

The following papers are not included in this thesis.

i Dyachuk, E., Goude A., Lalander, E., and Bernhoff H., “Influence ofIncoming Flow Direction on Spacing Between Vertical Axis MarineCurrent Turbines Placed in a Row”, In “Proceedings of the 31st In-ternational Conference on Offshore Mechanics and Arctic Engineering,OMAE 2012”, Rio de Janeiro, Brazil, July 2012.

ii Dyachuk, E., Goude A., and Bernhoff H., “Simulating Pitching BladeWith Free Vortex Model Coupled With Dynamic Stall Model for Condi-tions of Straight Bladed Vertical Axis Turbine”, In “Proceedings of the33rd International Conference on Offshore Mechanics and Arctic Engi-neering, OMAE 2014”, San Francisco, USA, June 2014.1

1This article was further modified and published as Paper III.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.1 Background history of wind energy and different wind turbines 131.2 Main features of vertical axis wind turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.3 Vertical axis wind turbines research at Uppsala University . . . . . . . . 161.4 Contribution of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.5 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Development of simulation models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.1 Streamtube model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2 Vortex model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3 Blade force modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3.1 Unsteady attached flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3.2 Condition of dynamic stall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3.3 Separated flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3.4 Application to the turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4 Measurements of aerodynamic forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.2 Data treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.3 Data treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.4 Measurement accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.5 Evaluation of the measurement method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.1 Pitching blade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.2 Turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.2.1 12 kW VAWT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.2.2 Sandia 17-m VAWT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

7 Suggestions for future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677.1 Simulation tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677.2 Experimental work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

8 Summary of papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

9 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

10 Sammanfattning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

11 Аннотация к научной диссертации . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

12 Анотація до наукової дисертації . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Nomenclature

Ablade m2 Blade areaAd m2 Cross-sectional area of a streamtubeAe m2 Streamtube area at the mid-disk of a turbineA∞ m2 Asymptotic cross-sectional area of a streamtubeCD – Drag coefficientCD0 – Drag coefficient at zero angle of attackCL – Lift coefficientCN – Normal force coefficientCNα

– Slope of normal force coefficient for attached flowC f

N – Normal force coefficient for trailing edge separationCv

N – Normal force coefficient during vortex convectionCT – Tangential force coefficientC f

T – Tangential force coefficient for static separation pointC f ′′

T – Tangential force coefficient for separated flowCT,scale – Scaling constant for tangential force coefficientCT,scale,r – Scaling factor for low pitch rateCT,scale,α – Scaling factor for low angles of attackCT,static – Static tangential force coefficientD f – Deficiency function for separation pointDα – Deficiency function for geometrical angle of attackE0 – Constant for tangential force coefficientΩ rad/s Rotational speed of a turbineF0 N Measured force by load cell 0F1 N Measured force by load cell 1F2 N Measured force by load cell 2F3 N Measured force by load cell 3FD N Drag forceFC N Centrifugal forceFL N Lift forceFN N Normal force∆FN N Maximum measured error of normal forceFN,norm N/m Normal force per unit span

∆FN,shape N Maximum measured error of the normal force shapeFT N Tangential force∆FT N Maximum measured error of tangential forceFT,norm N/m Tangential force per unit spanFx N Aerodynamic force on a blade in x-directionL1 m Horizontal distance between the load cellsLB m Distance from the load cells to the bladeLC m Distance between load cells and center of massNB – Number of turbine bladesP W Absorbed power by a turbineQ Nm Absorbed torque by a turbineR m Turbine radiusS1,S2 – Coefficients for separation pointT C Air temperatureTf – Empirical time constant for dynamic separation pointTα – Empirical time constant for delay in pressure responseV m/s Flow velocityVb m/s Blade velocityVd m/s Velocity at the turbine diskVe m/s Velocity at the turbine mid-diskVrel m/s Relative flow velocityVω m/s Velocity due to vorticesV∞ m/s Asymptotic wind velocityX ,Y,Z – Deficiency functions for unsteady attached flowc m Chord length of a bladef – Static flow separation pointf ′ – First-order delayed flow separation pointf ′′ – Dynamic flow separation pointg m/s2 Gravitational accelerationh %RH Relative humidityk – Reduced frequency of a pitching blade∆l m Length of a blade segment in a streamtubem kg Mass of the blade with support armspd,1 Pa Pressure directly in front of turbine diskpd,2 Pa Pressure directly behind turbine diskp∞ Pa Asymptotic pressurer m Arbitrary positionr0 – Critical reduced pitch raterk m Vortex position

rl – Lower scaling limit of reduced pitch ratern – Reduced pitch rateru – Upper scaling limit of reduced pitch rate∆s – Non-dimensional time-stept s Time∆t s Time-stepu – Velocity induction factorx – unit vector in x-directionx0 – Normalized blade attachment point∆z m Height of a blade segmentΓ m2/s Circulation of two-dimensional vortexΓblade m2/s Total two-dimensional circulation of a bladeα – Angle of attackα ′ – Delayed angle of attack due to pressure delayα rad/s Blade pitch rateα1 – Angle of attack for a breakpoint of flow separationαE – Effective angle of attackαcr – Critical angle of attackαds0 – Critical stall onset angleαl – Lower scaling limit of angle of attackαss – Static stall onset angleαu – Upper scaling limit of angle of attackβ – Angle between a blade and vertical axisγ – Angle between incoming flow and a streamtubeδ – Blade pitch angleε m Cutoff radius of Gaussian vortex kernelη – Efficiency factor for tangential force coefficientθ – Blade azimuth angleθ – Unit vector in the tangential direction∆θ – Angular width of a streamtubeλ – Tip speed ratioν m2/s Kinematic viscosityρ kg/m3 Air densityϕ – Angle of relative flow velocityφ m2/s Velocity potentialω 1/s Vorticity

Abbreviations

BEM Blade element momentum modelCFD Computational fluid dynamicsDS Dynamic stall model coupled with streamtube modelFEM Finite element methodFVM Finite volume methodHAWT Horizontal axis wind turbineLES Large eddy simulationNACA National Advisory Committee for AeronauticsNASA National Aeronautics and Space AdministrationRANS Reynolds-averaged Navier Stokes equationsTSR Tip speed ratioVAWT Vertical axis wind turbineVDS Dynamic stall model coupled with vortex model

1. Introduction

This thesis comprises studies on the aerodynamics of vertical axis wind tur-bines. The focus is on unsteady aerodynamic forces which are usually consid-ered as the major challenge for the development of vertical axis wind turbines.Within the presented work, simulation tools have been developed and experi-mental work has been conducted.

1.1 Background history of wind energy and differentwind turbines

The utilization of wind energy dates back to ancient times when people startedto use sails to propel boats and ships. Along with transport, the use of windenergy was adopted in agriculture to grind grain and pump water. The earliestdocumented wind mills were used by the Persians more than two thousandyears ago [1]. Those were drag-driven vertical axis mills with sails made ofreeds or wood. Later, knowledge about wind mills was brought from Persiaand the Middle East to Europe most likely by the Crusades [2]. In contrastto earlier vertical axis design, European wind mills had a horizontal axis ofrotation. The first wind turbine generating electricity was built in 1887 byScottish scientist and engineer Professor James Blyth [3, 4].

The development of wind turbines continued in the 1900s in Europe and theUSA. A Finnish inventor Sigurd Johannes Savonius in the early 1920s devel-oped a vertical axis wind turbine (VAWT) which was driven mainly by dragforce. Two US patents on this turbine were published, one in 1929 [5] andanother one in 1930 [6]. The invention was mentioned as the Savonius rotorin the patents. It was suggested that the turbine could be driven either by windor flowing water. Another type of VAWT was invented by a French aeronau-tical engineer Georges Jean Marie Darrieus and patented in 1931 [7]. Thepatent emphasizes simplicity of the turbine and covers turbines with curvedand straight blades. It is also suggested in the patent that the Darrieus turbinecan be utilized in tidal currents or rivers with low fall. The Darrieus turbine isa lift-based machine (torque is mainly generated by lift force) which is moreefficient than the Savonius turbine and requires less material to build it (seefigure 1.1).

Due to the cheap electricity from fossil fuels before the 1973 oil crisis, onlysmall scale wind turbines were used in remote locations which were not con-nected to an electrical grid. However, the oil crisis has led to the growing

13

interest in electricity production from renewable energy sources on a largerscale. Wind energy in the USA has received financial support from the gov-ernment. The research was focused on the development of new hardware,resource analysis and cost reduction techniques. Along with the developmentof horizontal axis wind turbines (HAWT) by the National Aeronautics andSpace Administration (NASA), Sandia National Laboratories from the 1970suntil the early 1990s focused the research on Darrieus turbines with curvedblades. Several companies have commercialized VAWTs with curved blades.One of these companies was FloWind, which deployed over 500 turbines inCalifornia.

In the UK, development of the VAWT technology was mostly applied toturbines with straight blades, (the H-rotor design, see figure 1.1), originallyproposed and developed in the late 1960s by British engineer and scientistPeter Musgrove [8]. A mechanism of feathering the blades at strong windswas later included in order to reduce the aerodynamic torque. The turbinewith feathering blades was later referred to as the Musgrove rotor. It wascommercialized by several companies; among them was VAWT Ltd, which inthe 1980s built several VAWTs with rated power up to 500 kW.

However, the knowledge about blade fatigue at that time was limited, andboth the American and the British turbines suffered from the blade failures.In particular, the blades of the Musgrove rotor turbines in UK broke due tomanufacture error. The curved blades of the FloWind turbines made of ex-truded aluminium had poor fatigue properties, which resulted in several bladefailures. Those blade failures led to a common perception that VAWTs aremore prone to fatigue than HAWTs. The British VAWT Ltd was closed in theearly 1990s, and the American FloWind was closed in the mid-1990s. Sincethen, the interest in VAWTs has decreased within the wind energy commu-nity. A more detailed history of the development of VAWTs can be found inRefs. [9–12].

1.2 Main features of vertical axis wind turbinesThe absolute majority of the currently deployed wind turbines are HAWTs.However, the VAWT design has several principal advantages over the HAWT.VAWTs are omni-directional, i.e. they work with winds from any directionand the yawing system is excluded. This is a considerable advantage, sincea large portion of the failures in HAWT occur within the yawing mecha-nism [13–15]. Another advantage is that the generator of VAWTs can beplaced at the ground level, which simplifies installation and maintenance. Ad-ditionally, this minimizes concerns about the size and the weight of the gener-ator, which is favourable for the installation of heavy direct driven generatorswith permanent magnets [16]. A concern for VAWTs is the cyclic blade forcesresulting in varying torque, which is inherent during operation. Although this

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Savonius rotor Darrieus rotor H-rotor

Figure 1.1. Different types of vertical axis wind turbines

problem was handled by Sandia National Laboratories and FloWind by addingcompliance to the shafts of their VAWTs [9], the varying torque is still consid-ered the major concern of VAWTs.

The accessibility of generators together with the low mass center of VAWTis specifically advantageous for offshore applications including floating tur-bines. Several studies on VAWTs for offshore use have been published [17–19]. Sandia National Laboratories has summarized properties of the Darrieusturbine from the perspective of the offshore deployment of VAWTs [9].

This thesis focuses on VAWTs with straight-bladed H-rotors, although ex-perimental data from a Sandia turbine with curved blades have been used (seechapter 5). The straight blades of the H-rotor are easier and cheater to man-ufacture than the curved blades of the Darrieus rotor. The bending momentson the shaft of the H-rotor are much smaller than on the shaft of the Darrieusturbine with curved blades, since the upper bearing can be placed close tothe turbine hub, reducing the bending moments in the axis. Instead, it is thetower in the H-rotor design that takes most of the bending loads from the tur-bine bearings. Hence, the shaft of the H-rotor is easier to manufacture than theshaft of the curve-bladed turbines with a small tower. Additionally, the H-rotordesign offers larger cross-sectional area due to the constant radius of the tur-bine. However, the blades of the H-rotor need support arms, which introduceadditional losses which affect the turbine performance. Another drawback ofthe H-rotor, compared to the design with curved blades, is the higher bendingmoments in the blades due to centrifugal forces.

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1.3 Vertical axis wind turbines research at UppsalaUniversity

The wind power research at the Division of Electricity at Uppsala Universityhas been conducted since 2002. Three H-rotor VAWTs have been built north ofUppsala by the division: a small one with rated power of 1.5 kW, one turbinefor telecom applications rated at 10 kW [20] and a 12 kW turbine, whichhas been used for the most of experiments, see Papers IV, V and Refs. [21–23]. A large VAWT with rated power of 200 kW has been constructed by thespinoff company Vertical Wind AB in Falkenberg and some published data areavailable for that turbine [24–26]. The research related to these turbines hasresulted in five doctoral theses [27–31]. Within this research, simulation toolsfor aerodynamics of VAWTs were developed by Anders Goude [30] and PaulDeglaire [28].

1.4 Contribution of the thesisThe focus of this thesis is on the aerodynamic blade forces of the VAWTs.The work comprises both development of simulation tools and force measure-ments. A model of unsteady forces on a pitching blade for the conditions ofVAWTs has been developed and assessed against existing experimental data.The model is commonly known as the dynamic stall model and it is publishedin Paper I. The dynamic stall model has been included into a simulation modelof the turbine and compared with existing measurements on a VAWT withcurved blades. This study is published in Paper II. A further development ofthe dynamic stall model is published in Paper III, where the model is coupledwith a free vortex model for a pitching blade.

Novel measurements of the aerodynamic blade forces on the straight-bladed12 kW VAWT have been conducted and the evaluation of experimental methodis published in Paper IV. The measurements of the aerodynamic normal forcesare published in Paper V. A modified vortex model coupled with the dynamicstall model is validated against the new measurements on the 12 kW VAWT.The validation is documented in Paper VI.

1.5 Outline of the thesisThe introduction of this thesis is followed by the background theory of theaerodynamics of VAWTs. The overview of simulation models is presented inchapter 3. This is followed by chapter 4, where the experimental method ofthe force evaluation is described. The assessment of the simulations againstthe experimental results is presented in chapter 5 along with the discussionsregarding applicability of the models. At the end of the thesis, the conclusionsand suggestions for future work are presented.

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2. Theory

The power of wind turbine can be expressed in terms of torque as

P = QΩ, (2.1)

where Q is the torque and Ω is the turbine rotational speed. The focus of thiswork is on lift-based turbines. The torque of these turbines is generated mainlyby lift force, and drag force contributes to losses. The structural loads on theturbine blades are often given by the normal force FN which is the resultant ofthe aerodynamic force in radial direction, figure 2.1. The tangential force FTis usually used to express the turbine torque during one revolution

Q = NB〈FT 〉R, (2.2)

where NB is the number of blades, R is the turbine radius and 〈FT 〉 is the aver-age tangential force during one revolution. If the turbine blades are supportedby struts (e.g. H-rotor) the tangential force comprises the contribution of theblades and support arms. Both the turbine geometry and operational condi-tions influence the tangential force. To illustrate this, assume that the blade islocated at the azimuth angle θ (see figure 2.1). The velocity of the blade is

~Vb = ΩRθ , (2.3)

where θ is the unit vector in the tangential direction which is positive in thecounter-clockwise direction. The vector of relative wind flow velocity ~Vrel atthe blade consists of the vector of incident wind flow at the turbine disk ~Vdand the velocity vector at the blade due to its own rotation −~Vb (negative signindicates that the direction of the flow is opposite to the direction of the blade)

~Vrel =~Vd−~Vb. (2.4)

Due to the extracted energy from the flow, the magnitude of the wind velocityat the turbine disk Vd is generally lower than the asymptotic velocity V∞. InCartesian coordinates, when the asymptotic velocity is aligned with x-axis andno flow expansion is assumed (i.e. ~V∞ =V∞x and ~Vd =Vd x), the magnitude ofthe relative flow velocity is

|~Vrel|=Vd

√(ΩRVd

+ sinθ

)2

+(cosθ)2, (2.5)

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Vd

φδ

α

FL

FN

FT

θ

θ = 0°

FD

→

V∞→

Vrel

→

Vb

→

Figure 2.1. Illustration of the velocity vectors and forces acting at the blade of aVAWT. Counter-clockwise direction is defined as positive for the angles. Hence, theangles ϕ and α have negative direction for the position of the blade shown in thisfigure. The normal force FN is positive when pointing outwards, i.e. FN is negative inthe figure.

and the angle of the relative wind is

ϕ = arctan

(cosθ

ΩRVd

+ sinθ

). (2.6)

The estimation of the relative wind angle according to equation (2.6) isonly valid assuming the blade as a point. The blade performs rotational mo-tion, indicating that there are flow curvature effects which change the effectiveangle ϕ . To account for the curvature effects, the relative wind angle is fur-ther modified. The assumption of the potential flow around a flat plate andthe Joukowski transformation together with the Kutta condition are used as byGoude [30]. The final expression of the relative wind angle including the flowcurvature effects is

ϕ = arctan

(cosθ

ΩRVd

+ sinθ

)− Ωx0c

Vrel− Ωc

4Vrel, (2.7)

where x0 is the normalized blade attachment point which varies from −0.5 to0.5 with x0 = 0 corresponding to the blade attachment at the middle of thechord length.

The angle of attack is the sum of the relative wind angle and the blade pitchangle

α = ϕ +δ . (2.8)

18

One important operational parameter for wind turbines is a tip speed ratio(TSR) λ , which is defined as the ratio between the speed of a blade tip and theasymptotic flow velocity

λ =ΩRV∞

. (2.9)

The term ΩRVd

in equation (2.7) should not be interpreted as the TSR, althoughΩRVd

is proportional to λ since Vd ∝ V∞. From equation (2.7) it follows that therelative wind angle ϕ increases with decreased TSR, and when the pitch angleδ is fixed

α ∝1λ. (2.10)

To relate this to the turbine torque, the expression for the tangential force FTcan be used

FT = FL sinϕ−FD cosϕ. (2.11)

The lift force FL is orthogonal to the vector~Vrel and the drag force FD is parallelto ~Vrel . The lift and drag forces can be estimated as

FL =12

ρAbladeV 2relCL, (2.12)

FD =12

ρAbladeV 2relCD, (2.13)

where ρ is the flow density, Ablade is the blade area, CL and CD are the lift anddrag coefficients, which mainly are the functions of the angle of attack, aspectratio, the Reynolds number and the airfoil profile. For airfoils, CL increaseswith increased α and CD is relatively constant. This holds until the point ofstall (associated with the flow separation), when CL starts to decrease and CDincreases. Thus, the maximum tangential force is obtained when the anglesof attack are close to the stall region. Consequently, the turbine should be de-signed to operate at TSRs corresponding to the nearly stall limit in order toobtain the highest power for the given wind conditions. Please note that dur-ing the operation of VAWT, the angle of attack and the relative wind velocitychange constantly, which causes the phenomenon dynamic stall. During dy-namic stall, the coefficients CL and CD will have a delay compared to the staticflow case. This should be taken into account when operating VAWT close tothe stall limit. The details regarding the dynamic stall event are presented insection 3.3.

19

3. Development of simulation models

The simulation models for aerodynamics of VAWTs can be divided into threegroups. The first group includes the models which solve the Navier-Stokesequations on a grid of the entire simulated volume. Due to its very high com-putational complexity, the Navier-Stokes equations are usually combined withturbulence models such as the Reynolds-averaged Navier-Stokes equations(RANS) [32–35] and large eddy simulations (LES) [36–38]. The most com-mon methods for this group of models are the finite element method (FEM)and the finite volume method (FVM) which solve the flow for a confined re-gion. There are commonly available FEM and FVM models, however due totheir high computational cost, these models are not used in the presented work.

The second method for simulating aerodynamics of VAWTs is the use ofthe vorticity equation, which is based on the Navier-Stokes equations. Manysimplifications can be applied within this method, and when a model for bladeforces is used, the computational time can be reduced substantially. A vortexmodel has been used within the work.

The third type of the models for the VAWT simulations is based on the mo-mentum conservation principle. These models are usually referred to as bladeelement momentum models (BEM), when simulating HAWT, and streamtubemodels, for simulating VAWT. The models assume steady flow, and the fulldescription of the flow through entire turbine is not obtained. However dueto their simplicity and high computational speed, the momentum models areused to quickly evaluate the performance of VAWTs.

This chapter presents the summary of the simulation tool development.More details concerning the development of each model is presented in Pa-pers I, II, III and VI.

3.1 Streamtube modelA double multiple streamtube model is used in this work, and the implementa-tion of Paraschivoiu [39] is applied. The difference between the implementedmodel and the version by Paraschivoiu is that the positive flow direction istowards right, see figure 3.1. The flow expansion by Read and Sharpe [40]is further added into the model. The present version of the model has beentested against the experimental data on a curve-bladed Darrieus turbine bySandia National Laboratories, see Paper II.

20

Ad,j Δθ

V∞ Vd,j Ve,j

θj

Ω

p∞ pd,1 pd,2 p∞

Figure 3.1. Illustration of the double multiple streamtube model. The horizontal linesrepresent streamtubes, and the vertical line in the middle of the turbine represents thetransition from the upwind to the downwind side. The height of a streamtube ∆z j isincluded into the streamtube area Ad, j. Note, that the flow expansion is not includedin this figure.

In the double multiple streamtube model, the turbine is divided into two ac-tuator disks, one upwind and one downwind, figure 3.1. It is assumed that thevelocity across the actuator disks remains constant and the pressure in the mid-dle of the turbine equals to the asymptotic pressure. With these assumptionsand by using the Bernoulli’s equation at the upwind disk

p∞ +12

ρV 2∞ = pd1 +

12

ρV 2d, j, (3.1)

pd2 +12

ρV 2d, j = p∞ +

12

ρV 2e, j, (3.2)

combined with the momentum conservation for the control volume of a singlestreamtube

ρA∞V 2∞−ρAe, jV 2

e, j = Ad, j(

pd1− pd2

), (3.3)

and with continuityA∞V∞ = Ad, jVd, j = Ae, jVe, j, (3.4)

it can be shown that the the velocity at the upwind disk is the average of theasymptotic velocity and the velocity at the middle of the turbine

Vd, j =V∞ +Ve, j

2. (3.5)

21

β

Δzj

Δlj

V∞

Figure 3.2. Illustration of the angle β for a curve-bladed VAWT.

The asymptotic flow velocity V∞, the velocity at the turbine disk Vd, j and thevelocity at the middle of the turbine Ve, j are illustrated in figure 3.1 togetherwith the asymptotic pressure p∞, the pressure directly in front of the disk pd1and directly behind the disk pd2 . The cross sectional areas of a streamtube inequations (3.3) to (3.5) are denoted as A∞, Ad, j and Ae, j.

To calculate the forces on the turbine, the lift and drag coefficients togetherwith the flow velocity have to be estimated. The expressions for the flowvelocity and the angle of attack including the curvature effects are given inequations (2.5), (2.7) and (2.8). For the case with the curve-bladed VAWTs(see figure 3.2), Vrel, j and α j become

Vrel, j =Vd, j

√(ΩRVd, j

+ sinθ j

)2

+(cosθ j cosβ j)2, (3.6)

α j = ϕ j +δ = arctan

cosθ j cosβ jΩRVd, j

+ sinθ j

−(Ωx0cVrel, j

+Ωc

4Vrel, j

)cosβ j +δ , (3.7)

where β j is the angle between the blade in a streamtube and vertical axis.The lift and drag coefficients are obtained from the blade force model,

which is described in section 3.3. The lift and drag forces are estimated as

FL, j =12

ρc∆l jV 2rel, jCL, j, (3.8)

FD, j =12

ρc∆l jV 2rel, jCD, j, (3.9)

where ∆l j is the blade length in a streamtube

∆l j =∆z j

cosβ j(3.10)

22

and ∆z j is the height of a streamtube, see figure 3.2. As mentioned in chapter 2,it is the normal force FN (giving the structural loads on the blades) and thetangential force FT (corresponding to the turbine torque) which are of interestwhen designing the turbine. The normal and tangential force coefficients aredefined as

CN j =CL j cosϕ j +CD j sinϕ j, (3.11)

CTj =CL j sinϕ j−CD j cosϕ j, (3.12)

which can be used to estimate the normal and tangential forces

FN j =12

ρc∆l jV 2rel, jCN j , (3.13)

FTj =12

ρc∆l jV 2rel, jCTj . (3.14)

Since the turbine has NB blades, and each blade passes a streamtube duringa fraction of time, the average force on the turbine blades in the x-direction is⟨

Fx, j⟩= NB

∆θ

2πFx, j, (3.15)

where Fx, j isFx, j = FN, j cosθ j cosβ j−FT, j sinθ j. (3.16)

Combining equations (3.10), (3.13), (3.14) and (3.16) with equation (3.15),the force

⟨Fx, j⟩

becomes

⟨Fx, j⟩=

NBρc∆z j∆θV 2rel, j

4π

(CN j cosθ j−CTj

sinθ j

cosβ j

). (3.17)

The force⟨Fx, j⟩

is equal to the pressure difference about the streamtube areaAd, j, equation (3.3). Thus, combining an expression for Ad, j

Ad, j = R∆z j∆θ |cosθ j|, (3.18)

with equations (3.3) and (3.5), it is possible to find the force⟨Fx, j⟩

from mo-mentum conservation⟨

Fx, j⟩= ρR∆z j∆θ |cosθ j|Vd, j (V∞−Ve, j) . (3.19)

Combining equations (3.17) and (3.19) with the velocity induction factor

u j =Vd, j

V∞

, (3.20)

and equation (3.5), the following expression is obtained

1u j−1 =

NBcV 2rel, j

8πR|cosθ j|V 2d, j

(CN j cosθ j−CTj

sinθ j

cosβ j

). (3.21)

23

Equation (3.21) is solved iteratively to obtain the induction factor u j for eachstreamtube. The velocity at the middle of the turbine is found based on equa-tions (3.5) and (3.20)

Ve, j = 2V∞ (1−u j) . (3.22)

To solve the downstream disk, the velocity Ve, j is used as an input, and theblade forces are calculated the same way as described above.

After the estimation of velocities at the both sides of the turbine, the flowexpansion model as by Read and Sharpe [40] is included. The model recalcu-lates the size of streamtubes by applying the continuity equation (3.4) at theupwind and downwind sides of the turbine. Please note, that the expansionis modeled in the horizontal plane only. The cross-section of a streamtube isshown in figure 3.3. It is assumed that the flow expands symmetrically aboutthe horizontal line which connects θ = 90 and θ = 270. An iterative processis required to find the size of each streamtube for known Vd, j at the upwind andthe downwind side. New azimuthal angle θ j is obtained as

θnew = θold− γ j, (3.23)

and the angular streamtube width ∆θ is recalculated. The angle γ j representsthe deviation of streamlines from the direction of the asymptotic flow due tothe difference between the streamtube area at the upwind and the downwindsides. After the size of each streamtube is estimated, the flow velocities andthe blade forces are calculated again.

3.2 Vortex modelThe motion of the fluid can be described with the continuity equation and theNavier-Stokes equations. The continuity equation

∂ρ

∂ t+∇ ·

(ρ~V)= 0, (3.24)

represents the conservation of mass. The flow is assumed to be incompressibledue to the low flow velocities at VAWTs, and equation (3.24) reduces to

∇ ·~V = 0. (3.25)

The Navier-Stokes equations are derived from the Newton’s second law, andin the case of incompressible flow, the equations are

∂~V∂ t

+(~V ·∇

)~V =−∇p

ρ+~g+ν∇

2~V , (3.26)

where ~g is the gravitational acceleration and ν is the kinematic viscosity.Equation (3.26) represents the system of non-linear partial differential equa-tions, which can be solved analytically only in rare cases. One method to

24

Vd,dw γ

Ad,dw

V∞

Ad,up

Vd,up

θup θdw

θ = 90° θ = 270°

Figure 3.3. Illustration of the flow expansion model. The doted line represents thecenter of a streamtube. Indices up and dw denote the upwind and the downwind sideaccordingly. Please, note that the counter-clockwise direction is defined positive forγ , i.e. γ is positive for the streamline in this figure.

solve the Navier-Stokes equations numerically is to use the finite element orfinite volume models, where the equations are solved on a grid. An alternativemethod is to use the vorticity equation, which is obtained by taking the curl ofequation (3.26)

∂~ω

∂ t+(~V ·∇

)~ω = (~ω ·∇)~V +ν∇

2~ω, (3.27)

where the vorticity is~ω = ∇×~V . (3.28)

In the two dimensional (2D) case, the vorticity is aligned with the z-axis andthe flow occurs in the x− y plane, i.e. ~ω = ω z and ~V =Vxx+Vyy. Thus in thetwo dimensional case, equation (3.27) reduces to

∂~ω

∂ t+(~V ·∇

)~ω = ν∇

2~ω, (3.29)

Since experimental data and a blade force model are used to obtain the liftand the drag forces (i.e. the boundary layer does not have to be resolved),viscosity in equation (3.29) can be neglected. This significantly increases thecomputational speed of the vortex method.

The flow velocity is obtained from

~V = ∇φ +~Vω , (3.30)

25

where ∇φ is the potential flow solution and ~Vω is the contribution from re-leased vortices. Under the assumption that VAWTs are not confined withboundaries and that the blades are approximated as single points, the poten-tial flow solution equals to the asymptotic flow velocity, i.e. ∇φ = V∞. Thecontribution from the vortices is calculated from the Biot-Savart law by usingcomplex variables instead of vectors, which in the 2D case becomes

Vω (r) =Nv

∑k=1

iΓk

2π

1(r− rk)

(1− e−

|r−rk|2ε2

), (3.31)

where r is the arbitrary position, rk is the position of vortex k, Γk is the circu-lation of vortex k (r and rk denote the complex conjugate of r and rk), and ε isthe smoothing parameter for the vortex when applying a Gaussian smoother.The vortices are allowed to drift with the flow velocity when using the La-grangian description. By neglecting the viscosity outside the boundary layersof the blades (due to the high Reynolds numbers), the vortices are propagatedaccording to

drk

dt=V (rk) . (3.32)

The velocities V (rk) at each vortex position are efficiently evaluated using thefast multipole method as described in Ref. [41].

The blade force model (described further in section 3.3) requires the flowvelocity and the angle of attack to calculate the forces at the blade. The flowcurvature effects have to be taken into account when calculating the flow ve-locity and the angles of attack. The flow curvature can be handled by usinglinear panels with linear distribution of vorticity to model the blade geometryaccording to Refs. [30, 42]. By applying the Kutta condition at the trailingedge, the circulation around the blade can be determined. The angle of attackcorresponding to the blade circulation can then be calculated as

αE = k(

arcsin(

Γblade

πcV

)−α0

), (3.33)

where k and α0 are two constants which are determined by curve-fitting froma set of static attached flow simulations at known angles of attack. The ref-erence flow velocity V in equation (3.33) is calculated with equations (3.30)and (3.31) at the quarter chord position for the blade, under the condition, thatthe panels for blades are not present during this calculation. This whole pro-cedure is described in detail and denoted as “explicit method” in Paper III.This method adds the released vortex into the panel equations and uses theassumption that the total circulation of the released vortex and the bound cir-culation is equal to the bound circulation form previous time-step, i.e. the totalcirculation is conserved.

The blade force model estimates the lift force coefficient CL for the knownangle of attack αE and flow velocity V . This lift force coefficient is used to

26

calculate a reduced circulation around the blade by using the Kutta-Joukowskilift formula

Γ =12

CLcV. (3.34)

Due to the conservation of circulation, a vortex has to be released at each time-step, with a strength corresponding to the change of circulation between thetime-steps.

To ensure the convergence of the model results, the simulations are per-formed for 100 revolutions. This value is chosen from the convergence stud-ies in Ref. [43]. Each revolution has 120 time-steps, i.e. one time-step corre-sponds to ∆θ = 3. The results from the last revolution are used in the modelassessment.

3.3 Blade force modelingThe blade forces in both the streamtube and the vortex models are estimatedwith lift and drag coefficients. The calculation of the lift and drag coefficientshave to account for the continuous change of the angle of attack and the rel-ative wind velocity, which are natural during the operation of VAWTs, seeequations (2.5), (2.7) and (2.8). For high wind speeds, where the rotationalvelocity is limited, the TSR decreases, which causes an increase in the ampli-tude of the angles of attack. Consequently, the blades fall into a stall, causinga drop in the lift force. Since the flow around the blades is not steady, dynamicstall is present, which is associated with the delay in the lift and drag forcecoefficients compared to the static values. Moreover, the amplitude of the os-cillations of the blade forces changes with dynamic stall, which is related tofatigue problems.

This section presents the dynamic stall model which is based on the modelby Leishman-Beddoes [44, 45]. The model is semi-empirical and requires thelocal Reynolds number, the rotational speed, the time-step, the angle of attack,the flow velocity and the airfoil profile. Additionally, the model uses experi-mental data on static lift and drag coefficients [46]. This model is describedin detail in Paper I and the modifications for the conditions of VAWTs arepresented in Papers II and IV. The model consists of three parts: unsteadyattached flow, stall onset and separated flow.

3.3.1 Unsteady attached flowThe time varying bound vortex during the unsteady attached flow is repre-sented by an effective angle of attack

αEn = αn−Xn−Yn−Zn, (3.35)

27

where α is the geometrical angle of attack and X , Y and Z are the deficiencyfunctions, which are empirically derived based on the flow velocity and thepitch rate, and they can be found in Paper I. Indices n and n− 1 in equa-tion (3.35) represent the current and previous time-steps. The delayed angleof attack due to the lag in pressure response is calculated as

α′n = αn−Dαn , (3.36)

where Dα is the deficiency function

Dαn = Dαn−1 exp(−∆s

Tα

)+(αn−αn−1)exp

(− ∆s

2Tα

), (3.37)

with an empirically derived time constant Tα , which is found in table 3.1. Thenon-dimensional time-step ∆s in equation (3.37) is calculated as

∆s =2V ∆t

c. (3.38)

Here and in equation (3.40), the velocity V stands for the relative flow velocityVrel from equation (3.6) for the streamtube model and for the relative flowvelocity |~V −~Vb| from equations (2.4) and (3.30) for the vortex model.

3.3.2 Condition of dynamic stallDue to flow reversal within the boundary layer, a leading edge vortex formsat the airfoil surface. The critical angle of attack αcr is used to define thecondition at which the dynamic stall may begin

αcrn =

αds0 |rn| ≥ r0,

αss +(αds0−αss)|rn|r0|rn|< r0,

(3.39)

where the reduced pitch rate rn is

rn =αnc2V

. (3.40)

Here, α is the pitch rate, r0 is the reduced pitch rate which delimits the quasi-steady stall and the dynamic stall; r0 = 0.01 for symmetrical NACA-airfoils,see Paper I. The static stall onset angle αss and the critical stall onset angleαds0 are listed in table 3.1. The following dynamic stall condition is used∣∣α ′∣∣> αcr→ stall. (3.41)

3.3.3 Separated flowThe effects of separated flow are divided into two groups: trailing edge sep-aration and leading edge vortex convection. The trailing edge separation is

28

associated with the time delay in the movement of the boundary separationpoint, and it is obtained via Kirchhoff’s flow approximation

f ′n =

1−0.4exp

(|α ′n|−α1

S1

)|α ′n|< α1,

0.02+0.58exp(

α1−|α ′n|S2

)|α ′n| ≥ α1,

(3.42)

where f ′ is the delayed separation point and α1, S1 and S2 are constants basedon the airfoil profile and the local Reynolds number, found in Paper I. Theboundary layer around the blade itself is time dependent, which is superim-posed on the pressure response delay. This additional delay is represented bythe dynamic separation point

f ′′n = f ′n−D fn . (3.43)

Here, the deficiency function D fn is

D fn = D fn−1 exp(−∆s

Tf

)+(

f ′n− f ′n−1)

exp(− ∆s

2Tf

), (3.44)

where Tf is an empirically derived time constant, table 3.1. The normal forcecoefficient for the unsteady conditions before the dynamic stall onset is ob-tained as

C fNn

=CNααEn

(1+√

f ′′n2

)2

. (3.45)

After the stall condition is met, the leading edge vortex convects over thesurface of the airfoil towards the trailing edge. During this process, a signifi-cant increase in the normal force is present

CvN = B1

(f ′′− f

)Vx, (3.46)

where CvN is the normal force coefficient during the vortex convection (so-

called “vortex lift”), Vx and B1 are parameters based on the local Reynoldsnumber and the airfoil profile and are found in Paper I. The normal forcedecreases rapidly when the vortex passes the trailing edge. The total normalcoefficient is estimated as

CN =C fNn

+CvN . (3.47)

Figure 3.4 shows an example of the normal force coefficient of a pitchingblade, simulated by the dynamic stall model, and the features described aboveare present.

The calculation of the tangential force coefficient is based on the Kirch-hoff’s flow relation using the dynamic separation point with the modificationby Sheng et al. [47]

CT, f ′′ = ηCNαα

2E

(√f ′′−E0

), (3.48)

29

0 5 10 15 20 250.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Angle of attack α [deg]

Nor

mal

forc

e co

effic

ient

CN Formation of

leading edge vortex,dynamic stall onset

Vortex convection

Vortex passestrailing edge,full stall

Start of the flowreattachment

Figure 3.4. Normal force coefficient for the pitching NACA0021 airfoil of a periodicalmotion α = 12+10sin(αt), α = 7.20 rad/s, V∞ = 33 m/s, c = 0.55 m.

where η and E0 are the empirical constants listed in table 3.1.For low angles of attack, the values of CN approach the static values accord-

ing to equations (3.45) and (3.47). However, the tangential force coefficientCT, f ′′ does not approach the static value at low angles of attack when calcu-lated with equation (3.48). Therefore, a modification is applied to solve thislimitation. The coefficient CT, f is calculated for the static separation point f

CT, f = ηCNαα

2E

(√f −E0

). (3.49)

Depending on the absolute value of the geometrical angle of attack α and thereduced pitch rate rn, the tangential force coefficient CT is estimated as

CT =CT, f ′′+(CT,static−CT, f

)CT,scale, (3.50)

where CT,static is the tangential force coefficient from the static lift and dragdata [46]. The scaling constant CT,scale includes the scaling factor for the angleof attack CT,scale,α

CT,scale,α =

0 |α|> αu,

1 |α|< αl,αu−|α|αu−αl

αl ≤ |α| ≤ αu,

(3.51)

30

Table 3.1. Empirical constants for the dynamic stall model model

Airfoil Tα αss [deg] αds0 [deg] B1 η E0

NACA0012 3.90 14.95 18.73 0.75 1 0.25NACA0015 5.78 14.67 17.81 0.50 1 0.25NACA0018 6.22 14.68 17.46 0.50 1 0.20NACA0021 6.30 14.33 17.91 0.50 0.975 0.15NACA0025 6.95 13.59 17.22 0.50 0.90 0.18

and the scaling factor for the pitch rate CT,scale,r

CT,scale,r =

0 |rn|> ru,

1 |rn|< rl,ru−|rn|ru−rl

rl ≤ |rn| ≤ ru.

(3.52)

The scaling constant CT,scale in equation (3.49) is defined as the product of thescaling factors

CT,scale =CT,scale,αCT,scale,r. (3.53)

The parameters αu and αl in equation (3.51) together with ru and rl in equa-tion (3.52) represent the upper and the lower scaling limits of the angle ofattack and the reduced pitch rate respectively. These limits are used to ensurethe smooth transition of the tangential force coefficient to its static values atlow angles of attack and low pitch rate. This method is similar to the one pub-lished in Paper IV with the exception that the pitch rate is taken into accountin the present implementation. The scaling limits of the angle of attack areαu = αss and αl = 0.5αss and the limits of the reduced pitch rate are ru = 2r0and rl = r0. Please note that these values of the scaling limits have been cal-ibrated for the simulation cases in the current work. Further investigation ofthe scaling limits is recommended if data for other turbines become available.

3.3.4 Application to the turbinesDue to the circulatory motion of the blades during the turbine operation, theblade vortices may be released faster at the downwind disk. Figure 3.5 showsthe vortex shedding structure of a Darrieus turbine at low TSR. This figureis based on the velocity measurements of the straight-bladed Darrieus turbineoperating at a TSR of 2.14 in a water channel, obtained by Brochier et al. [48].It is shown that the dynamic stall vortices are released and swept away whenthe blade passes Quadrant III; thus, the flow is fully separated. This can beboth due to the circulatory motion of the blade and due to the highly turbulentflow, as noted in Ref. [48].

An additional modification to the dynamic stall model is applied to accountfor the vortex shedding. This is done by modeling fast release of the trail-ing and leading edge vortices: the delays in the angle of attack, the leading

31

0º

90º

180º

4

5

4

5

1 1

2

3

3

2

a

a

a' a

a'

aa

b

bb

b

b

cIVI

IIIII

V∞

Figure 3.5. Flow visualization in the dynamic stall condition at λ = 2.14, taken fromRef. [48]. a, a′, b and c denote vortices.

edge separation point and the vortex lift are set to zero when the blade passesQuadrant III:

Quadrant III→ α′ = αE ,Cv

N = 0 (3.54)

When the normal and tangential force coefficients are found, the lift anddrag coefficients are estimated as

CL =CN cosϕ +CT sinϕ, (3.55)

CD =CN sinϕ−CT cosϕ +CD0

(1−CT,scale

), (3.56)

where CD0 is the drag coefficient at the zero angle of attack. Note that CD0is already taken into account in CT for low angles of attack and low pitchrate (CT,scale = 1) through CT,static according to equation (3.50). To assure asmooth transition at increased angles of attack and pitch rate, CD0 is linearlyinterpolated to avoid applying the value twice.

The integration of the dynamic stall model is different between the stream-tube and the vortex model. The angles of attack and the flow velocities inthe streamtube model are calculated for the whole revolution forming an inputmatrix to the dynamic stall model. Then, the dynamic stall model runs twoloops, which is sufficient to approach the convergence, see Paper I. The liftand drag coefficients from the dynamic stall model are then used within thestreamtube model to estimate the blade forces.

Since the vortex model is time-dependent, the dynamic stall model is usedevery time-step. Hence, the inputs to the dynamic stall model are from thecurrent and the previous time-steps. Note, that the changes of flow velocity

32

due to the vortex contribution and the effective angle of attack are calculatedwithin the vortex model. Thus, the attached flow part of the dynamic stallmodel (equation (3.35)) is omitted when combined with the vortex model.

33

4. Measurements of aerodynamic forces

This chapter summarizes the force measurements on the 12 kW VAWT withthe diameter of 6.48 m which is operated at an open site north of Uppsala,Sweden (N5955′32”, E1735′12”). The measurement campaign is describedin detail in Paper IV and the experimental data on the turbine forces is pub-lished in Paper V.

4.1 Experimental setupThe measurements of the forces were conducted on the VAWT designed andconstructed at Uppsala University in 2006. The measurements of the powercoefficient on this turbine were conducted in 2009. The results have shownthat the power coefficient had the maximum value of 0.29 at the TSR of 3.3[21]. In 2014, this turbine was renovated and equipped with force sensors,see figure 4.1. Due to the installed force sensors, the radius of the turbine hasincreased from 3 to 3.2 m. The force sensors are single-axis load cells whichmeasure tension and compression and are installed between the turbine huband support arms of one blade. Spacers of equal radial distance and weightare installed between the hub and the support arms of other two blades. Theload cells assembly is depicted in figure 4.2. The specifications of the turbinewith installed load cells are listed in table 4.1.

4.2 Data treatmentFour load cells are used to measure forces on one blade and its support arms.The sum of the measured forces F0, F1, F2 and F3 represents the radial force.

Table 4.1. Nominal parameters of the 12kW VAWT used for the force measurements

Number of blades 3Turbine radius 3.24 mHub height 6 mBlade length 5 mBlade profile NACA0021Chord length 0.25 mBlade pitch angle 2

34

Figure 4.1. The 12 kW turbine, designed and built at the Division of Electricity atUppsala University. The turbine is equipped with load cells used for the force mea-surements. The torbine location is N5955′32′′, E1735′12′′.

The turbine rotational speed can be kept constant by controlling the electricload of the generator, see Ref. [21] and Paper IV. When the speed of the tur-bine is nearly constant, it can be assumed that the rate of change in angularmomentum approaches zero, and centrifugal force on the rotor is constant.The normal force is the difference between the radial force and the centrifugalforce

FN = F0 +F1 +F2 +F3−FC, (4.1)

and the centrifugal force FC is estimated as

FC = mΩ2LC, (4.2)

where m is the mass of the blade and support arms and LC is center of massof the blade and support arms. The tangential force on the blade and supportarms is related to the difference between F0+F2 and F1+F3, and based on theassumption of the constant centrifugal force, it can be estimated as

FT =L1

2LB(F0 +F2−F1−F3) . (4.3)

Dimensions L1, LB and LC in equations (4.2) and (4.3) are illustrated in fig-ure 4.2 and specified in table 4.2.

35

R

LC

center of massof blade andsupport arms

Load cells setup

FN

FT

LB

(a)

F3

F2

F1

F0

L1

spacers

(b)

Figure 4.2. The load cells assembly on the 12 kW VAWT used for the force measure-ments. (a) Notation of the normal and tangential force; (b) Notation of the measuredloads.

36

Table 4.2. Dimensions of the load cell assembly and the maximum measured errors

Dimension Value Maximum error

Center of mass LC = 1.83 m ∆LC =±0.01 mDistance from sensor to blade LB = 2.99 m ∆LB =±0.01 mHorizontal distance between sensors L1 = 0.200 m ∆L1 =±0.0005 mMass of blade and support arms m = 35.79 kg ∆m =±0.05 kg

θ = 0°

θ = 90°

FN ≈ 0

FN ≈ 0

streamlines

downwind diskupwind disk

middle disk

V∞

Figure 4.3. Definition of the blade azimuth angle θ from the normal force.

The correlation between the blade position and the incoming wind flowdirection was not determined during the measurements. Therefore, the normalforce response was used to estimate the azimuthal angle of the blade θ . Thenormal force on the blade is close to zero when the blade passes from theupwind to the downwind side of the rotor, see figure 4.3. It was assumedthat during constant rotational speed, the blade is at θ = 90 when passing themiddle of the upwind disk during one revolution. The start position of theblade θ was updated on every revolution.

4.3 Data treatmentThe measurement data were obtained from September to December 2014.Both wind speed and wind direction had large variations at the measurementsite. Therefore, conditions of steady flow were defined to extract data bins.The time span of the steady conditions was divided into two parts: the time

37

0 5 10 15 16 20 244

5

6

7

time [s]

V ∞ [m

/s]

wake stabilization(wake time)RSD of V∞ is ≤ 10 %

steady flow operation(disk time)RSD of V∞ is ≤ 5 %

Figure 4.4. Allowed variations of the asymptitic wind speed during steady conditions.Illustration of the wake time and disk time.

required to build stable wake (further referred as the ”wake time”) followedby the time of steady flow operation (further referred as the ”disk time”). Thewake time was set to 16 s (corresponding to 10 revolutions at 40 rpm), and thedisk time was set to 8 s (5 revolutions at 40 rpm). The relative standard devia-tion (RSD) was used when defining the steady flow condition. The followingexpression for the RSD was used

RSD =

(1n

n

∑j=1

(x j−〈x〉)2

) 12 1〈x〉×100%, (4.4)

where 〈x〉 is

〈x〉= 1n

n

∑j=1

x j. (4.5)

The flow was considered steady when the RSD of the asymptotic wind speedV∞ during the wake time was ≤ 10% and the RSD of V∞ was ≤ 5% during thedisk time, see Figure 4.4. The RSD of wind direction and the turbine rotationalspeed was ≤ 1% during both wake and disk time.

Measured forces were analysed for different sets of data at the steady con-ditions. A wide range of TSRs from λ = 1.7 to λ = 4.6 was obtained. Thevalue of TSR was estimated using the average rotational speed Ω and windvelocity V∞

λ =〈Ω〉R〈V∞〉

, (4.6)

where the averages 〈Ω〉 and 〈V∞〉 are taken over time with the steady condi-tions.

38

The air density was estimated based on the measured weather parameters

ρ =1T

( p287.1

−2.7 ·10−10hexp(0.06318(T +273.15))), (4.7)

with the air temperature T in C, the air pressure p in Pa and the relativehumidity h in %RH. The kinematic viscosity was obtained as the function ofthe air temperature according to Ref. [49].

4.4 Measurement accuracyEstimation of maximum error of a value based on measurements can be per-formed as

|∆Z|=∣∣∣∣∂Z

∂x∆x∣∣∣∣+ ∣∣∣∣∂Z

∂y∆y∣∣∣∣+ . . . (4.8)

where x, y, . . . are measurements and ∆x, ∆y, . . . are the maximum errorsof the measurements. The normal and tangential forces depend on severaldifferent measurements (m, L1, LB and LC) as shown in equations (4.1) to (4.3).Therefore, it is chosen to use the maximum errors to quantify the measurementaccuracy.

The maximum errors ∆m, ∆L1, ∆LB and ∆LC are listed in table 4.2. Pleasenote that the measured forces F0, F1, F2 and F3 include the no-load forces,which are documented together with their maximum errors in Paper IV. Theexpressions for the maximum errors of the normal and tangential forces areobtained by applying equation (4.8) to equations (4.1) to (4.3)

∆FN =±(0.0049Ω2rpm +0.072Ωrpm +23), (4.9)

∆FT =±(0.0058|FT |+1.1). (4.10)

The turbine rotational speed Ωrpm in equation (4.9) is expressed in rpm. Themaximum error of the normal force ∆FN is dependent on the rotational speedof the turbine. The maximum error of the tangential force ∆FT depends on itsown value FT , though the dependence is rather weak. Note, that the maximumvalue of the shape of the normal force is independent of the centrifugal forceFC and its value is ∆FN,shape =±23 N.

4.5 Evaluation of the measurement methodThe measured normal and tangential forces are presented in figure 4.5. Theexperimental values are compared with the simulated FN and FT to show thevalidity of the measurement method. Please note that the detailed compari-son of the experimental and simulation data is presented further in chapter 5.

39

The simulations in figure 4.5 are performed with the streamtube model de-scribed in section 3.1. The behaviour of the measured normal force is consis-tent and agrees with the model. The response of the measured tangential forceis disturbed by oscillations which are not expected according to the simulationmodel. It is concluded in Paper IV that the measurements of the tangentialforce are disturbed by the turbine dynamics, which is the subject of futurestudies. Hence, the tangential forces measured on the 12 kW VAWT are notfurther compared to the simulations.

40

Figure 4.5. Force response from the 12 kW VAWT at λ = 3.6, 〈Ω〉 = 50.6 rpm. (a)the normal force; (b) the tangential force.

41

(a)

(b)

5. Results and discussion

This chapter presents the comparison between experimental data and the de-veloped simulation tools. First, the performance of the blade force model ispresented for the motion of a pitching blade. This test shows the applicabilityof the developed dynamic stall model. After the presentation of the resultson a pitching blade, the developed simulation models are tested for the wholeturbine. The models are assessed against the force measurements on two tur-bines. The first one is the 12 kW VAWT at Uppsala University (described inchapter 4) and the second turbine is the 17 m tall VAWT with curved bladeswhich was operated by Sandia National Laboratories in the 1980s [50].

5.1 Pitching bladeAs discussed in chapter 2, the variation of the angle of attack is natural duringthe operation of a VAWT. The magnitude of the angle of attack increases whenthe turbine is operated at low TSRs and dynamic stall is taking place. Theunsteady flow effects which are present during the dynamic stall are handledwithin the blade force model. This model has been tested before the couplingit with the turbine models.

Experimental data on pitching airfoils were used to assess the blade forcemodel. The measurement data were obtained from Glasgow University [51,52] where the experiments on symmetrical NACA airfoils were conducted.The chord length of the airfoil was 0.55 m and the span was 1.61 m. Dur-ing the experiments, the airfoils were pitched about the quarter chord and thevariations of the angles of attack were similar to those of VAWTs

α = arctan(

cosθ

λ + sinθ

). (5.1)

The experiments were conducted in a wind tunnel at different Reynolds num-bers depending on the asymptotic wind velocity. The reduced frequency k wasused to express the blade pitch frequency (which corresponds to the turbine ro-tational speed)

k =Ωc2V∞

. (5.2)

Two versions of the blade force model have been tested. The first one refersto the dynamic stall model used with the streamtube model and it is further de-noted as the DS model. The second model represents the dynamic stall model

42

for the vortex model and it is referred to as the VDS model. The difference be-tween the DS and VDS models is in the modeling of the attached flow, whichis described in section 3.3.4. The normal and tangential force coefficients CNand CT from the DS and VDS models are compared to the experimental data.The assessment is done for the range of TSRs from 2.60 to 4.73 to cover awide range of operational conditions of VAWTs, including the dynamic stallregion. The performance of the models is presented for two symmetrical air-foils NACA0015 and NACA0021. The reduced frequency of the pitching isk = 0.05 (corresponding to Ω = 5.1 rad/s) and the incoming flow velocity isV∞ = 28.4 m/s resulting in the Reynolds number of approximately 1,000,000,which is a reasonable value for an operating VAWT.

Normal and tangential force coefficientsAt high TSRs, the results of CN by the DS and VDS models are similar andclose to the experimental data, figures 5.1 and 5.2. The peaks of CN and CTsimulated by the VDS model are closer to the experimental values than the DSmodel. The difference between the results for the NACA0015 and NACA0021profiles is rather small. The maximum magnitude of the angle of attack is12.20 and 13.80 for the TSR of 4.73 and 4.19 respectively. This comparisonshows that the VDS model is sufficient for the modeling of the attached flow.

At the TSR of 3.34, the difference between the VDS and DS models ismore evident, figures 5.3 and 5.4. The maximum magnitude of the angle ofattack is 17.40 which is in the stall region. The delay in the flow reattachmentis evident from the “loop” of the CN and CT response, which is a distinctivefeature of the dynamic stall phenomenon. Peak values of the CN response areoverestimated by both models, although the result of the VDS model is closerto the measured data. The shape and the peak values of the CT -curve predictedby the VDS model are close to the experimental data.

The maximum magnitude of the angles of attack for the TSR of 2.60 is22.60, which corresponds to the deep stall region, figures 5.5 and 5.6. Atthis condition, the “loop” of the force coefficients is wider than at the TSRof 3.34 and the reattachment of the flow is further delayed. Results of thesimulated CN response are similar by both the DS and VDS models, althoughthe VDS model is slightly closer to the experimental data. The predictions ofthe CT response by the VDS model have considerably higher agreement withthe experimental data than the DS model. The simulated CT values are higherthan measured during the return from the stall for both the NACA0015 andNACA0021 airfoils.

DiscussionsThe presented results show that both blade force models can reproduce thedynamic stall “loop” corresponding to the separated flow. However, the VDSmodel agrees better with the experimental data. This introduces an additionaladvantage of the VDS model in terms of the simplicity of the model. Since

43

(a)

(b)

Figure 5.1. Force response during the pitching motions of NACA0015 airfoil witha maximum magnitude of 12.20 (corresponding to λ = 4.73). (a) Normal forcecoefficient; (b) tangential force coefficient.

44

(a)

(b)


45

(a)

(b)


46

(a)

(b)


47

(a)

(b)


48

(a)

(b)


49

Figure 5.7. Comparison of the angle of attack during the motion of the pitching bladeand the blade of a VAWT at two different TSRs. The angle of attack during the pitch-ing motion is estimated according to equation (5.1). The streamtube model is usedto simulate α of the blade of the VAWT with the zero pitch angle, and the motion isgiven by equation (3.7). Please note that the curvature effects are included into thecalculation of α within the steamtube model.

the attached flow part is omitted in the VDS model, the number of empiricalconstants in the model is reduced, compared to the DS model.

Both the VDS and the DS models use Kirchhoffs’s flow approximationof the flow over a thin plane when calculating the separation point, equa-tion (3.45). The actual point of the flow separation may not be represented us-ing this assumption, and this is a common limitation of the Leishman-Beddoesbased models. Additionally, it should be emphasized that the flow around thepitching blade does not represent the flow around the blade in a turbine, al-though the variations of the angle of attack are similar. The angle of attack forthe presented pitching motion follows equation (5.1) and λ is used, which canbe compared to equation (2.6) where ΩR

Vdis used. Due to the extracted energy,

the incoming flow velocity is not constant at different azimuthal angles θ forthe blades of VAWTs. The history of angles of attack for the pitching bladeis compared to the angles of attack for the blade of a VAWT, see figure 5.7.The angles of attack for low TSR are similar between the pitching motionand the motion in a VAWT. The difference in the angles of attack history be-tween the pitching blade and the blade of VAWT increases with increasedTSR. Please note that the magnitude of the angle of attack at the downwindregion of VAWT is lower due to the lower flow velocity, while the flow veloc-ity at the pitching blade remains constant.

50

5.2 Turbines5.2.1 12 kW VAWTThis section presents the assessment of the streamtube and the vortex mod-els to the measurements on the 12 kW VAWT. The simulation models aredescribed in chapter 3 and the experiments on this turbine are presented inchapter 4. The results of the normal force are presented at a range of TSRsfrom 1.84 to 4.57. The maximum measurement error is presented for everydata set (please note that the maximum error of the shape of the normal forceis lower as specified in section 4.4). The measured normal force is the av-erage over at least five turbine revolutions at the steady flow conditions (thedefinition of the steady conditions is given in chapter 4).

Normal forceThe vortex and streamtube models show similar results at low TSRs, see fig-ures 5.8 and 5.9. The slope of the normal force at 0 < θ < 45 is overesti-mated by both models. The conditions of λ = 1.84 and λ = 2.55 correspondto dynamic stall, and correct estimate of the relative flow velocity Vrel is crit-ical for the dynamic stall modeling, see section 3.3. Since the measurementsare in 3D and the simulation models are in 2D, the estimated value of Vrel cancause an earlier onset of the dynamic stall in the upwind region. The maxi-mum magnitude of the FN response at the upwind region is overestimated byboth models. The offset at θ = 0 is mainly due to the blade pitch, and thevalue of the offset is well predicted by both models.

The difference between the streamtube and the vortex model is clearer atλ = 3.06, see figure 5.10. The FN-slope at 0 < θ < 45 is overestimatedsimilarly to λ = 1.84 and λ = 2.55. The peak of the simulated normal forceby the vortex model matches the experimental data at the upwind side, whilethe FN response by the streamtube model is shifted. The models show similarbehaviour at the downwind region.

The results at λ = 3.44 are presented in figure 5.11. This is close to the opti-mal TSR (λ = 3.3 corresponding to the power coefficient of 0.29), which wasmeasured for this turbine before the increase in radius due to the installationof the load cells. The slope of the normal force at 0 < θ < 45 predicted bythe vortex model is very close to the one from the experimental data. The vor-tex model shows better agreement with experimental data than the streamtubemodel. There is a drop in the FN-curve observed at 230 < θ < 330, whichis not predicted by the models. This drop is also present at higher TSRs, andthe discussions regarding the FN-drop are found further in this section.

The normal force at λ = 3.74 is presented in figure 5.12. The results by thevortex model are in a good agreement with experimental data except the FN-drop at the downwind, which is missed by the model. The streamtube modelproduces higher flow expansion rate than the vortex model. Otherwise, theresults of the models are similar to each other.

51

Figure 5.8. The normal force response at λ = 1.84 and Ω= 40.29 rpm. The maximummeasured error is ∆FN =±34 N.

Figure 5.9. The normal force response at λ = 2.55 and Ω= 49.89 rpm. The maximummeasured error is ∆FN =±39 N.

52

Figure 5.10. The normal force response at λ = 3.06 and Ω = 65.36 rpm. The maxi-mum measured error is ∆FN =±49 N.


53


The results at the high TSRs of 4.09 and 4.57 are presented in figures 5.13and 5.14. At these conditions, the flow expansion strongly affects the turbineaerodynamics. The maximum magnitude of the FN response at the upwind sideby the vortex model is matching the experimental value, while the streamtubemodel slightly overestimates it. The maximum magnitudes at the downwindside in the experimental data are higher than the simulated values. The afore-mentioned FN-drop in the downwind is clearly observed at λ = 4.57.

DiscussionsThe presented simulations are in 2D, while the measured data are in 3D, andthe contribution of the support arms is included in the measured forces. There-fore, it is expected that the models do not reproduce the experimental resultsin a great detail. Additionally, the simulated flow expansion is limited to thehorizontal plane only since the vertical expansion is not included in the mod-els.

Over the whole range of the presented data, the models perform better atthe upwind side. This is expected, since the dynamic stall vortex interactionis not implemented in the simulation models. Furthermore, since the supportarms are not included in the models, the interaction between the blades andthe released vortices from the support arms can not be reproduced.

The performance of the model at low TSRs is highly dependent on the bladeforce model, which has its limitations. The point of flow separation in the

54



55

dynamic stall model is calculated assuming that the blade is a flat plate andthe incoming flow velocity is constant. These assumptions should be takeninto account when evaluating the performance of the models for the wholeturbine.

One interesting phenomenon observed in the experimental data is the dropof the normal force at the downwind region at λ > 3.4. Similar drop is ob-served at low TSRs in the data on the normal force coefficient from mea-surements on the curve-bladed VAWT operated by Sandia National Laborato-ries [50], which is discussed in section 5.2.2. However, the drop at high TSRshas not been noted before. This drop is not expected to be caused by the towerwake, since the tower diameter is considerably smaller than the region of theFN-drop. It might be that the FN-drop is caused the blade tip vortices or by thesupport arms.

The effect of the blade pitching is noted in the experimental data for all thetested TSRs. This effect is observed as an offset of the normal force at θ = 0.The offset of FN is reproduced well by both models.

The maximum measurement error is presented for all the tested conditions.As discussed in section 4.4, the shape of the normal force is likely to remain,though the measurement error can change the scale of the magnitudes of FN .Therefore, the measurement error has to be considered when assessing thesimulation models against the experiments.

The major advantage of the presented models is the computation speed.The streamtube model is faster than the vortex model. However, the vortexmodel gives a more detailed picture of the flow and is more accurate. Onesimulation of the vortex model with 100 revolutions is in the order of minuteson the single core machine, which is much faster than simulations with the 2DCFD models. Simulations in 3D can overcome the limitations of the presentedmodels, however the computational time of a single simulation in 3D withCFD models can take a few months [37].

5.2.2 Sandia 17-m VAWTSeveral VAWTs with curved blades were studied by Sandia National Labora-tories in the 1980s – 1990s in Albuquerque, New Mexico, USA [50,53,54] andthe Sandia 17-m was one of them. It was a two-bladed VAWT with radius andheight of approximately 17 m as depicted in figure 5.15. Force measurementson the Sandia 17-m turbine [50] were used in this work due to the lack ofexperimental data on the full-size straight-bladed VAWTs operated at an opensite at high Reynolds numbers. In particular, it was important to compare thesimulations against the tangential forces, since the tangential force responsefrom the 12 kW VAWT was distorted and only the normal force could be usedfor this type of analysis. Please note that the measuring resolution on this tur-bine was lower than on the 12 kW VAWT (presented in section 5.2.1). The

56

R = 8.36 mH = 16.72 m

Figure 5.15. Basic illustration of the Sandia 17-m VAWT with two blades.

blade profile of the Sandia 17-m was NACA0015 and the chord length wasc = 0.61 m. Pressure transducers for the force measurements were installedat the midspan of the blades. Thus, it is important to emphasize that the dataon this turbine are in 2D (contrary to the force measurements on the 12 kWVAWT, which are in 3D). The normal and tangential force coefficients wereobtained by using the pressure distribution and the incident flow velocity Vrelat the blade

CN,T =FN,T,norm

0.5ρVrelc, (5.3)

where FN,norm and FT,norm are the measured normal and tangential forces perunit span. The turbine rotational speed was kept at the constant level ofΩ = 38.7rpm. The average Reynolds number was approximately 1.4× 106

depending on the asymptotic wind velocity.

Normal and tangential force coefficientsThe data presented in this work covers a range of TSRs from 2.20 to 4.60. Dataat λ = 2.20, λ = 2.33 and λ = 2.49 show that the dynamic stall is predictedearlier than in the experimental data, figures 5.16 to 5.18. The slope of theCN curve is overestimated by both models. This deviation may originate fromthe static lift and drag data, which are used by the blade force model for boththe streamtube and the vortex models. The overestimation of the magnitudeof the CN response is present and it is similar to the normal force responsefrom the data on the 12 kW VAWT. There is a drop in the CN-curve centredat θ ≈ 225 at λ = 2.20, λ = 2.33 and λ = 2.49, which is not reproducedby the simulation models. This peak is likely to be due to the collision ofthe blade with the released dynamic stall vortices from the upwind side. Thewake interaction has also been noted when the experiment was conducted, seeRef. [50]. To hit the blade at θ ≈ 225, the impacting vortex would originateat θ ≈ 135, similarly to the illustration of the vortex shedding in figure 3.5.

57

There is a drop in the CN response at θ ≈ 135, which should correspond tothe vortex release. This CN-drop is less evident at increased TSR, and it isnot present at higher TSRs. Since the dynamic stall vortex interaction is notimplemented in the present simulation models, the CN-drop is not modeled.

The experimental values of the tangential force coefficient do not approachzero values at θ = 180, when the blade moves directly towards the downwindside, and the angle of attack approaches zero. This is due to a measurement er-ror in the pressure response from the sensors, and it is reported by Akins [50].Such a behaviour is observed for all the presented TSRs, and hence, the simu-lated results should not be considered erroneous in this region.

The results of the tangential force coefficient by the vortex model are closerto the experimental data than the streamtube model. The maximum magnitudeof the CT response by the vortex model is very close to the experimental val-ues in the upwind region at λ = 2.20, λ = 2.33 and λ = 2.49. For both thestreamtube and the vortex models, the agreement between the simulated datain the upwind region is better than in the downwind region. Similarly to theCN response, this is expected since the flow is highly disturbed by the wake atthe downwind side in general and due to the dynamic stall vortex interactionin particular.

The dynamic stall is less pronounced at λ = 3.09 and the wake interactionat the downwind region is no longer evident, figure 5.19. However, the modelsstill overestimate the CN values at the upwind region. The measured CT curveis shifted from the simulated results. However, the shape of the CT responseby the vortex model is very similar to the shape of the measured CT .

The comparison at λ = 3.70 and λ = 4.60 is presented in figures 5.20and 5.21. The flow expansion is overestimated by the streamtube model. Thevortex model estimates the normal force coefficient quite well, although over-shoots of the CN response are present at θ = 90. The shape of tangentialforce coefficient by the vortex model is close to the experimental values. How-ever, the magnitude of the experimental CT -curve is higher than the simulated,which is also observed at λ = 3.09.

The comparison to the Sandia 17-m turbine shows that the streamtube andthe vortex model can predict both the tangential and the normal forces. Thevortex model agrees better with the experimental data and the shape of thetangential force by the vortex model is close to the measurements.

58

(a)

(b)

Figure 5.16. Force response at λ = 2.20. (a) Normal force coefficient; (b) tangentialforce coefficient.

59

(a)

(b)


60

(a)

(b)


61

(a)

(b)


62

(a)

(b)


63

(a)

(b)


64

6. Conclusions

The challenge of modeling unsteady aerodynamic blade forces on VAWTs hasbeen addressed within this thesis. Significant progress has been achieved in theaerodynamics of VAWTs within both the development of simulation modelsand the measurements on VAWTs. New simulation tools were developed andassessed against existing experimental data on a pitching blade as well as onVAWTs operating at an open site. Additionally, new force measurements on a12 kW VAWT have been conducted and used when validating the models.

All the simulation models used in this thesis have been assessed againstexperimental data for a wide range of TSRs to cover the regions of dynamicstall, nominal operation and the region of high flow expansion.

The flow expansion model as by Read and Sharpe [40] together with themodified version of the Leishman-Beddoes [44,45] dynamic stall model wereincluded into the streamtube model. The comparison against experimentaldata has shown clear improvement of the streamtube model compared to itsprevious versions, see Paper II.

The accuracy of the dynamic stall model has been further improved by cou-pling a free vortex model to the dynamic stall model for a pitching blade. InPaper III, it has been shown that unsteady attached flow solution can be ob-tained within the vortex model, which has reduced the number of empiricalconstants in the dynamic stall model. Additionally, the efficient way of mod-eling the blade with a point vortex has increased the computational speed.

Measurements of the blade forces using load cells have been conducted andthe experimental method has been evaluated in Paper IV. The measurementshave shown that it is possible to obtain the aerodynamic normal force with thepresented method. However, the tangential force is disturbed by the dynamicsof the turbine and the load cell assembly. Additionally, a numerical estimationof the maximum measurement error has been done and the sources of the errorhave been comprehensively described.

The measured aerodynamic normal forces acting on the VAWT blade havebeen analyzed in Paper V. The measurements show new trends which have notbeen observed before in experimental data on VAWTs with high operationalReynolds number. These are the flow expansion at high TSRs and the drop ofthe normal force at the downwind side.

The performance of the streamtube and the vortex models has been evalu-ated by comparing simulation results to existing and to recently measured dataon blade forces. Both models are stable even at low TSRs. The models per-form better at the upwind side, which is expected as the flow at the downwind

65

side is highly disturbed by the wake and dynamic stall vortex interaction aswell as 3D effects of the wake are not implemented. The streamtube modeloverestimates the flow expansion at high TSRs, when compared with the vor-tex model and experimental data. The vortex model agrees quite well with themeasurements at TSRs of 3 and higher. However, the aforementioned drop inthe normal force at the downwind side at high TSRs is not reproduced.

The vortex model agrees better with the experiments than the streamtubemodel does. Although it does not reproduce the experimental results in a greatdetail, the vortex model can be used to simulate the turbine with reasonableaccuracy at low computational cost.

66

7. Suggestions for future work

Further improvements can be achieved within the presented simulation toolsand the experimental work. Thus, the recommendations for future studies aredivided into two groups.

7.1 Simulation toolsA double-wake model for the dynamic stall will introduce interaction of thetrailing edge vortices with the dynamic stall vortices. That should significantlyimprove the accuracy of the vortex model at the downwind region, which ex-cludes one of its major limitations. A way to represent the separation pointof the dynamic stall vortex should be found, since the present dynamic stallmodel assumes that the blade is a flat plate. The vortex model is then requiredto handle an additional vortex released from a blade. This should be a decentstart for further model development. A complete model of the double wakeshould first be tested for the motions of a pitching blade against the identicalexperimental data used in this thesis. This should be done to ensure that themodel performs well before coupling with a more complex vortex model forthe turbine. Additionally, a way to calculate the lift force coefficient from theblade circulation can be implemented to reduce the dependence of the modelon empirical data.

Since the vortex model is two-dimensional, the tip vortices are not takeninto account. Other effects not included in the vortex models are three-dimen-sional wake effects and the influence of support arms and tower. It should beinvestigated whether it is possible to develop an efficient method of includingthose three-dimensional effects into the vortex model, without developing acompletely new complex three-dimensional vortex model.

7.2 Experimental workA way to measure an actual blade position relative to the wind flow directionshould be developed for the 12 kW turbine. For steady wind flow, this rela-tively simple measurement will significantly improve the analysis of the flowexpansion.

It has been observed that the tangential force measured with the presentload cells assembly has dynamic disturbances. To reduce this problem, the

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measurements can instead be done with sensors placed close to the turbineblades. Investigations of the turbine dynamics could be performed with newforce sensors and the existing load cell assembly. Additional studies on thedrivetrain of the 12 kW turbine can provide the mechanical efficiency as afunction of torque, rotational speed and probably the temperature of the bear-ings. With knowledge of turbine dynamics and the drivetrain efficiency, itshould be possible to conduct further studies on both the aerodynamics andthe interaction between the turbine and the generator.

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8. Summary of papers

This section presents all papers that are included in the thesis.

Paper IDynamic Stall Modeling for the Conditions of Vertical Axis WindTurbinesThis paper aims at finding a dynamic stall model that is suitable for simula-tions of VAWT. The baseline model is the one by Leishman-Beddoes whichwas originally designed for the rotors of helicopters, and therefore tuned foroperation at relatively high Mach numbers. Three versions of this model aredescribed in a great detail and the models are recreated from a number of ref-erences. Additionally, one version is further modified for sign changes of theangle of attack which are inherent during the operation of VAWT. Measure-ment data on pitching airfoils obtained from the wind tunnel experiments atGlasgow University, Scotland, are used when evaluating the performance ofthe models. The tested motions are periodic motions similar to those of theblades of VAWTs. A version of the dynamic stall model which is modified forlow Mach numbers is the most accurate throughout the tests.

The author has implemented the codes, has obtained the results and haswritten the article.

Paper IISimulating Dynamic Stall Effects for Vertical Axis Wind TurbinesApplying a Double Multiple Streamtube ModelTwo dynamic stall models are compared: the widely-used Gormont model andthe dynamic stall model which has performed best according to Paper I. Themodels have been included into a double multiple streamtube model for thesimulations of the whole turbine. The modeling of the dynamic stall effectshas been further modified by taking into account the circular motion of theblades of VAWTs. Additionally, the effects of flow curvature and the flowexpansion have been implemented in the double multiple streamtube model.The simulations are assessed against the data on blade force measurementson a curve-bladed VAWT which was operated by Sandia National Laborato-ries. The results show that the modified dynamic stall model outperforms theGormont model for all tested conditions.

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The author has implemented the code for the Leishman-Beddoes based dy-namic stall model and the flow expansion model, has obtained the results andhas written the article.

Paper IIISimulating Pitching Blade With Free Vortex Model Coupled WithDynamic Stall Model for Conditions of Straight Bladed Vertical AxisTurbineThis paper describes the integration of the dynamic stall model from Paper Iwith a free vortex model. The attached flow part is solved within a vortexmodel and the separated flow part is solved within the dynamic stall model.The model is tested for a blade, pitching with a similar motion as the bladesof a vertical axis turbine. The assessment of the new model is performed ata wide range of conditions, including the region of dynamic stall. The agree-ment of the dynamic stall model with experimental data has been significantlyimproved compared to previous versions of the model. Additionally, a newefficient way of modeling the blade with vortices has been verified, which hasincreased the computational speed of the vortex model.

The author has implemented parts of the code for a pitching blade motionand has written the separated flow part. The author has also obtained theresults and has written most of the article.

Paper IVEvaluation of a Blade Force Measurement System for a Vertical AxisWind Turbine Using Load CellsA unique blade force measurement system is evaluated in this paper. A systemfor measuring aerodynamic blade forces has been designed and implementedon a 12 kW VAWT operated at an open site north of Uppsala. The forces weremeasured using four load cells mounted at the hub of the VAWT. Weather con-ditions were measured 15 m away from the turbine. The system has initiallybeen designed to measure both the tangential and the normal force. However,the tangential force was disturbed by the changed turbine dynamics. The nor-mal force response is in agreement with simulations. This paper describesthe measurement system in a great detail together with the analysis of the tur-bine dynamics. Additionally, a comprehensive method of measurement errorestimation is presented and the major sources of the errors are highlighted.

The author has contributed to the experimental setup and to obtaining theresults. The author has also contributed to the data analysis and interpretationand has written parts of the article.

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Paper VMeasurements of the Aerodynamic Normal Forces on a 12-kWStraight-Bladed Vertical Axis Wind TurbineThis paper presents the results of measured aerodynamic normal forces onthe 12 kW VAWT, which were obtained with the experimental method fromPaper IV. The measurements were done for the range of TSR from 1.7 to 4.6,which covers the dynamic stall region, nominal operation and the region ofhigh flow expansion. Additionally, the maximum error of the measurement hasbeen quantified and presented. The behaviour of the measured forces has beenanalyzed. The data in this paper can be used to validate simulation models andwhen designing VAWTs.

The author has contributed to the estimation of measurement error. Theauthor has obtained the results and has written the article.

Paper VINumerical Validation of a Vortex Model Against Experimental Data on aStraight-Bladed Vertical Axis Wind TurbineThe performance of a two-dimensional vortex model is presented in this pa-per. The dynamic stall model which has been tested in Paper III is includedinto the vortex model for the simulations of the whole turbine. The effect ofthe dynamic stall vortex release has been taken into account by modifying thedynamic stall model. The simulations are assessed against the measurementsfrom the 12 kW VAWT, described in Papers IV and V. Additionally, new setsof data on the aerodynamic normal force are presented. The validation is donefor a range of TSR from 1.8 to 4.6. The simulation model shows good agree-ment with experimental data for the region of nominal operation and for theregion of high flow expansion. Although it can not reproduce experimentaldata in a great detail, it is concluded that the model can be used when dimen-sioning the turbine for maximum loads.

The author has written the dynamic stall model for separated flow. Theauthor has obtained the results and written most parts of the article.

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9. Acknowledgements

I am glad I have reached the point of writing acknowledgements, the last andthe most enjoyable part to write. I would like to thank the Swedish EnergyAgency and STandUP for Energy for financing my work. The measurementson the turbine in Marsta were performed with the financial support of theJ. Gust Richert foundation and STandUP for Energy. Those measurementswould not be possible without Kristian Trolin and Senad Apelfröjd who madesubstantial grounds for the load cell project. Big thanks to Dana Salar for allyour practical help in the workshop. My thanks to Wanan Sheng for providingme with the Glasgow University data for the dynamic stall model. I wouldlike to give a credit for people reviewing my thesis: spasibo to Marcus Bergfor your excellent language check, thanks to Sandra Eriksson for good tips onhighlighting the contribution of my work and of course thanks go to AndersGoude for his patience of writing each of I won’t say how many comments.

I would like to thank my formal supervisor at Mälardalen University inVästerås, professor Erik Dahlquist, who has given an obviously good recom-mendation for my employment at Uppsala University. Thanks to my supervi-sor, professor Hans Bernhoff, for the given trust and freedom. I would like tothank professor Mats Leijon for making it possible to have cool things at ourdivision. Meeting The Crown Princess of Sweden Victoria at our workshop,the summer party 2011 in Fjällnora and many other memories stay with me.Thanks to Senad for collaborating in arranging our office the way we wanted.Although I have not seen you much in the office since then, it was good tobe your office mate. Many thanks to all colleagues at our division, and windpower group in particular.

One of the main reasons why I started PhD studies was the possibility ofgoing to conferences and study visits abroad. Thanks to Simon Lindroth forsharing his ideas on how to reach that. Saman, grazie mille for your tips onfinding the travel grants. The ÅForsk Foundation, the C. F. Liljevalchs Foun-dation and the Anna Maria Lundins Foundation are very much acknowledgedfor financing my conference trips and study visits.

During my work, many of the colleagues became my friends. In no partic-ular order, thanks for an awesome time to Valeria, Johan, Erik, Foppa, Wei,Jose, Francisco, Ling, Liselotte, Malin, Maria, Flore, Saman, Kaspars, Victorand all the rest, you know who you are. The time I had with you was great funand sometimes very relaxing and enjoyable. Appreciate you all. Additionalcredit goes to Johan for your wise advices on writing articles, big thanks forthat.

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Big love to my mama Valentina and papa Valeriy, my sis Lida and mybabushka Alla and dedushka Vanya. I can not describe in words what youmean for me and how much I appreciate having you. All my bodies in Ukraine,yo to Lviv and Mariupol. You are my roots, you keep my back. Cheers to allmy people in Sweden and abroad. You know me, one big love.

Last but not least, I would like to acknowledge people that were of the mostimportance in my research work during the last years: Anders Goude andMorgan Rossander. I am pleased I worked with you. Can not appreciate yourhelp and support more.

Morgan, I am glad we worked together on the turbine in Marsta. Goodmemories of climbing up and down the tower, all those tests followed by fikain Ulva Kvarn that sunny summer 2014 stay with me. I am proud of the workwe have done there. When writing articles together, I appreciated your thor-oughness and patience and those hot discussions we had when working withmanuscripts, you know what I mean. Big thanks for that.

Anders, thank you for all the support I have been receiving throughout theentire PhD time. That is just invaluable. All the coding and fixing our modelsis highly appreciated as well as your suggestions and advices when writingarticles. You showed me a good academic approach on solving problems. Iam so much happy that you were supervising me.

Now in the end, I actually feel good about my work. Although things couldalways be a bit better, I am happy of what has happened. It has been awesome5 years to share with you all, the memories stay with me. In my heart.

Eduard

August 2015

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10. Sammanfattning

Vindkraft är idag en av de mest exploaterade förnybara energikällorna ochbidrar redan till hållbar elproduktion. Från ett ekonomiskt perspektiv är detelkostnaden som är en avgörande faktor när man bedömer konkurrenskraftenhos en energikälla. Idag är horisontalaxlade vindkraftverk (som ser ut sompropellrar) det helt dominerande konceptet på marknaden. Anledningen tilldetta är att den horisontalaxlade teknologin har dominerat utveckling och for-skning i mer än 20 år. Fokus för denna avhandling är ett alternativt vind-kraftverkskoncept, nämligen vertikalaxlade vindkraftverk (VAVK) vilka po-tentiellt kan minska energikostnaden.

Konceptet att använda VAVK för elproduktion är inte nytt. Under 1920-och 1930-talen utvecklades två olika designer som blev de största grundkon-cepten för VAVK. Ett koncept utvecklades av den finske uppfinnaren Sig-urd Johannes Savonius. Det blev patenterat år 1929 i USA under namnetSavonius-rotor. En fransk ingenjör vid namn George Jean Marie Darrieusutvecklade ett andra VAVK-koncept och patenterade det år 1931 i USA (Dar-rieusturbin). Efter oljekrisen år 1973 bedrevs omfattande studier kring Dar-rieusturbiner av Sandia National Laboratories i New Mexico, USA. Forskn-ing på flera Darrieusturbiner genomfördes från 1970- till 1990-talet. Mät-ningarna från de vindkraftverken används fortfarande när man verifierar simu-leringsmodeller. Konceptet kommersialiserades av bl.a. det amerikanska före-taget FloWind, vilket installerade mer än 500 VAVK i Kalifornien. Underdessa tider var dock kunskapen om materialutmattning i vindkraftverkens bladinte omfattande nog. Turbinernas bågformiga aluminiumblad hade dessutomdålig prestanda vad gäller materialutmattning. Detta resulterade i bladhaver-ier på flera VAVK och har lett till den spridda uppfattningen att VAVK ärsämre än horisontalaxlade vindkraftverk vad avser materialutmattning. Efteren viss framgång i utvecklingen av kompositmaterial blev vindkraftverkensblad lättare och tåligare mot materialutmattning. Men nya blad var fortfarandedyra och därför var det mest horisontalaxlade vindkraftverk som använde dem,eftersom intresset för VAVK minskade inom vindkraftsbranschen efter de tidi-ga haverierna.

VAVK-konceptet har flera fördelar jämfört med horisontalaxlade vindkraft-verk. VAVK är riktningsoberoende, d.v.s. de behöver ingen girmekanism somriktar hela turbinen mot vinden. Det är en viktig fördel därför att stor an-del av felen i horisontalaxlade vindkrafverk sker i girmekanismen. En annanfördel är att generatorn till en VAVK kan placeras på marken, vilket förenklarinstallation och underhåll. Dessutom är varken vikten eller storleken hos en

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generator för en VAVK kritisk, vilket gör att tunga generatorer med perma-nentmagneter kan installeras. Däremot gör de cykliska krafterna på bladen attvridmomentet varierar under drift. Även om detta problem har behandlats avSandia National Laboratories och FloWind tidigare, anses det fortfarande somen stor angelägenhet.

På avdelningen för elektricitetslära vid Uppsala Universitet påbörjades for-skning om VAVK år 2002. Sedan dess har tre VAVK designats och byggtsvid Uppsala Universitet: ett 1.5 kW vindkraftverk, ett 10 kW vindkraftverkmed kompositblad för telekom-tillämpningar och ett 12 kW vindkraftverk medkompositblad. Det sistnämnda verket har använts i de flesta av de hittills ut-förda experimenten. Ett stort 200 kW vindkraftverk med kompositblad ochträtorn har kosntruerats och byggts av avknoppningsföretaget Vertical WindAB i samarbete med universitetet. Alla dessa turbiner har raka blad medbärarmar (ett s.k. H-rotorkoncept). Jämfört med Darrieusturbinen med båg-formiga blad, är de raka bladen på en H-rotor lättare att tillverka Dock behövsbärarmar, vilket inkräktar på vindflödet genom turbinen. En annan fördel medH-rotorn är att energiabsorptionen per markyta maximeras på grund av denkonstanta rotorbredden.

Denna avhandling omfattar studier på aerodynamiken för VAVK. Arbetetkombinerar simuleringsverktyg med mätningar. Numeriska modeller av deaerodynamiska krafterna på turbinens blad har utvecklats och validerats motexperimentella data. Studierna i avhandlingen berör VAVK som befinner sig iöppen terräng.

Ett komplext fenomen under VAVK-drift är dynamisk överstegring (s.k. dy-namisk stall), vilken sker när turbinen drivs vid låga löptal. En befintlig modellför dynamisk stall som ursprungligen har designats för helikoptrar har mod-ifierats för förutsättningarna vid VAVK-drift. Denna modell har kopplats tillen strömrörsmodell och har validerats mot kraftmätningar på VAVK med båg-formiga blad som genomfördes av Sandia National Laboratories. Denna valid-ering har visat att strömrörsmodellens noggrannhet har förbättrats jämfört medtidigare versioner av modellen.

Modellen för dynamisk stall har ytterligare utvecklats genom koppling tillen virvelmodell. Den nya modellen har blivit mindre beroende av empiriskakonstanter och har visat bättre överensstämmelse med experimentella data förvinklade blad. Dessutom har ett effektivt sätt att modellera blad bekräftats,vilket har lett till minskade beräkningstider.

Unika mätningar av krafter på ett blad genomfördes på 12 kW- vindkraftver-ket norr om Uppsala. Krafterna på turbinen har mätts av fyra kraftgivare ochsignalen har skickats trådlöst till mätsystemet. En omfattande analys av nog-grannheten av dessa mätningar har genomförts och olika källor till mätfel haranalyserats.

Uppmätta aerodynamiska normalkrafter har presenterats och analyserats förett brett spektrum av driftsförhållanden inkluderande dynamisk stall, nominelldrift och hög flödesexpansion. En förbättrad virvelmodell har validerats mot

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data från de nya mätningarna. Modellen överensstämmer ganska väl med mät-ningar under nominell drift respektive hög flödeexpansion. Även om modelleninte reproducerar alla mätresultat på en detaljerad nivå, föreslås det att den nu-varande virvelmodellen kan användas för preliminära beräkningar av krafternapå VAVK-blad på grund av dess höga beräkningshastighet och acceptabla nog-grannhet.

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11. Аннотация к научной диссертации

На сегодняшний день ветроэнергетика является одним из наиболее экс-плуатированных видов возобновляемой энергии, способствуя устойчиво-му развитию. С экономической точки зрения, стоимость единицы произ-веденной энергии является решающим фактором при оценке конкуренто-способности того или иного вида энергетики. Абсолютное большинствоустановленных ветрогенераторов на сегодняшний день имеют ветряныетурбины с горизонтальной осью вращения (ВТГОВ, крыльчатый либопропеллерный тип). В большей степени это обусловлено тем, что техно-логическое развитие именно ВТГОВ было доминирующим в отрасли напротяжении более 20 лет. Альтернативный тип ветрогенератора являетсяобъектом этой диссертации – ветряная турбина с вертикальной осью вра-щения (ВТВОВ), которая потенциально может снизить стоимость произ-веденной энергии.Идея производства электроэнергии по средствам ВТВОВ не является

новой. В 1920 – 1930-х годах были разработаны два основных дизайнаВТВОВ. Первый был запатентован в США в 1929 году финским изобре-тателем Зигмундом Йоханнесом Савониусом (ротор Савониуса). Второйконцепт был разработан французским инженером Жоржем Дарье и за-патентован в США в 1931 году (ротор Дарье). После нефтяного кризиса1973 года в национальной лаборатории министерства энергетики СШАSandia началась исследовательская кампания, в ходе которой изучаласьработа ветрогенераторов с ротором Дарье. С поздних 1970-х до середи-ны 1990-х годов исследовались несколько ВТВОВ с изогнутыми лопастя-ми. Полученные результаты тех исследований до сих пор используютсяв имитационном моделировании при оценке точности моделей. КонцептВТВОВ был коммерциализирован несколькими компаниями, в том числеамериканской компанией FloWind, которая установила более 500 турбинв Калифорнии. Однако на то время знания об усталости материала былинедостаточны. Изогнутые лопасти ВТВОВ из экструзионного алюминияобладали плохой стойкостью к усталости материала, что стало причинойих повреждения. Это привело к распространенному мнению о том, чтоВТВОВ являются менее стойкими к циклическим нагрузкам по сравне-нию с ВТГОВ. С прогрессом в области композитных материалов лопастиветряных турбин стали легче и устойчивее к периодическим нагрузкам.Однако, из-за высокой стоимости композитных материалов, возможностьустанавливать композитные лопасти получили лишь ВТГОВ, так как ин-терес в сфере ветроэнергетики к ВТВОВ упал в связи с предыдущимиповреждениями их лопастей.

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Конструктивные особенности ВТВОВ представляют несколько прин-ципиальных преимуществ над ВТГОВ. Работа ВТВОВ не зависит от на-правления ветра, и, следовательно, механизм устройства ориентации тур-бины относительно ветрового потока не требуется. Это является суще-ственным преимуществом над ВТГОВ, так как большая часть сбоев вВТГОВ приходится именно на механизм ориентации турбины. Еще од-ним преимуществом ВТВОВ является то, что электрогенератор может на-ходится на уровне земли, что упрощает установку и техническое обслу-живание. Более того, это дает возможность устанавливать тяжелые элек-трогенераторы с постоянными магнитам, так как масса и размер электро-генератора не критичны дляВТВОВ.Однако циклические нагрузки на ло-пасти ВТВОВ приводят к периодическим колебаниям вращательного мо-мента турбины, что является естественным при работе ВТВОВ. Несмот-ря на то, что эта проблема была решена лабораторией Sandia и компани-ей FloWind по средствам усовершенствований осевого вала, периодиче-ские колебания вращательного момента считаются основным недостат-ком ВТВОВ.Исследования ВТВОВ на кафедре электричества в Университете Уп-

псалы проводятся с 2002 года. Были спроектированы и построены триВТВОВ: ветрогенератор с мощностью 1.5 кВт, ветрогенератор мощно-стью 10 кВт для телекоммуникаций и ВТВОВ мощностью 12 кВт. По-следний ветрогенератор имеет композитные лопасти и использовался вбольшинстве проведенных экспериментов. Крупная ВТВОВ мощностью200 кВт с композитными лопастями и башней из древесно-слоистых плитбыла спроектирована и построенашведской компаниейVerticalWind. Всевыше-перечисленные ВТВОВ имеют прямые лопасти, поддерживаемыеопорами (так называемый Н-ротор). По сравнению с изогнутыми лопа-стями ротора Дарье, прямые лопасти Н-ротора проще производить, од-нако опоры для лопастей интерферируют с воздушным потоком внутритурбины. Еще одним преимуществом Н-ротора является то, что произве-денная энергия на единицу использованной площади земли максимизи-рована из-за того, что радиус турбины – постоянный по всей ее высоте.Аэродинамика ВТВОВ является объектом исследований этой диссерта-

ции. Этот проект объединяет разработку имитационных моделей с экспе-риментальной работой. Точность имитационных моделей аэродинамиче-ских сил, действующих на лопасти ВТВОВ, подтверждена измерениями.Исследования проведены на ВТВОВ, работающих в открытой местности.Одним из сложных феноменов при работе ВТВОВ является динами-

ческое сваливание. Оно имеет место при работе ВТВОВ с низким соот-ношением скорости вращения лопастей к асимптотической скорости вет-ра. Модель динамического сваливания, которая изначально была созда-на для лопастей вертолетов, была модифицирована для условий работыВТВОВ. Эта модель была комбинирована с моделью векторной трубкипотока (далее – модель трубки), и ее результаты были сравнены с измере-

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ниями сил на изогнутые лопасти ВТВОВ, которая исследовалась в лабо-ратории Sandia. Анализ результатов показал, что точность модели трубкибыла улучшена относительно предыдущих версий этой модели.Модель динамического сваливания быламодифицирована посредством

ее комбинирования с вихревой моделью. Новая модель стала менее зави-симой от эмпирических констант, а также более точной при ее оценкес экспериментальными данными колебаний аэродинамической лопасти.К тому же был испытан эффективный метод моделирования лопастей, ивычислительная производительность модели была улучшена.Были произведены уникальные измерения сил на лопастиВТВОВ.Дан-

ная ВТВОВ с мощностью 12 кВт находится на севере Уппсалы. Сенсорынагрузки были использованы для измерений сил на турбину. Был произ-веден полноценный анализ точности измерений, а также были выявленыосновные источники погрешности.Измерения аэродинамических радиальных сил были проанализирова-

ны в широком диапазоне рабочих условий ВТВОВ, включая режим дина-мического сваливания, номинальной работы и высокого уровня расшире-ния ветрового потока. Было показано, что результаты модели достаточноблизки к экспериментальным результатам в регионе номинальной рабо-ты и высокого уровня расширения потока. Не смотря на то, что модельне воспроизводит результаты измерений на очень глубоком уровне, ис-пользование данной модели целесообразно при предварительных расче-тах сил на лопасти ВТВОВ в связи с высокой вычислительной произво-дительностью и относительно высокой точностью.Результаты пяти из шести исследований были опубликованы автором в

научных журналах, и их обобщение представлено в данной диссертации.

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12. Анотація до наукової дисертації

Вітроенергетика є одним з найбільш експлуатованих видів відновлюва-ної енергетики та вже сьогодні сприяє сталому розвитку. З економічноїточки зору, вартість одиниці виробленої енергії є вирішальним факторомщодо оцінки конкурентоспроможності виду енергетики. Абсолютна біль-шість побудованих на сьогодні вітроелектростанцій мають вітроколеса згоризонтальної віссю обертання (ВКГВО, пропелерний тип). Це обумов-лено більшою мірою тим, що технологічний розвиток саме ВКГВО бувдомінантний у галузі протягом більш ніж 20 років. Альтернативний типвітроелектростанції є у фокусі цієї дисертації – вітроколесо з вертикаль-ною віссю обертання (ВКВВО), що потенційно може знизити вартість ви-робленої енергії.Ідея виробництва електроенергії за допомогою ВКВВО не є новою. У

1920 – 1930-х роках було розроблено два основних дизайни ВКВВО. Пер-ший був запатентований у США у 1929 році фінським винахідником Зи-гмундом Йоханнесом Савоніусом (ротор Савоніуса). Другий концепт буврозроблений французьким інженеромЖоржем Дар’є та запатентований уСША у 1931 році (ротор Дар’є). Після нафтової кризи 1973 року у націо-нальній лабораторії міністерства енергетики США Sandia було розпоча-то дослідження, у яких вивчалася робота вітроелектростанції з роторомДар’є. З пізніх 1970-х до середини 1990-х років було досліджено кількаВКВВО з вигнутими лопатями. Отримані результати цих досліджень досівикористовуються в імітаційному моделюванні для оцінювання точностімоделей. Концепт ВКВВО було комерціалізовано декількома компанія-ми, у тому числі американською компанією FloWind, яка встановила по-над 500 турбін у Каліфорнії. Однак знання щодо втоми матеріалу булиобмеженими у той час. Це стало причиною пошкодження вигнутих ло-патей ВКВВО з екструзійного алюмінію, що мали погану стійкість що-до втоми матеріалу. Це призвело до поширеної думки про те, що лопатіВКВВО є менш стійкими до циклічних навантажень, ніж ВКГВО. З про-гресом в області композитних матеріалів лопаті вітроколес стали легши-ми та більш стійкими щодо періодичних навантажень. Однак у зв’язкуз високою вартістю композитних матеріалів, можливість встановлюватикомпозитні лопаті у вітроколесах отримали саме ВКГВО, оскільки інте-рес у сфері вітроенергетики до ВКВВО впав через попередні пошкодже-ння їхніх лопатей.Концепт ВКВВО має кілька принципових переваг над ВКГВО. Робо-

та ВКВВО не залежить від напрямку вітру, та отже механізм пристрою

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орієнтації турбіни щодо вітрового потоку не потрібен. Це є суттєвою пе-ревагою над ВКГВО, тому що більшість збоїв у ВКГВО припадає самена механізм орієнтації турбін. Ще однією перевагою ВКВВО є те, щоелектрогенератор може знаходиться на рівні землі, що спрощує установ-ку та технічне обслуговування. Це дає можливість установки громіздкихелектрогенераторів з постійними магнітами, через те що маса і розмірелектрогенератора не критичні. Однак циклічні навантаження на лопатіВКВВО призводять до періодичних коливань обертального моменту тур-біни, що є присутнім у роботі всіх ВКВВО. Попри те, що цю проблемубуло вирішено національною лабораторією Sandia та компанією FloWindудосконаленням осьового валу, періодичні коливання обертального мо-менту вважаються головним недоліком ВКВВО.Дослідження ВКВВО на кафедрі електрики Університету Уппсали про-

водяться з 2002 року. Три ВКВВО було спроектовано та побудовано: ві-троелектростанція потужністю 1.5 кВт, ВКВВО потужністю 10 кВт длятелекомунікацій та ВКВВОпотужністю 12 кВт. Остання ВКВВОмає ком-позитні лопаті та використовується у більшості експериментів. ВеликуВКВВО потужністю 200 кВт з композитними лопатями та дерев’яноювежею було спроектовано та побудовано шведською компанією VerticalWind. Усі вищезазначені ВКВВО мають прямі лопаті, підтримувані опо-рами (так званий Н-ротор). У порівнянні з вигнутими лопатями роторуДар’є, прямі лопаті Н-ротора простіше виробляти, однак підтримуваль-ні опори є частковою перешкодою вітровому потокові усередині турбіни.Ще однією перевагою Н-ротора є те, що вироблена енергія на одиницювикористаної площі землі максимізовано у зв’язку зі сталим радіусом наусій висоті турбіни.Аеродинаміка ВКВВО є об’єктом дослідження цієї дисертації. Цей про-

ект об’єднує розробку імітаційних моделей з експериментальною робо-тою. Точність імітаційних моделей аеродинамічних сил, що діють на ло-паті ВКВВО, підтверджено експериментами. Дослідження були проведе-ні на ВКВВО, що працюють у відкритій місцевості.Одним зі складних феноменів у роботі ВКВВО є динамічне звалюван-

ня, яке відбувається у роботі ВКВВО з низьким співвідношеннямшвидко-сті обертання лопаті до асимптотичної швидкості вітрового потоку. Існу-ючу модель динамічного звалювання, яку вперше було створено для ло-патей вертольотів, було модифіковано для умов роботи ВКВВО. Ця мо-дель була комбінована з моделлю векторної трубки потоку (далі – модельтрубки), та її результати було порівняно з вимірами сил на вигнуті лопатіВКВВО, що було досліджено національною лабораторією Sandia. Резуль-тати досліджень показали, що точність моделі трубки було покращено впорівнянні до попередніх версій цієї моделі.Модель динамічного звалювання було згодом модифіковано завдяки її

об’єднанню з вихровою моделлю. Нова модель стала менш залежною відемпіричних констант, а також точнішою при її оцінці з експерименталь-

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ними даними коливань аеродинамічної лопаті. До того ж було випробу-вано ефективний метод моделювання лопатей, та було покращено обчи-слювальну продуктивність моделі.Було проведено унікальні виміри сил, що діють на лопаті ВКВВО. Да-

на ВКВВО потужністю 12 кВт знаходиться на півночі Уппсали. Для ви-мірювання сил на турбіну були встановлені сенсори навантаження. Булопроведено досконалий аналіз точності вимірювань, та головні джерелапохибки були вказані.Здобуті вимірювання аеродинамічних радіальних сил було проаналізо-

вано у широкому діапазоні робочих умов ВКВВО, включаючи режимидинамічного звалювання, номінальної роботи та регіону з високим рів-нем розширення вітрового потоку. Точність удосконаленої вихрової мо-делі було оцінено у порівнянні з даними нових вимірів. Було виявлено, щорезультати моделі є досить близькими до експериментальних результатів.Не зважаючи на те, що дана вихрова модель не відтворює результати ви-мірювань на дуже глибокому рівні, використання цієї моделі є доцільниму прелімінарних розрахунках сил на лопаті ВКВВО у зв’язку з її високоюобчислювальною продуктивністю та відносно високою точністю.Результати п’яти з шести досліджень опубліковано автором у наукових

журналах, та їх узагальнення представлено у цій дисертації.

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