# Fluid Dynamics and Wind Energy

By khozoliozo
Jan 14, 2013
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UNIVERSITÁ DEGLI STUDI DI UDINE Dottorato in Tecnologie Chimiche ed Energetiche

FLUID DYNAMIC MODELLING OF WIND TURBINES

sec. D

Vr 0 D Vt

Vz Vr Vt 3

Relatori: Prof.Ing. Lorenzo BATTISTI Prof.Ing. Piero PINAMONTI

Dottorando: Dott.Ing. Luca ZANNE

Udine 21 Maggio 2010

Summary

Introduction PART I : HAWT analysis HAWT Fluid dynamics A turbomachinery approach Inverse design

Summary

PART II : VAWT analysis VAWT fluid dynamics VAWT experimental analysis VAWT free vortex wake Results and conclusions

Introduction

Wind energy market (EWEA) Installed capacity

Offshore WE market (EWEA)

Aim of the thesis & thesis outline

The aim of the thesis is to analyze the fluid dynamic models of wind energy conversion systems, pointing out the limitations of current engineering models and proposing innovative solutions from the design point of view The research activities have been divided in two main parts, following the different rotor – flow interaction characteristics: 1. Horizontal axis wind turbines - HAWT; 2. Crossflow wind turbines, as vertical axis wind turbines - VAWT.

Part I : HAWT analysis HAWT fluid dynamics

HAWT fluid dynamics is mainly based on the actuator disk concept

HAWT fluid dynamics Actuator disk concept

The turbine generates mechanical work from the kinetic energy of the fluid flow The work exchange between the fluid and the shaft is done by is done by the rotor, which can be modelled as an actuator disk The bladed rotor can be represented with equivalent forces distribuited over a permeable, immaterial disk Infinite number of blades Infinite rotational velocity

HAWT fluid dynamics Actuator disk – momentum theory

Froude applied for the first time the actuator disk concept to a rotor in open flow. He applied it with the 1D momentum balance in axial direction Momentum equation T = ∆p ⋅ Am = ρVz ,3 A3 (Vz ,0 − Vz ,3 )

Energy conservation

Weul = ∆p

Mass conservation

Vz ,m Am = Vz ,3 A3 Vz ,1 ≅ Vz ,2 ≅ Vz ,m

ρ

=

Vz2 − Vz2 ,0 ,3 2

Froude result!

Vz ,m = Vz ,0 + Vz ,3 2

Actuator disk Blade element – momentum theory

Drzewiecki first applied Froude result dividing the rotor in different annular streamtubes : Non uniform loading Vz ,m = Vz ,0 + Vz ,3 2

Raero FN Lift φflow

With the blade element airfoil theory rotor performances can be easily calculated The annuli interaction is neglected No swirl flow, (wake expansion?) Ok lightly loaded rotors

Wind.=[ -a·V0; -a’·ωr ]

FT

Drag z

chord line

-ωr θpitch+βtwist φflow αattack V0 Vrel. y

rotor plane

HAWT fluid dynamics General momentum theory

The general momentum theory should overcome the issues of the swirl flow modelling Momentum equation : axial T = ∫ ( p1 − p2 ) dA = ∫ ρVz ,3 (Vz ,0 − Vz ,3 ) + ( p0 − p3 ) dA Am A3

tangential

M = ∫ ρVθ ,3Vz ,3 r3 dA

A3

radial

p3 − p0

ρ

=

p3 ( r3 ) − p3 rtip ,3

(

ρ

)=−

∫r3

2 rtip ,3 Vθ ,3

r3

dr3

1 + 1 2Vθ ,3 Ωr3 1 + 1 2Vθ ,2 Ωr 2 1 ρ ∫A3 (Vz ,0 − Vz ,3 ) dA = ρΩ ∫A3Vz ,3Vθ ,3r3 − dA 2 Vz ,3 Vz ,m

Solutions: • De Vries • Differential

• GM theory is an integral formulation • It needs the wake solution

8 V2 Weul = ⋅ 0 9 2

Actuator disk – momentum theory limitations

Actuator strip Wake states

Conway exact solution

HAWT fluid dynamics Vortex theory

Vortex theory calculates the flow field of the rotor wake by using the fluid dynamic laws of vorticity (BiotSavart law, Kelvin’s theorem, Helmholtz’s laws) Introduced by Joukowski – Betz – Prandtl Most widespread for propeller analysis and design (both for aerodynamic and marine propellers) and for helicopter rotor performance prediction • Prescribed vortex wake • Free vortex wake

Vortex theory Prescribed vortex wake

Vz ,m =

Vz ,0 + Vz ,3 2

Axial velocity

d Γ = 2π ⋅ d ( rVθ ,2 )

gθ ,m = d Γ r Ω + Vθ ,2 2 Vz , m 2π r d Γ r3Ω + Vθ ,3 2π r3 Vz ,3

Radial velocity

Vr ( r ,0 ) = − 1 r ∂Vz r ( r ,0 ) dr r ∫0 ∂z

gθ gθ + r 2r 2π ( r − r ) gθ

2

∂Vz ∂z ( r ,0 ) =

gθ ,3 =

vz ,m = gθ 2

vz ,3 = gθ

1 1 gθ ( r − r ) ∂Vz ∂z ( r ,0 ) = − − 2 4 2π 2π ( r − r ) r r5

Part I : HAWT analysis A turbomachinery approach

V ∂rVθ 1 ∂p 0 ∂V ∂V = Fr + θ − Vz r − z ρ ∂r r ∂r ∂r ∂z

Vz

∂Vθ Vr ∂rVθ + = Fθ ∂z r ∂r

∂Vθ + Vθ ∂z

1 ∂p 0 ∂V ∂V = Fz + Vr r − z ρ ∂z ∂r ∂z

∂rVr ∂rVz + =0 ∂r ∂z

A turbomachinery approach Stoke’s stream function

ωθ =

∂Vr ∂Vz − ∂z ∂r

1 ∂ψ ∂ 2ψ + = −rωθ r ∂r ∂z 2

d ( rVθ ) dψ − r dp 0 ρ dψ

∂ 2ψ ∂r 2

−

ωθ = Vθ

∂ 2ψ ∂r 2

−

1 ∂ψ ∂ 2ψ + =0 r ∂r ∂z 2

∂ 2ψ ∂r 2

−

d ( rVθ ) r 2 dp 0 1 ∂ψ ∂ 2ψ + 2 = −rωθ = −rVθ + r ∂r ∂z dψ ρ dψ

Linearized solution : Horlock actuator disk solution

∂ 2ψ 1 ∂ψ ∂ 2ψ − + = − F (r ) ∂r 2 r ∂r ∂z 2 Vz ,3 − Vz ,0 kz Vz ( r , z ) = Vz ,0 + e 2 Vr ( r , z ) = − 1 r Vz ,3 ( r ) − Vz ,0 kz kr e dr r ∫0 2

Froude result

A turbomachinery approach Motion in region II

The flow is determined by

rVθ

p0

Euler equation

1

ρ

(p

0 2

0 − p1 = ΩrVθ = Weul

)

Wu equation

∂ 2ψ

0 0 1 ∂ψ ∂ 2ψ p2 − p1 − + = − ∂r 2 r ∂r ∂z 2 Ω2

(

)

ρ

+r

2

0 d ( rVθ ) 1 dp2 = Ωr 2 − rVθ ρ dψ dψ

(

)

The angular momentum distribution can be assigned

Vθ = k1r n + k2 r

rVθ = k1r n +1 + k 2

Free vortex distribution

rVθ = const

The radial equilibrium theory applied to wind turbines

Radial momentum equilibrium

V ∂rVθ 1 ∂p 0 ∂V ∂V = Fr + θ − Vz r − z ρ ∂r r ∂r ∂r ∂z

dV 1 dp 0 Vθ d ( rVθ ) = + Vz z ρ dr r dr dr

ISRE

Sections 1 - 2

Vz2 − Vz2hub = ,

(p ρ

2

0

0 − pr ,hub − 2 ∫

)

r

r , hub

Vθ ∂rVθ r ∂Vr dr + 2 ∫ V dr − Vr2 − Vr2hub , r , hub z ∂z r ∂r ψ

(

)

Wu hypothesis

∂Vr ,1 ∂z =− ∂Vr ,2 ∂z

Wu hypothesis on a streamline

∂Vr ,1 ∂Vr ,2 = − ∂z ψ ∂z ψ 0 dVz ,m dVr ,m 1 dp2 Vθ ,2 d ( rVθ ,2 ) = + 2Vz ,m + 2Vr ,m ρ dr r dr dr dr

0 dVz ,m 1 dp2 Vθ ,2 d ( rVθ ,2 ) = + 2Vz ,m ρ dr r dr dr

The radial equilibrium theory results and comments

Radial equilibrium solution for a uniformly loaded disk λ=8 8 V2 Weul = ⋅ 0 9 2

λ=2

8 V2 Weul = ⋅ 0 9 2

The radial equilibrium theory results and comments

Mikkelsen actuator disk – CFD solution for a uniformly highly loaded disk (wind turbine state) 8 V2 Weul = ⋅ 0 9 2

Conway actuator disk – vortex theory exact solution for a (almost) parabolic highly loaded disk (propeller state) CT = 3.147

The radial equilibrium theory results and comments

Power and thrust coefficients for the different flow field solution models with a constant work extraction

Conway velocity at the centre of the disk for a propeller

The radial equilibrium theory on a streamline

Radial equilibrium with meridional velocity

0 dVs ,m 1 dp2 Vθ ,2 d ( rVθ ,2 ) = + 2Vs ,m ρ dr r dr dr

Vs2m = Vz2m + Vr2m , , ,

Denton / Cumpsty approach

∂V V2 1 ∂ 2 1 ∂p 0 1 ∂ 2 2 Vs ∂ Vs = + Vs s sin ( ε + δ ) + s cos ( ε + δ ) − 2 r Vθ + ( rVθ ) tan γ + Fd 2 ∂q ∂s rs r ∂s ρ ∂q 2r ∂q

(

)

∂V V2 1 ∂Vs2 1 ∂p 0 1 ∂ 2 2 = + Vs s sin ε − s cos ε − 2 r Vθ ρ ∂r ∂s 2 ∂r rs 2r ∂q

(

)

( )

Coning / yaw effects Turbulence wake state / stall Tip effects Unsteady dynamics

∂Vs2m , ∂r

=

0 1 ∂Vs ,m 1 ∂p2 1 1 ∂ 2 2 + 2Vr ,m − Vs2m cos ε + − r Vθ , rs ,1 rs ,2 2r 2 ∂q ∂s ρ ∂r

Considerations on the turbomachinery approach

• The theory handles an expanding and rotating wake. • Only the disk station has to be solved to obtain the information needed to compute CP and CT. • The method is simple and robust also for low tip speed ratios • Arbitrary disk loading can be analyzed. • The mathematics involved are comparable with those of the usual actuator disk approaches. • The actual velocities distribution are qualitatively assessed even though more work has to be carried out to better understand the fluid flow in the neighborhood of the disk and in the wake. • The radial velocity gradients along the streamlines at the disk have to be better described to reduce the axial velocity overestimation at the disk inner portion.

Part I : HAWT analysis Inverse Design

Inverse design and direct design methods

The turbine close field structure The blade architecture

Blade forces

Fθ , Z = ρ ⋅ Vz ,m ⋅ s ⋅ (Vθ ,2 − Vθ ,1 ) = ρ ⋅ V 2 z ,m ⋅ s ⋅ ( tan α 2 − tan α1 ) 0 0 Fz , Z = ( p1 − p2 ) ⋅ s +

1 ρ ⋅ Vθ2,2 ⋅ s 2

Weul = U ⋅ (Vθ ,2 − Vθ ,1 ) = U ⋅ Vz ,m ⋅ (tan α 2 − tan α1 ) = U ⋅ Vz ,m ⋅ (tan β 2 − tan β1 )

k Weul = rω ⋅ k1r n + 2 r

Flow angles

β1 = tan −1

U + Vθ ,1 Vz ,1

β m = tan −1

U + Vθ ,m Vz ,m

β 2 = tan −1

U + Vθ ,2 Vz ,2

Vz ,m

Vz ,1 Vz ,2

The blade architecture

Cy = Fy Fy ,max = 2⋅ s ( tan β2 − tan β1 ) cos2 β2 cz cz = c ⋅ cos β m

s CL = 2 (tan β 2 − tan β1 ) cos β m c

C y = 0.8

Zweifel Lieblein

θ=

π

2

− β m − sen −1 (

CL, ID 2π

)

Dloc =

Wmax − W2 Wmax

Inverse Design Results and discussion

Gaia turbine

Inverse Design Results and discussion

1 W / 1/2V 2 0 1 VDz / V0

Flow characteristics

0.5 0 0 0.2 0.4 r/R 0.6 0.8 1

0.5

0

0

0.2

0.4 r/R

0.6

0.8

1

40 20 0 1.5

beta1-beta2 [deg]

The blade architecture and loads

C/R 0.2 0.1 0 0 0.2 0.4 r/R Fn / q0R 0.6 0.8 1 betam [deg] 80 60 40 20 0 2 1 0 0.6 Mn / q0R3 0.4 0.2 0 P / 1/2rhoAV 3 0 0.6 0.4 0.2 0 0 0.2 0.4 r/R 0.6 0.8 1 0 0.2 0.4 r/R 0.6 0.8 1 0 0.2 0.4 r/R 0.6 0.8 1

alpha2 [deg]

40 20 0

0

0.2

0.4 r/R

0.6

0.8

1

0

0.2

0.4 r/R

0.6

0.8

1

1 p1-p2 / q0 1 Psi 0.5 0 0.5

2 Ft / q0R

0 0.2 0.4 r/R 0.6 0.8 1

1 0

0

0.2

0.4 r/R

0.6

0.8

1

0

0

0.2

0.4 r/R

0.6

0.8

1

0

0.2

0.4 r/R

0.6

0.8

1

0.6 Mt / q0R3 0.4 0.2 0 0 0.2 0.4

λ=6,Z=3

dCp / d(r/R)

0.6 r/R

0.8

1

1 0.5 0

0

0.2

0.4 r/R

0.6

0.8

1

Inverse Design Results and discussion

λ=6,Z=3

Inverse Design Results and discussion

λ = 1.5 , Z = 3

Part II : VAWT analysis VAWT fluid dynamics

Darrieus eggbeater – Darrieus H/V – Gorlov type Building environment Offshore multi Mega Watt

CL =

dL 1 ρ W 2 c dh 2 0

VAWT fluid dynamics The double disk BEM for VAWT

Blade element forces

C N = CL cos β + C D sin β

2

Flow characteristics

β = tan −1

Vsenϑ cos δ ( V cosϑ + Ωr ) cos γ

2

W 2 = ( Vcosϑ + Ωr ) cos γ + ( Vsenϑ cos δ ) Re = cW

CT = C L sin β − C D cos β

ν0

dL 1 ρ W 2 c dh 2 0 CD = dD 1 ρ W 2 cdh 2 0

1 dh dFN = ρ0 W2 c CN 2 cos δ dFT = 1 dh ρ0 W 2 c CT 2 cos δ

CL =

Shaft torque/power

dM = dFT Ω

1 ϑ dM Ω Nϑ ∫ ∫ ∫ MΩ CP = = 1 ρ A V3 1 ρ A V3 0 sw 0 2 2 0 sw 0

N BL

VAWT fluid dynamics The double disk BEM for VAWT

Blade element

dFx = dFT cos ϑ cos βc cos γ + dFN sin ϑ cos δ

dFx = B 2

CTH =

∆ϑ

π

dFx

dFx 1 2 ρ0 V0 dAs 2

dAs = dh r dϑ sinϑ

Momentum theory

α=

V V0

dFx = 2ρ dA s V(V0 -V)

CTH =

dFx 1 2 ρ V0 dA s 2

=

2ρ dA s V(V0 -V) = 4α (1 − α ) 1 2 ρ V0 dA s 2

The double disk BEM for VAWT Corrections

Glauert correction Tip losses

CTH =

26 4 (1 − α ) + 15 15

Post stall airfoil performance correction Flow curvature Dynamic stall Streamtubes expansion

VAWT fluid dynamics Validation and results

Sandia 5m Darrieus 3blades NACA0015

Four geometric characteristics

VAWT fluid dynamics Validation and results

Blade tangential and normal force coefficients Shaft forces and torque Mean value and fluctuations

VAWT fluid dynamics Validation and results

Shaft torque and forces diagrams

2-bladed 3-bladed 3-bladed 2-bladed

presents the best power performance presents lower forces fluctuations Gorlov type presents the lowest fatigue loads (complex geometry) a 90° reduces the loads fluctuations but needs rotor balancig

VAWT fluid dynamics Limitations of VAWT BEM codes

• The circular path is simplified in two actuator disks • The momentum equilibrium is applied only in axial direction • The axial expansion is generally neglected or not correctly/completely implemented • The turbulent wake state correction is taken from HAWT corrections • No (or weak) interaction between streamtubes • Tip losses correction is of doubtful application for VAWT • Complex geometry not resolvable from a fluid dynamic point of view • Unsteady fluid dynamic effects are of difficult implementation

Part II : VAWT analysis VAWT experimental analysis

VAWT experiments in controlled conditions The Politecnico di Milano Large Wind Tunnel High speed test section: 4x3.84m Wind speed up to 55m/s Possibility to work in open/close test section 2 different rotor prototypes designed by Tozzi Nord Wind Turbines: PDF1 – research purpose PDF3 – commercial turbine

The turbines layout and the instrumentations

PDF1 3 Blades H = 1.46m D = 1.03m NACA0021 Solidity 0.25 Rotor position Torque Support loads PDF3 3 Blades - Gorlov H = 2.5m D = 1.78m P = 1.5kW H(tower) = 3.5m Rotor position Torque (electric) Support loads Aerodynamics Directional pneumatic 5 holes probe Single sensor hot wire anemometer

VAWT experimental analysis PDF1 rotor - Performance

Blockage : 0.097 close test section Blockage effects up to 20-30% for CP and 10-20% for CT Reynolds numbers very important on power performance for Re < 200000

VAWT experimental analysis PDF1 rotor - Aerodynamics

λ = 1.6

λ = 2.6

Wake non symmetric and deformed turnwise (in particular at low tip speed ratios) In closed wind tunnel there is an higher velocity due to blockage effects

λ = 1.6

VAWT experimental analysis PDF1 rotor - Aerodynamics

Wind tunnel blockage

1 1 1 1 2 2 T = AD p0 + ρV02 − ρVD − p3 + ρV32 − ρVD 2 2 2 2 V0' VD C = + T V V0 V0 4 D V0

1D momentum theory doesn’t seem the best model for blockage effects

VAWT experimental analysis PDF1 rotor - Aerodynamics

Unsteady flow field

λ = 2.5

VAWT experimental analysis PDF3 rotor - Dynamics

Dynamic analysis and modelling

Part II : VAWT analysis 2D Free vortex wake

Bound and shed vorticity

L = Cl

1 ρW 2 c = ρW Γ B 2

1 Γ B = ClWc 2

δΓ S = −

d ΓB δθ dθ

Induced velocitites (Biot-Savart)

u=−

( y − y0 ) Γ 2π ( x − x0 )2 + ( y − y0 )2 + h 2

v=

( x − x0 ) Γ 2π ( x − x0 )2 + ( y − y0 )2 + h 2

Flow characteristics

W 2 = ΩR + (U 0 + uC ) cos (θ ) + vC sin (θ ) + (U 0 + uC ) sin (θ ) − vC cos (θ ) 2 2

Shed vortex position

xS ,i = xS ,i −1 + U 0 + uS ( xS ,i −1 , yS ,i −1 ) ⋅ dt yS ,i = yS ,i −1 + vS ( xS ,i −1 , yS ,i −1 ) ⋅ dt

xS ,i = xS ,i −1 + U 0 + 0.5 ⋅ uS ( xS ,i , yS ,i ) + uS ( xS ,i −1 , yS ,i −1 ) ⋅ dt

(

)

(U 0 + uC ) sin (θ ) − vC cos (θ ) φ = tan −1 − ΩR + (U 0 + uC ) cos (θ ) + vC sin (θ ) α =φ − β

(

)

yS ,i = yS ,i −1 + 0.5 ⋅ vS ( xS ,i , yS ,i ) + vS ( xS ,i −1 , yS ,i −1 ) ⋅ dt

(

)

VAWT 2D Free vortex wake Validation and results

Comparison with Shen et al. actuator surface – CFD computations of a 2bladed rotor • Flow characteristics are qualitatively well assessed • Viscosity is quite important

VAWT 2D Free vortex wake Validation and results

• The angle of attack is well reproduced • Airfoil database are very important • Normal force coefficient peak not well reproduced: dynamic stall model to be improved

Validation and results Ferreira panel model

The angle of attack is reproduced sufficiently well

The efficiency seems slightly lower than HAWT

Drag!

Conclusions - HAWT

• HAWT analysis : actuator disk – momentum theory • Shortcomings : swirl flow, wake expansion, tip effects • General momentum theory can’t overcome these issues • Turbomachinery approach • Radial equilibrium • Radial equilibrium in meridional flow • Turbomachinery approach + inverse design • Innovative dsign should be found

Conclusions - VAWT

• VAWT complex 3D geometry, working in his own wake • VAWT analysis : double moultiple streamtubes – BEM model • DMS-BEM limitations • 2D free vortex wake • Airfoil database + DS + tip correction • Slightly lower efficiency • Blockage effects and Reynolds numbers • 1D momentum theory is not suited for VAWT - unsteady • Structural dynamics : aeroelastic codes + free wake codes

References - HAWT

1. Glauert H. Airplane Propellers (Div L) in Aerodynamic Theory (Vol 4). Durand WF ed. Springer: Berlin, 1935. 2. Horlok JH. Axial Flow Turbines. Butterworths: London, England, 1966. 3. Wilson RE, Lissaman PBS. Applied Aerodynamics of Wind-power Machines. Corvallis: Oregon State University, 1974. 4. Horlock JH. Actuator Disk Theory – Discontinuities in thermo-fluid dynamics. McGraw-Hill: New York, 1978. 5. Acton O. Turbomacchine Macchine a Fluido (vol 4). UTET: Torino, 1986. 6. Eppler R. Airfoil Design and Data Springer Verlag: Berlin/New York, 1990 7. Johnson W. Helicopter Theory. Dover Publications: New York, 1994. 8. Lewis RI. Turbomachinery Performance Analysis. Arnold: London, 1996. 9. Cebeci T. An Engineering Approach to the Calculation of Aerodynamic Flows. Horizons Publishing, 1999. 10. Burton T, Sharpe D, Jenkins N, Bossanyi E. Wind Energy Handbook. John Wiley & Sons: Chichester, 2001. 11. Osnaghi C. Teoria delle turbomacchine. Società Editrice Esculapio, 2002. 12. Cumpsty NA. Compressor Aerodynamics. 2nd ed. Krieger scientific: New York, 2004. 13. Leishman JG. Principles of Helicopter Aerodynamics. 2nd ed. Cambridge University Press: Cambridge, 2006. 14. Hansen MOL. Aerodynamics of Wind Turbines 2nd ed. Earthscan: London, 2008. 15. Rankine WJM. On the mechanical principles of the action of propellers. Transaction of the Institute of Naval Architects 1865; 6 :13-30. 16. Froude W. On the elementary relation between pitch, slip and propulsive efficiency. Transaction of the Institute of Naval Architects 1878; 19 : 47. 17. Froude RE. On the part played in propulsion by difference in pressure. Transaction of the Institute of Naval Architects 1889; 30 : 390-423. 18. Drzewiecki S. Méthode pour la détermination des eléments mécaniques des propulseurs hélicoidaux. Bullet. de l’Ass. Technique Maritime 1892. 19. Betz A. with Appendix by Prandtl L. Schraubenpropellermit Geringstem Energieverlust. Göttinger Nachrichten 1919; 193–217. 20. De Bothezat G. The general theory of blade screws. NACA-TR-29, 1920. 21. Goldstein S. On the vortex theory of screw propellers. Proc. Royal Soc. 1929; 123 : 440-465. 22. Theodorsen T. The theory of propellers. NACA-TR-775-776-777-778, 1944.

References - HAWT

23. Zweifel O. The spacing of turbomachine blading, especially with large angular deflections Brown Boweri Rev. 1945 Dec. 436-44 24. Wu C, Wolfenstein L. Application of radial equilibrium condition to axial-flow compressor and turbine design. NACA-TR-955, 1950. 25. Wu C. A general theory of three-dimensional flow in subsonic and supersonic turbomachines of axial-, radial-, and mixed-flow types. NACA-TN2604, 1952. 26. Marble FE, Michelson I. Analytical investigation of some three-dimensional flow problems in turbomachines. NACA-TN-2614, 1952. 27. Hawthorne WR, Horlock JH. Actuator disc theory of the incompressible flow in axial compressors. Proc. Instn. Mech. Engrs. 1962; 176 : 789-814. 28. Wu TY. Flow through a heavily loaded actuator disc. Schifftechnik 1962; 9 : 134 138. 29. Creveling HF, Carmody RH. Axial flow compressor design computer programs incorporating full radial equilibrium. NASA-CR-54532, 1968. 30. Greenberg MD, Powers SR. Nonlinear actuator disk theory and flow field calculations, including nonuniform loading. NASA-CR-1672, 1970. 31. Stoddard FS. Momentum theory and flow states for windmills. Wind Tech. J. 1977; 1 : 3-9. 32. Hütter U. Optimum wind-energy conversion system. Ann. Rev. Fluid Mech. 1977; 9 : 399-419. 33. Denton JD. Throughflow calculations for axial flow turbines. Trans. ASME, J. Eng.Power. 1978; 100. 34. De Vries O. Fluid dynamic aspects of wind energy conversion. AGARDograph AG-243, 1979. 35. Milborrow DJ. 1982 Performance, blade loads and size limits for horizontal axis wind turbines 4th BWEA Wind Energy Conversion(Cranfield: BHRA) 36. De Vries o. On the theory of the horizontal-axis wind turbines. Ann. Rev. Fluid Mech. 1983; 15 : 77-96. 36. Lee JHW, Greenberg MD. Line momentum source in shallow inviscid fluid. J. Fluid Mech. 1984; 145 : 287-304. 37. Kerwin JE. Marine propellers. Ann. Rev. Fluid Mech. 1986; 18 : 367-403. 38. Øye S. A simple vortex model. Proc. of the Third IEA Symposium on the Aerodynamics of Wind Turbines, ETSU, Harwell 1990, 4.1-5.15. 39. Van Kuik GAM. On the limitations of Froude’s actuator disc concept. PhD Thesis dissertation 1991, Technical University of Eindhoven. 40. Hansen C, Butterfield CP. Aerodynamics of horizontal axis wind turbines Ann. Rev. Fluid Mech. 1993; 25 : 115-149. 41. Sørensen JN. A survey of CFD methods in rotor aerodynamics. 7th IEA Symp. On Aerodynamics of Wind Turbines, Lyngby, November 1993. 42. Conway JT. Analytical solutions for the actuator disk with variable radial distribution of load. J. Fluid Mech. 1995; 297 : 327-355. 43. Sørensen JN, Kock CW. A model for unsteady rotor aerodynamics. J. Wind Eng. Ind. Aerodyn. 1995; 58 : 259-275.

References - HAWT

44. Snel H, van Holten Th. Review of recent aerodynamical research on wind turbines with relevance to rotorcraft. Aerodynamics and Aerocoustics of Rotorcraft 1995; AGARD CP 552 : 7-11. 45. Sijtsma P, Sparenberg JA. On the equivalence of a dipole layer of constant strength and a concentrated vortex along its edge. ZAMM Z. angew. Math. Mech. 1996; 76 : 480-482. 46. Colinsk AT. Modern helicopter aerodynamics. Ann. Rev. Fluid Mech. 1997; 29 : 515-567. 47. Conway JT. Exact actuator disk solutions for non-uniform heavy loading and slipstream contraction. J. Fluid Mech. 1998; 365 : 235-267. 48. Snel H. Review of the present status of rotor aerodynamics. Wind Energy 1998; 1 : 46-69. 49. Sørensen JN, Shen WZ, Munduate X. Analysis of wake states by a full-field actuator disc model. Wind Energy 1998; 1 : 73-88. 50. Magnusson M. Near-wake behaviour of wind turbines aerodynamics. J. Wind Eng. Ind. Aerodyn. 1999; 80 : 147-167. 51. Sparenberg JA, de Jager EM. Concentrated force acting in an inviscid and incompressible parallel flow. Math. Meth. Appl. Sci. 2000, 23 : 16371654. 52. Corten GP. Flow separation on wind turbine blades. PhD Thesis dissertation 2001, University of Utrecht. 53. Chaney K, Eggers Jr AJ. Expanding wake induction effects on thrust distribution on a rotor disc. Wind Energy 2002; 5 : 213-226. 54. Leishman JG. Challenges in modeling the unsteady aerodynamics of wind turbines. Wind Energy 2002; 5 : 85-132. 55. Mikkelsen R. Actuator disc methods applied to wind turbines. PhD Thesis dissertation 2003, Technical University of Denmark. 56. Van Kuik GAM. An inconsistency in the actuator disc momentum theory. Wind Energy 2003; 7 : 9-19. 57. Veermer LJ, Sørensen JN, Crespo A. Wind turbine wake aerodynamics. Progr. in Aerospace Sci. 2003; 39 : 467-510. 58. Spalart PR. On the simple actuator disk. J. Fluid Mech. 2003; 494 : 399-405. 59. Sharpe DJ. A general momentum theory applied to an energy extracting actuator disc. Wind Energy 2004; 7 : 177-188. 60. Medici D. Wind turbine wakes - control and vortex shedding. TR KTH Mechanics Royal 2004, Institute of Technology of Stockholm. 61. Shen WZ, Mikkelsen R, Sørensen JN and Bak C. Tip loss corrections for wind turbine computations Wind Energy 2005; 8: 457-475 62. Wald QR. The aerodynamics of propellers. Progr. in Aerospace Sci. 2006; 42 : 85-128. 63. Bak C. Research in aeroelasticity EFP-2005 Risø National Laboratory Wind Energy Department 2006; Risø-R-1559(EN).

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64. Hansen MOL, Sørensen JN, Voutsinas S, Sørensen N, Madsen HAa. State of the art in wind turbine aerodynamics and aeroelasticity. Progr. in Aerospace Sci. 2006; 42 : 285-330. 65. Crawford C. Re-examining the precepts of the blade element momentum theory for coning rotors disc. Wind Energy 2006; 9 : 457-478. 66. Sant T. Improving BEM based aerodynamic models in wind turbine design codes. PhD Thesis dissertation 2007, Delft University of Tecnology. 67. Okulov VL, Sørensen JN. Stability of helical tip vortices in a rotor far wake. J. Fluid Mech. 2007; 576 : 1-25. 68. Wood DH. Including swirl in the actuator disk analysis of wind turbines. Wind Eng. 2007; 31 : 317-323. 69. Battisti L, Soraperra G. Analysis and application of pre-design methods for HAWT rotors. Proc. of EWEC (Milan, 7-10 May 2007). 70. Simon JF. Contribution to throughflow modelling for axial flow turbomachines. PhD Thesis dissertation 2007, Université de Liège. 71. L Battisti, G Soraperra, R Fedrizzi and L Zanne. Inverse design-momentum, a method for the preliminary design of horizontal axis wind turbines. Proc. The Science of making Torque from Wind (Lyngby, Denmark, 28-31 August 2007) (J. of Physics: Conference Series Vol 75) IOP Publishing Ltd 72. Okulov VL, Sørensen JN. Refined Betz limit for rotors with a finite number of blades. Wind Energy 2008; 11 : 415-426. 73. Casey M, Robinson C. A new streamline curvature throughflow method for radial turbomachinery. Proc. ASME Turbo Expo (Berlin, 9-13 June 2008). 74. Ramakrishna PV, Govardhan M. Study of sweep and induced dihedral effects in subsonic axial flow compressor passages – Part I: design considerations – changes in incidence, deflection, and streamline curvature. I.J.Rotating Machinery 2009; 2009. 75. www.gaia-wind.com

References - VAWT

1. Glauert H. Airplane Propellers - Aerodynamic Theory, W. F. Durand ed, Chapter XI. Berlin:Springer Verlag, 1935. 2. Glauert H. The Elements of Aerofoil and Airscrew Theory. 2nd ed. Cambridge University Press: Cambridge, England, 1947. 3. Abbott IH, Von Doenhoff AE. Theory of Wing Sections. Dover Publications Inc.: New York, 1959. 4. Horlok JH. Axial Flow Turbines. Butterworths: London, England, 1966. 5. Wilson RE, Lissaman PBS. Applied Aerodynamics of Wind-power Machines. Corvallis: Oregon State University, 1974. 6. Clancy LJ. Aerodynamics, John Wiley & Sons: New York, 1975. 7. Barlow JB, Rae WH, Pope A. Low Speed Wind Tunnel Testing. 3rd ed. John Wiley & Sons Inc.: New York, 1999. 8. Katz J, Plotkin A. Low Speed Aerodynamics. 2nd ed. Cambridge University Press: Cambridge, 2000. 9. Paraschivoiu I. Wind Turbine Design - With Emphasis on Darrieus Concept. Polytechnic International Press, 2002. 10. Leishman JG. Principles of Helicopter Aerodynamics. 2nd ed. Cambridge University Press: Cambridge, 2006. 11. Hansen MOL. Aerodynamics of Wind Turbines 2nd ed. Earthscan: London, 2008. 12. Glauert H. A General Theory of the Autogyro, ARCR R&M, No. 1111, 1926. 13. Darrieus GJM. Turbine having its rotating shaft transverse to the flow of the current. US patent 1,835,018, 8-12-1931. 14. Jacobs E, Sherman A. Airfoil characteristics as affected by variations of the Reynolds number. NACA Report 586, 1937. 15. Riegels FW. Aerofoil sections: results from wind-tunnel investigations, Theoretical foundation, Ch.7, London, Butterworths Ed., 1961. 16. Maskell EC. A Theory of the blockage effects on bluff bodies and stalled wings in an enclosed wind tunnel. ARC/R&M-3400, 1963. 17. Strickland JH. The Darrieus turbine: A performance prediction model using multiple streamtubes, SAND75-0431, 1975. 18. Muraca RJ, Stephens MV, Dagenhart JR. Theoretical performance of cross-wind axis turbines with results for a catenary vertical axis configuration, NASA TMX-72662, 1975. 19. Blackwell BF, Sheldal RE, Feltz LV. Wind tunnel performance data for the Darrieus wind turbine with NACA0012 blades, Sandia National Laboratories, Albuquerque, New Mexico, SAND76-0130, 1976. 20. Sheldahl RE, Blackwell BF. Free-air performance tests of a 5-metre-diameter Darrieus turbine. Sandia National Laboratories, Albuquerque, New Mexico, SAND77-1083, 1977. 21. De Vries O. Fluid dynamic aspects of wind energy conversion. AGARDograph AG-243, 1979.

References - VAWT

22. Strickland J, Webster B, Nguyen T. A vortex model of the Darrieus turbine: an analytical and experimental study. Sandia National Laboratories, Albuquerque, New Mexico, SAND79-7058, 1979. 23. Read S, Sharpe DJ. An extended multiple streamtube theory for vertical axis wind turbines, 2nd BWEA Workshop (April 1980). 24. Sheldahl RE, Klimas PC. Aerodynamic characteristics of 7 symmetrical airfoil sections through 180-degree angle of attack for use in aerodynamics analysis of vertical axis wind turbine. Sandia National Laboratories: SAND80-2114, 1980. 25. Sheldahl RE, Klimas PC, Feltz LV. Aerodynamic performance of a 5-metre-diameter Darrieus turbine with extruded NACA-0015 blades. SAND800179, 1980. 26. Strickland J, Webster B, Nguyen T. A vortex model of the Darrieus turbine: an analytical and experimental study. Sandia National Laboratories, Albuquerque, New Mexico, SAND81-7071, 1981. 27. Viterna LA, Corrigan RD. Fixed pitch rotor performance of large horizontal axis wind turbines, NASA CP-2230, 1981. 28. Madsen HAa. The actuator cylinder. A flow model for vertical axis wind turbines. Aalborg University Centre: Aalborg, Denmark, 1982. 29. Carne TG, Nord AR. Modal testing of a rotating wind turbine. Sandia National Laboratories, Albuquerque, New Mexico, SAND82-0631, 1982. 30. Oler JW, Strickland JH et Al. Dynamic stall regulation of the Darrieus turbine. Sandia National Laboratories, Albuquerque, New Mexico, SAND837029, 1983. 31. Wilson RE, Walker SN. Performance analysis of horizontal axis wind turbines, Oregon State Univ., Corvallis, OR, 1984. 32. Loeffler AL Jr, Steinhoff JS. Computation of wind tunnel wall effects in ducted rotor experiments. AIAA Journal of Aircraft 1985; 22, n.3: 188-192 33. Marini M, Massardo A, Satta A, Zamparo G. Theoretical aerodynamic methods for VAWT analysis. Energy Conversion Engineering Conference, 1989. 34. Homicz GF. Numerical simulation of VAWT stochastic aerodynamic loads produced by atmospheric turbulence: VAWT-SAL code. Sandia National Laboratories, Albuquerque, New Mexico, SAND91-1124, 1991. 35. Tangler LJ, Ostowari C. Horizontal axis wind turbine post stall airfoil characteristics synthetization. Solar Energy Research Institute. SERI/TP-2574400 - UC Cathegory 261 - DE91002198. 36. Mandal C, Burton JD. The effects of dynamic stall and flow curvature on the aerodynamics of darrieus turbines applying the Cascade model. Wind Eng 1994; 18 (6): 267–282. 37. Allet A, Paraschivoiu I. Viscous flow and dynamic stall effects on vertical-axis wind turbines. International Journal of Rotating Machinery 1995; 2, n.1: 1-14

References - VAWT

38. Abdel Azim El-Sayed AF, Hirsch C and Derdelinckx R. Dynamics of vertical axis wind turbines (Darrieus Type), International Journal of Rotating Machinery 1995; 2, n.1: 33-41. 39. Fortunato B, Dadone A, Trifoni V. A two-dimensional methodology to predict vertical axis wind turbine performance. Journal of Solar Energy Engineering 1995; 117:187-193. 40. Mercker E, Wiedemann J. On the correction of the interference effects in open jet wind tunnels. SAE - 960671, 1996. 41. Allet S, Halle I, Paraschivoiu I. Numerical simulation of dynamic stall around an airfoil in darrieus motion. Journal of Solar Energy Engineering, 1999; 121: 69-76. 42. Lindenburg C. Stall coefficients. Aerodynamic airfoil coefficients at large angles of attack. IEA symposium on the aerodynamics of wind turbines. (NREL, CO, USA: December 4-5, 2000). 43. Corten GP. Flow separation on wind turbine blades. PhD Thesis dissertation 2001. Utrecht University, The Netherlands. 44. Fujisawa N, Shibuya S. Observations of dynamic stall on Darrieus wind turbine blades. Journal of Wind Engineering and Industrial Aerodynamics 2001; 89, n. 2: 201–214. 45. Mikkelsen R, Sørensen JN. Modelling of wind tunnel blockage. Proc. CD-ROM Global Windpower Conference and Exhibition (2002). 46. Mertens S, van Kuik G, van Bussel G. Performance of a H-Darrieus in the skewed flow on a roof. Journal of Solar Energy Engineering 2003; 125: 433–440. 47. Timmer WA, van Rooij RPJOM. Summary of the Delft University wind turbine dedicated airfoils. AIAA-2003-0352, 2003. 48. Grignoux T, Gibert R et Al. Eoliennes en milieu urbain – Etat de l’art. ARENE, Ile-de-France, 2004. 49. Van Bussel GJW, Mertens S et Al. TURBY®: concept and realisation of a small VAWT for the built environment. The Science of making Torque from Wind (Delft : 19-21 April 2004). 50. Hansen MH, Gaunaa M, Madsen HA. A Beddoes-Leishman type dynamic stall model in state-space and indicial formulations. Risø National Laboratory. Risø-R-1354, 2004. 51. Montgomerie B. Methods for root effects, tip effects and extending the angle of attack range to ±180°, with application to aerodynamics for blades on wind turbines and propellers. FOI Swedish Defence Research Agency. FOI-R—1305—SE, 2004. 52. Mertens S. Wind energy in the build environment. PhD Thesis dissertation 2006. Delft University of Technology, The Netherlands. 53. Van Der Tempel J. Design of support structures for offshore wind turbines. PhD Thesis dissertation 2006. Delft University of Technology, The Netherlands.

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54. Ferreira CS, van Kuik G, van Bussel G. Wind tunnel hotwire measurements, flow visualization and thrust measurement of a VAWT in skew. AIAA/ASME Wind Energy Symposium (2006). 55. Sørensen JN, Shen WZ and Mikkelsen R. Wall correction model for wind tunnels with open test section, AIAA Journal 2006; 44, n.8. 56. Claessens MC. The design and testing of airfoils for application in small vertical axis wind turbines. MSc Thesis dissertation 2006. Delft University of Technology, The Netherlands. 57. Sant T. Improving BEM based aerodynamic models in wind turbine design codes. PhD Thesis dissertation 2007, Delft University of Tecnology. 58. Ferreira CS, van Bussel GJW et Al. 2D PIV visualization of dynamic stall on a vertical axis wind turbine. AIAA/ASME Wind Energy Symposium, (2007). 59. Fitzgerald RE. Wind tunnel blockage corrections for propellers. MS Thesis 2007. University of Maryland, Department of Aerospace Engineering, College Park MD. 60. Islam M, Ting D, Fartaj A. Aerodynamic models for Darrieus-type straight-bladed vertical axis wind turbines. Renewable & sustainable energy reviews 2008. 12: 1087-1109. 61. Dixon C, Ferreira CS et Al. A 3D unsteady panel method for vertical axis wind turbines. EWEC Brussels (31 March – 3 April 2008). 62. Hofemann C, Ferreira CS, et Al. 3D Stereo PIV study of tip vortex evolution on a VAWT. EWEC, Brussels, (2008). 63. Vita L, Paulsen US et Al. “A novel floating offshore wind turbine concept” EWEC (Marseille : 16 - 19 March 2009). 64. Battisti L, Brighenti A, Zanne L. Analisi dell’effetto della scelta dell’architettura palare sulle prestazioni di turbine eoliche ad asse verticale. 64° Congresso Nazionale ATI. L'Aquila. (6 - 11 September 2009). 65. Shen WZ, Zhang JH, Sørensen JN. The actuator surface model: a new Navier–Stokes based model for rotor computations. Journal of Solar Energy Engineering 2009; 131. 66. Battisti L, Zanne L et Al. Aerodynamic measurements on a vertical axis wind turbine in a large scale wind tunnel. Proc. of ASME Turbo Expo 2010. Glasgow, UK (14-18 June 2010). 67. Ferreira CS. The near wake of the VAWT. PhD Thesis dissertation 2009, Delft University of Tecnology. 68. The Eurocode 1, Part 2-4: Wind actions (ENV 1991-2-4: 1994). 69. http://www.sandia.gov/wind/topical.htm#VAWTARCHIVE 70. http://www.tozzinord.it/

FLUID DYNAMIC MODELLING OF WIND TURBINES

sec. D

Vr 0 D Vt

Vz Vr Vt 3

Relatori: Prof.Ing. Lorenzo BATTISTI Prof.Ing. Piero PINAMONTI

Dottorando: Dott.Ing. Luca ZANNE

Udine 21 Maggio 2010

Summary

Introduction PART I : HAWT analysis HAWT Fluid dynamics A turbomachinery approach Inverse design

Summary

PART II : VAWT analysis VAWT fluid dynamics VAWT experimental analysis VAWT free vortex wake Results and conclusions

Introduction

Wind energy market (EWEA) Installed capacity

Offshore WE market (EWEA)

Aim of the thesis & thesis outline

The aim of the thesis is to analyze the fluid dynamic models of wind energy conversion systems, pointing out the limitations of current engineering models and proposing innovative solutions from the design point of view The research activities have been divided in two main parts, following the different rotor – flow interaction characteristics: 1. Horizontal axis wind turbines - HAWT; 2. Crossflow wind turbines, as vertical axis wind turbines - VAWT.

Part I : HAWT analysis HAWT fluid dynamics

HAWT fluid dynamics is mainly based on the actuator disk concept

HAWT fluid dynamics Actuator disk concept

The turbine generates mechanical work from the kinetic energy of the fluid flow The work exchange between the fluid and the shaft is done by is done by the rotor, which can be modelled as an actuator disk The bladed rotor can be represented with equivalent forces distribuited over a permeable, immaterial disk Infinite number of blades Infinite rotational velocity

HAWT fluid dynamics Actuator disk – momentum theory

Froude applied for the first time the actuator disk concept to a rotor in open flow. He applied it with the 1D momentum balance in axial direction Momentum equation T = ∆p ⋅ Am = ρVz ,3 A3 (Vz ,0 − Vz ,3 )

Energy conservation

Weul = ∆p

Mass conservation

Vz ,m Am = Vz ,3 A3 Vz ,1 ≅ Vz ,2 ≅ Vz ,m

ρ

=

Vz2 − Vz2 ,0 ,3 2

Froude result!

Vz ,m = Vz ,0 + Vz ,3 2

Actuator disk Blade element – momentum theory

Drzewiecki first applied Froude result dividing the rotor in different annular streamtubes : Non uniform loading Vz ,m = Vz ,0 + Vz ,3 2

Raero FN Lift φflow

With the blade element airfoil theory rotor performances can be easily calculated The annuli interaction is neglected No swirl flow, (wake expansion?) Ok lightly loaded rotors

Wind.=[ -a·V0; -a’·ωr ]

FT

Drag z

chord line

-ωr θpitch+βtwist φflow αattack V0 Vrel. y

rotor plane

HAWT fluid dynamics General momentum theory

The general momentum theory should overcome the issues of the swirl flow modelling Momentum equation : axial T = ∫ ( p1 − p2 ) dA = ∫ ρVz ,3 (Vz ,0 − Vz ,3 ) + ( p0 − p3 ) dA Am A3

tangential

M = ∫ ρVθ ,3Vz ,3 r3 dA

A3

radial

p3 − p0

ρ

=

p3 ( r3 ) − p3 rtip ,3

(

ρ

)=−

∫r3

2 rtip ,3 Vθ ,3

r3

dr3

1 + 1 2Vθ ,3 Ωr3 1 + 1 2Vθ ,2 Ωr 2 1 ρ ∫A3 (Vz ,0 − Vz ,3 ) dA = ρΩ ∫A3Vz ,3Vθ ,3r3 − dA 2 Vz ,3 Vz ,m

Solutions: • De Vries • Differential

• GM theory is an integral formulation • It needs the wake solution

8 V2 Weul = ⋅ 0 9 2

Actuator disk – momentum theory limitations

Actuator strip Wake states

Conway exact solution

HAWT fluid dynamics Vortex theory

Vortex theory calculates the flow field of the rotor wake by using the fluid dynamic laws of vorticity (BiotSavart law, Kelvin’s theorem, Helmholtz’s laws) Introduced by Joukowski – Betz – Prandtl Most widespread for propeller analysis and design (both for aerodynamic and marine propellers) and for helicopter rotor performance prediction • Prescribed vortex wake • Free vortex wake

Vortex theory Prescribed vortex wake

Vz ,m =

Vz ,0 + Vz ,3 2

Axial velocity

d Γ = 2π ⋅ d ( rVθ ,2 )

gθ ,m = d Γ r Ω + Vθ ,2 2 Vz , m 2π r d Γ r3Ω + Vθ ,3 2π r3 Vz ,3

Radial velocity

Vr ( r ,0 ) = − 1 r ∂Vz r ( r ,0 ) dr r ∫0 ∂z

gθ gθ + r 2r 2π ( r − r ) gθ

2

∂Vz ∂z ( r ,0 ) =

gθ ,3 =

vz ,m = gθ 2

vz ,3 = gθ

1 1 gθ ( r − r ) ∂Vz ∂z ( r ,0 ) = − − 2 4 2π 2π ( r − r ) r r5

Part I : HAWT analysis A turbomachinery approach

V ∂rVθ 1 ∂p 0 ∂V ∂V = Fr + θ − Vz r − z ρ ∂r r ∂r ∂r ∂z

Vz

∂Vθ Vr ∂rVθ + = Fθ ∂z r ∂r

∂Vθ + Vθ ∂z

1 ∂p 0 ∂V ∂V = Fz + Vr r − z ρ ∂z ∂r ∂z

∂rVr ∂rVz + =0 ∂r ∂z

A turbomachinery approach Stoke’s stream function

ωθ =

∂Vr ∂Vz − ∂z ∂r

1 ∂ψ ∂ 2ψ + = −rωθ r ∂r ∂z 2

d ( rVθ ) dψ − r dp 0 ρ dψ

∂ 2ψ ∂r 2

−

ωθ = Vθ

∂ 2ψ ∂r 2

−

1 ∂ψ ∂ 2ψ + =0 r ∂r ∂z 2

∂ 2ψ ∂r 2

−

d ( rVθ ) r 2 dp 0 1 ∂ψ ∂ 2ψ + 2 = −rωθ = −rVθ + r ∂r ∂z dψ ρ dψ

Linearized solution : Horlock actuator disk solution

∂ 2ψ 1 ∂ψ ∂ 2ψ − + = − F (r ) ∂r 2 r ∂r ∂z 2 Vz ,3 − Vz ,0 kz Vz ( r , z ) = Vz ,0 + e 2 Vr ( r , z ) = − 1 r Vz ,3 ( r ) − Vz ,0 kz kr e dr r ∫0 2

Froude result

A turbomachinery approach Motion in region II

The flow is determined by

rVθ

p0

Euler equation

1

ρ

(p

0 2

0 − p1 = ΩrVθ = Weul

)

Wu equation

∂ 2ψ

0 0 1 ∂ψ ∂ 2ψ p2 − p1 − + = − ∂r 2 r ∂r ∂z 2 Ω2

(

)

ρ

+r

2

0 d ( rVθ ) 1 dp2 = Ωr 2 − rVθ ρ dψ dψ

(

)

The angular momentum distribution can be assigned

Vθ = k1r n + k2 r

rVθ = k1r n +1 + k 2

Free vortex distribution

rVθ = const

The radial equilibrium theory applied to wind turbines

Radial momentum equilibrium

V ∂rVθ 1 ∂p 0 ∂V ∂V = Fr + θ − Vz r − z ρ ∂r r ∂r ∂r ∂z

dV 1 dp 0 Vθ d ( rVθ ) = + Vz z ρ dr r dr dr

ISRE

Sections 1 - 2

Vz2 − Vz2hub = ,

(p ρ

2

0

0 − pr ,hub − 2 ∫

)

r

r , hub

Vθ ∂rVθ r ∂Vr dr + 2 ∫ V dr − Vr2 − Vr2hub , r , hub z ∂z r ∂r ψ

(

)

Wu hypothesis

∂Vr ,1 ∂z =− ∂Vr ,2 ∂z

Wu hypothesis on a streamline

∂Vr ,1 ∂Vr ,2 = − ∂z ψ ∂z ψ 0 dVz ,m dVr ,m 1 dp2 Vθ ,2 d ( rVθ ,2 ) = + 2Vz ,m + 2Vr ,m ρ dr r dr dr dr

0 dVz ,m 1 dp2 Vθ ,2 d ( rVθ ,2 ) = + 2Vz ,m ρ dr r dr dr

The radial equilibrium theory results and comments

Radial equilibrium solution for a uniformly loaded disk λ=8 8 V2 Weul = ⋅ 0 9 2

λ=2

8 V2 Weul = ⋅ 0 9 2

The radial equilibrium theory results and comments

Mikkelsen actuator disk – CFD solution for a uniformly highly loaded disk (wind turbine state) 8 V2 Weul = ⋅ 0 9 2

Conway actuator disk – vortex theory exact solution for a (almost) parabolic highly loaded disk (propeller state) CT = 3.147

The radial equilibrium theory results and comments

Power and thrust coefficients for the different flow field solution models with a constant work extraction

Conway velocity at the centre of the disk for a propeller

The radial equilibrium theory on a streamline

Radial equilibrium with meridional velocity

0 dVs ,m 1 dp2 Vθ ,2 d ( rVθ ,2 ) = + 2Vs ,m ρ dr r dr dr

Vs2m = Vz2m + Vr2m , , ,

Denton / Cumpsty approach

∂V V2 1 ∂ 2 1 ∂p 0 1 ∂ 2 2 Vs ∂ Vs = + Vs s sin ( ε + δ ) + s cos ( ε + δ ) − 2 r Vθ + ( rVθ ) tan γ + Fd 2 ∂q ∂s rs r ∂s ρ ∂q 2r ∂q

(

)

∂V V2 1 ∂Vs2 1 ∂p 0 1 ∂ 2 2 = + Vs s sin ε − s cos ε − 2 r Vθ ρ ∂r ∂s 2 ∂r rs 2r ∂q

(

)

( )

Coning / yaw effects Turbulence wake state / stall Tip effects Unsteady dynamics

∂Vs2m , ∂r

=

0 1 ∂Vs ,m 1 ∂p2 1 1 ∂ 2 2 + 2Vr ,m − Vs2m cos ε + − r Vθ , rs ,1 rs ,2 2r 2 ∂q ∂s ρ ∂r

Considerations on the turbomachinery approach

• The theory handles an expanding and rotating wake. • Only the disk station has to be solved to obtain the information needed to compute CP and CT. • The method is simple and robust also for low tip speed ratios • Arbitrary disk loading can be analyzed. • The mathematics involved are comparable with those of the usual actuator disk approaches. • The actual velocities distribution are qualitatively assessed even though more work has to be carried out to better understand the fluid flow in the neighborhood of the disk and in the wake. • The radial velocity gradients along the streamlines at the disk have to be better described to reduce the axial velocity overestimation at the disk inner portion.

Part I : HAWT analysis Inverse Design

Inverse design and direct design methods

The turbine close field structure The blade architecture

Blade forces

Fθ , Z = ρ ⋅ Vz ,m ⋅ s ⋅ (Vθ ,2 − Vθ ,1 ) = ρ ⋅ V 2 z ,m ⋅ s ⋅ ( tan α 2 − tan α1 ) 0 0 Fz , Z = ( p1 − p2 ) ⋅ s +

1 ρ ⋅ Vθ2,2 ⋅ s 2

Weul = U ⋅ (Vθ ,2 − Vθ ,1 ) = U ⋅ Vz ,m ⋅ (tan α 2 − tan α1 ) = U ⋅ Vz ,m ⋅ (tan β 2 − tan β1 )

k Weul = rω ⋅ k1r n + 2 r

Flow angles

β1 = tan −1

U + Vθ ,1 Vz ,1

β m = tan −1

U + Vθ ,m Vz ,m

β 2 = tan −1

U + Vθ ,2 Vz ,2

Vz ,m

Vz ,1 Vz ,2

The blade architecture

Cy = Fy Fy ,max = 2⋅ s ( tan β2 − tan β1 ) cos2 β2 cz cz = c ⋅ cos β m

s CL = 2 (tan β 2 − tan β1 ) cos β m c

C y = 0.8

Zweifel Lieblein

θ=

π

2

− β m − sen −1 (

CL, ID 2π

)

Dloc =

Wmax − W2 Wmax

Inverse Design Results and discussion

Gaia turbine

Inverse Design Results and discussion

1 W / 1/2V 2 0 1 VDz / V0

Flow characteristics

0.5 0 0 0.2 0.4 r/R 0.6 0.8 1

0.5

0

0

0.2

0.4 r/R

0.6

0.8

1

40 20 0 1.5

beta1-beta2 [deg]

The blade architecture and loads

C/R 0.2 0.1 0 0 0.2 0.4 r/R Fn / q0R 0.6 0.8 1 betam [deg] 80 60 40 20 0 2 1 0 0.6 Mn / q0R3 0.4 0.2 0 P / 1/2rhoAV 3 0 0.6 0.4 0.2 0 0 0.2 0.4 r/R 0.6 0.8 1 0 0.2 0.4 r/R 0.6 0.8 1 0 0.2 0.4 r/R 0.6 0.8 1

alpha2 [deg]

40 20 0

0

0.2

0.4 r/R

0.6

0.8

1

0

0.2

0.4 r/R

0.6

0.8

1

1 p1-p2 / q0 1 Psi 0.5 0 0.5

2 Ft / q0R

0 0.2 0.4 r/R 0.6 0.8 1

1 0

0

0.2

0.4 r/R

0.6

0.8

1

0

0

0.2

0.4 r/R

0.6

0.8

1

0

0.2

0.4 r/R

0.6

0.8

1

0.6 Mt / q0R3 0.4 0.2 0 0 0.2 0.4

λ=6,Z=3

dCp / d(r/R)

0.6 r/R

0.8

1

1 0.5 0

0

0.2

0.4 r/R

0.6

0.8

1

Inverse Design Results and discussion

λ=6,Z=3

Inverse Design Results and discussion

λ = 1.5 , Z = 3

Part II : VAWT analysis VAWT fluid dynamics

Darrieus eggbeater – Darrieus H/V – Gorlov type Building environment Offshore multi Mega Watt

CL =

dL 1 ρ W 2 c dh 2 0

VAWT fluid dynamics The double disk BEM for VAWT

Blade element forces

C N = CL cos β + C D sin β

2

Flow characteristics

β = tan −1

Vsenϑ cos δ ( V cosϑ + Ωr ) cos γ

2

W 2 = ( Vcosϑ + Ωr ) cos γ + ( Vsenϑ cos δ ) Re = cW

CT = C L sin β − C D cos β

ν0

dL 1 ρ W 2 c dh 2 0 CD = dD 1 ρ W 2 cdh 2 0

1 dh dFN = ρ0 W2 c CN 2 cos δ dFT = 1 dh ρ0 W 2 c CT 2 cos δ

CL =

Shaft torque/power

dM = dFT Ω

1 ϑ dM Ω Nϑ ∫ ∫ ∫ MΩ CP = = 1 ρ A V3 1 ρ A V3 0 sw 0 2 2 0 sw 0

N BL

VAWT fluid dynamics The double disk BEM for VAWT

Blade element

dFx = dFT cos ϑ cos βc cos γ + dFN sin ϑ cos δ

dFx = B 2

CTH =

∆ϑ

π

dFx

dFx 1 2 ρ0 V0 dAs 2

dAs = dh r dϑ sinϑ

Momentum theory

α=

V V0

dFx = 2ρ dA s V(V0 -V)

CTH =

dFx 1 2 ρ V0 dA s 2

=

2ρ dA s V(V0 -V) = 4α (1 − α ) 1 2 ρ V0 dA s 2

The double disk BEM for VAWT Corrections

Glauert correction Tip losses

CTH =

26 4 (1 − α ) + 15 15

Post stall airfoil performance correction Flow curvature Dynamic stall Streamtubes expansion

VAWT fluid dynamics Validation and results

Sandia 5m Darrieus 3blades NACA0015

Four geometric characteristics

VAWT fluid dynamics Validation and results

Blade tangential and normal force coefficients Shaft forces and torque Mean value and fluctuations

VAWT fluid dynamics Validation and results

Shaft torque and forces diagrams

2-bladed 3-bladed 3-bladed 2-bladed

presents the best power performance presents lower forces fluctuations Gorlov type presents the lowest fatigue loads (complex geometry) a 90° reduces the loads fluctuations but needs rotor balancig

VAWT fluid dynamics Limitations of VAWT BEM codes

• The circular path is simplified in two actuator disks • The momentum equilibrium is applied only in axial direction • The axial expansion is generally neglected or not correctly/completely implemented • The turbulent wake state correction is taken from HAWT corrections • No (or weak) interaction between streamtubes • Tip losses correction is of doubtful application for VAWT • Complex geometry not resolvable from a fluid dynamic point of view • Unsteady fluid dynamic effects are of difficult implementation

Part II : VAWT analysis VAWT experimental analysis

VAWT experiments in controlled conditions The Politecnico di Milano Large Wind Tunnel High speed test section: 4x3.84m Wind speed up to 55m/s Possibility to work in open/close test section 2 different rotor prototypes designed by Tozzi Nord Wind Turbines: PDF1 – research purpose PDF3 – commercial turbine

The turbines layout and the instrumentations

PDF1 3 Blades H = 1.46m D = 1.03m NACA0021 Solidity 0.25 Rotor position Torque Support loads PDF3 3 Blades - Gorlov H = 2.5m D = 1.78m P = 1.5kW H(tower) = 3.5m Rotor position Torque (electric) Support loads Aerodynamics Directional pneumatic 5 holes probe Single sensor hot wire anemometer

VAWT experimental analysis PDF1 rotor - Performance

Blockage : 0.097 close test section Blockage effects up to 20-30% for CP and 10-20% for CT Reynolds numbers very important on power performance for Re < 200000

VAWT experimental analysis PDF1 rotor - Aerodynamics

λ = 1.6

λ = 2.6

Wake non symmetric and deformed turnwise (in particular at low tip speed ratios) In closed wind tunnel there is an higher velocity due to blockage effects

λ = 1.6

VAWT experimental analysis PDF1 rotor - Aerodynamics

Wind tunnel blockage

1 1 1 1 2 2 T = AD p0 + ρV02 − ρVD − p3 + ρV32 − ρVD 2 2 2 2 V0' VD C = + T V V0 V0 4 D V0

1D momentum theory doesn’t seem the best model for blockage effects

VAWT experimental analysis PDF1 rotor - Aerodynamics

Unsteady flow field

λ = 2.5

VAWT experimental analysis PDF3 rotor - Dynamics

Dynamic analysis and modelling

Part II : VAWT analysis 2D Free vortex wake

Bound and shed vorticity

L = Cl

1 ρW 2 c = ρW Γ B 2

1 Γ B = ClWc 2

δΓ S = −

d ΓB δθ dθ

Induced velocitites (Biot-Savart)

u=−

( y − y0 ) Γ 2π ( x − x0 )2 + ( y − y0 )2 + h 2

v=

( x − x0 ) Γ 2π ( x − x0 )2 + ( y − y0 )2 + h 2

Flow characteristics

W 2 = ΩR + (U 0 + uC ) cos (θ ) + vC sin (θ ) + (U 0 + uC ) sin (θ ) − vC cos (θ ) 2 2

Shed vortex position

xS ,i = xS ,i −1 + U 0 + uS ( xS ,i −1 , yS ,i −1 ) ⋅ dt yS ,i = yS ,i −1 + vS ( xS ,i −1 , yS ,i −1 ) ⋅ dt

xS ,i = xS ,i −1 + U 0 + 0.5 ⋅ uS ( xS ,i , yS ,i ) + uS ( xS ,i −1 , yS ,i −1 ) ⋅ dt

(

)

(U 0 + uC ) sin (θ ) − vC cos (θ ) φ = tan −1 − ΩR + (U 0 + uC ) cos (θ ) + vC sin (θ ) α =φ − β

(

)

yS ,i = yS ,i −1 + 0.5 ⋅ vS ( xS ,i , yS ,i ) + vS ( xS ,i −1 , yS ,i −1 ) ⋅ dt

(

)

VAWT 2D Free vortex wake Validation and results

Comparison with Shen et al. actuator surface – CFD computations of a 2bladed rotor • Flow characteristics are qualitatively well assessed • Viscosity is quite important

VAWT 2D Free vortex wake Validation and results

• The angle of attack is well reproduced • Airfoil database are very important • Normal force coefficient peak not well reproduced: dynamic stall model to be improved

Validation and results Ferreira panel model

The angle of attack is reproduced sufficiently well

The efficiency seems slightly lower than HAWT

Drag!

Conclusions - HAWT

• HAWT analysis : actuator disk – momentum theory • Shortcomings : swirl flow, wake expansion, tip effects • General momentum theory can’t overcome these issues • Turbomachinery approach • Radial equilibrium • Radial equilibrium in meridional flow • Turbomachinery approach + inverse design • Innovative dsign should be found

Conclusions - VAWT

• VAWT complex 3D geometry, working in his own wake • VAWT analysis : double moultiple streamtubes – BEM model • DMS-BEM limitations • 2D free vortex wake • Airfoil database + DS + tip correction • Slightly lower efficiency • Blockage effects and Reynolds numbers • 1D momentum theory is not suited for VAWT - unsteady • Structural dynamics : aeroelastic codes + free wake codes

References - HAWT

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References - HAWT

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References - VAWT

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22. Strickland J, Webster B, Nguyen T. A vortex model of the Darrieus turbine: an analytical and experimental study. Sandia National Laboratories, Albuquerque, New Mexico, SAND79-7058, 1979. 23. Read S, Sharpe DJ. An extended multiple streamtube theory for vertical axis wind turbines, 2nd BWEA Workshop (April 1980). 24. Sheldahl RE, Klimas PC. Aerodynamic characteristics of 7 symmetrical airfoil sections through 180-degree angle of attack for use in aerodynamics analysis of vertical axis wind turbine. Sandia National Laboratories: SAND80-2114, 1980. 25. Sheldahl RE, Klimas PC, Feltz LV. Aerodynamic performance of a 5-metre-diameter Darrieus turbine with extruded NACA-0015 blades. SAND800179, 1980. 26. Strickland J, Webster B, Nguyen T. A vortex model of the Darrieus turbine: an analytical and experimental study. Sandia National Laboratories, Albuquerque, New Mexico, SAND81-7071, 1981. 27. Viterna LA, Corrigan RD. Fixed pitch rotor performance of large horizontal axis wind turbines, NASA CP-2230, 1981. 28. Madsen HAa. The actuator cylinder. A flow model for vertical axis wind turbines. Aalborg University Centre: Aalborg, Denmark, 1982. 29. Carne TG, Nord AR. Modal testing of a rotating wind turbine. Sandia National Laboratories, Albuquerque, New Mexico, SAND82-0631, 1982. 30. Oler JW, Strickland JH et Al. Dynamic stall regulation of the Darrieus turbine. Sandia National Laboratories, Albuquerque, New Mexico, SAND837029, 1983. 31. Wilson RE, Walker SN. Performance analysis of horizontal axis wind turbines, Oregon State Univ., Corvallis, OR, 1984. 32. Loeffler AL Jr, Steinhoff JS. Computation of wind tunnel wall effects in ducted rotor experiments. AIAA Journal of Aircraft 1985; 22, n.3: 188-192 33. Marini M, Massardo A, Satta A, Zamparo G. Theoretical aerodynamic methods for VAWT analysis. Energy Conversion Engineering Conference, 1989. 34. Homicz GF. Numerical simulation of VAWT stochastic aerodynamic loads produced by atmospheric turbulence: VAWT-SAL code. Sandia National Laboratories, Albuquerque, New Mexico, SAND91-1124, 1991. 35. Tangler LJ, Ostowari C. Horizontal axis wind turbine post stall airfoil characteristics synthetization. Solar Energy Research Institute. SERI/TP-2574400 - UC Cathegory 261 - DE91002198. 36. Mandal C, Burton JD. The effects of dynamic stall and flow curvature on the aerodynamics of darrieus turbines applying the Cascade model. Wind Eng 1994; 18 (6): 267–282. 37. Allet A, Paraschivoiu I. Viscous flow and dynamic stall effects on vertical-axis wind turbines. International Journal of Rotating Machinery 1995; 2, n.1: 1-14

References - VAWT

38. Abdel Azim El-Sayed AF, Hirsch C and Derdelinckx R. Dynamics of vertical axis wind turbines (Darrieus Type), International Journal of Rotating Machinery 1995; 2, n.1: 33-41. 39. Fortunato B, Dadone A, Trifoni V. A two-dimensional methodology to predict vertical axis wind turbine performance. Journal of Solar Energy Engineering 1995; 117:187-193. 40. Mercker E, Wiedemann J. On the correction of the interference effects in open jet wind tunnels. SAE - 960671, 1996. 41. Allet S, Halle I, Paraschivoiu I. Numerical simulation of dynamic stall around an airfoil in darrieus motion. Journal of Solar Energy Engineering, 1999; 121: 69-76. 42. Lindenburg C. Stall coefficients. Aerodynamic airfoil coefficients at large angles of attack. IEA symposium on the aerodynamics of wind turbines. (NREL, CO, USA: December 4-5, 2000). 43. Corten GP. Flow separation on wind turbine blades. PhD Thesis dissertation 2001. Utrecht University, The Netherlands. 44. Fujisawa N, Shibuya S. Observations of dynamic stall on Darrieus wind turbine blades. Journal of Wind Engineering and Industrial Aerodynamics 2001; 89, n. 2: 201–214. 45. Mikkelsen R, Sørensen JN. Modelling of wind tunnel blockage. Proc. CD-ROM Global Windpower Conference and Exhibition (2002). 46. Mertens S, van Kuik G, van Bussel G. Performance of a H-Darrieus in the skewed flow on a roof. Journal of Solar Energy Engineering 2003; 125: 433–440. 47. Timmer WA, van Rooij RPJOM. Summary of the Delft University wind turbine dedicated airfoils. AIAA-2003-0352, 2003. 48. Grignoux T, Gibert R et Al. Eoliennes en milieu urbain – Etat de l’art. ARENE, Ile-de-France, 2004. 49. Van Bussel GJW, Mertens S et Al. TURBY®: concept and realisation of a small VAWT for the built environment. The Science of making Torque from Wind (Delft : 19-21 April 2004). 50. Hansen MH, Gaunaa M, Madsen HA. A Beddoes-Leishman type dynamic stall model in state-space and indicial formulations. Risø National Laboratory. Risø-R-1354, 2004. 51. Montgomerie B. Methods for root effects, tip effects and extending the angle of attack range to ±180°, with application to aerodynamics for blades on wind turbines and propellers. FOI Swedish Defence Research Agency. FOI-R—1305—SE, 2004. 52. Mertens S. Wind energy in the build environment. PhD Thesis dissertation 2006. Delft University of Technology, The Netherlands. 53. Van Der Tempel J. Design of support structures for offshore wind turbines. PhD Thesis dissertation 2006. Delft University of Technology, The Netherlands.

References - VAWT

54. Ferreira CS, van Kuik G, van Bussel G. Wind tunnel hotwire measurements, flow visualization and thrust measurement of a VAWT in skew. AIAA/ASME Wind Energy Symposium (2006). 55. Sørensen JN, Shen WZ and Mikkelsen R. Wall correction model for wind tunnels with open test section, AIAA Journal 2006; 44, n.8. 56. Claessens MC. The design and testing of airfoils for application in small vertical axis wind turbines. MSc Thesis dissertation 2006. Delft University of Technology, The Netherlands. 57. Sant T. Improving BEM based aerodynamic models in wind turbine design codes. PhD Thesis dissertation 2007, Delft University of Tecnology. 58. Ferreira CS, van Bussel GJW et Al. 2D PIV visualization of dynamic stall on a vertical axis wind turbine. AIAA/ASME Wind Energy Symposium, (2007). 59. Fitzgerald RE. Wind tunnel blockage corrections for propellers. MS Thesis 2007. University of Maryland, Department of Aerospace Engineering, College Park MD. 60. Islam M, Ting D, Fartaj A. Aerodynamic models for Darrieus-type straight-bladed vertical axis wind turbines. Renewable & sustainable energy reviews 2008. 12: 1087-1109. 61. Dixon C, Ferreira CS et Al. A 3D unsteady panel method for vertical axis wind turbines. EWEC Brussels (31 March – 3 April 2008). 62. Hofemann C, Ferreira CS, et Al. 3D Stereo PIV study of tip vortex evolution on a VAWT. EWEC, Brussels, (2008). 63. Vita L, Paulsen US et Al. “A novel floating offshore wind turbine concept” EWEC (Marseille : 16 - 19 March 2009). 64. Battisti L, Brighenti A, Zanne L. Analisi dell’effetto della scelta dell’architettura palare sulle prestazioni di turbine eoliche ad asse verticale. 64° Congresso Nazionale ATI. L'Aquila. (6 - 11 September 2009). 65. Shen WZ, Zhang JH, Sørensen JN. The actuator surface model: a new Navier–Stokes based model for rotor computations. Journal of Solar Energy Engineering 2009; 131. 66. Battisti L, Zanne L et Al. Aerodynamic measurements on a vertical axis wind turbine in a large scale wind tunnel. Proc. of ASME Turbo Expo 2010. Glasgow, UK (14-18 June 2010). 67. Ferreira CS. The near wake of the VAWT. PhD Thesis dissertation 2009, Delft University of Tecnology. 68. The Eurocode 1, Part 2-4: Wind actions (ENV 1991-2-4: 1994). 69. http://www.sandia.gov/wind/topical.htm#VAWTARCHIVE 70. http://www.tozzinord.it/