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# Fluid Dynamics and Wind Energy

By khozoliozo Jan 14, 2013 5982 Words
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

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

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
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

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

λ=2

8 V2 Weul = ⋅ 0 9 2

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

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

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
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

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]

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
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
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

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

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

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

References - HAWT
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References - HAWT
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