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Zanne2010.pdf
```UNIVERSITÁ DEGLI STUDI DI UDINE
Dottorato in Tecnologie Chimiche ed Energetiche
FLUID DYNAMIC MODELLING OF
WIND TURBINES
sec. D
Vz
Vr
Vr
Vt
0
Vt
D
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
represented with equivalent
forces distribuited over a
permeable, immaterial disk
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
ρ
=
Mass conservation
Vz2,0 − Vz2,3
Vz ,m Am = Vz ,3 A3
2
Vz ,1 ≅ Vz ,2 ≅ Vz ,m
Froude result!
Vz ,m =
Vz ,0 + Vz ,3
2
Actuator disk
Drzewiecki first applied Froude result
dividing the rotor in different annular
streamtubes :
Vz ,m =
FN
Lift
φflow
FT
z
θpitch+βtwist
φflow
V0
αattack
Vrel.
chord line
Drag
-ωr
y
2
With the blade element airfoil theory
rotor performances can be easily
calculated
Raero
Wind.=[ -a·V0; -a’·ωr ]
Vz ,0 + Vz ,3
rotor plane
The annuli interaction is neglected
No swirl flow, (wake expansion?)
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
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


• GM theory is an
integral formulation
• It needs the wake
solution
p3 − p0
ρ
=
(
p3 ( r3 ) − p3 rtip ,3
ρ
Solutions:
• De Vries
• Differential
8 V2
Weul = ⋅ 0
9 2
)=−
2
rtip ,3 Vθ ,3
∫r3
r3
dr3
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
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 =
Axial velocity
d Γ = 2π ⋅ d ( rVθ ,2 )
gθ ,m =
gθ ,3 =
d Γ r Ω + Vθ ,2 2
Vz , m
2π r
d Γ r3Ω + Vθ ,3
2π r3 Vz ,3
vz ,m = gθ 2
vz ,3 = gθ
Vr ( r ,0 ) = −
1 r ∂Vz
r
( r,0 ) dr
r ∫0 ∂z
∂Vz ∂z ( r,0 ) =
gθ
gθ
+
r
2r 2π ( r − r )
∂Vz ∂z ( r,0 ) =
 1 1  gθ ( r − r )
−
 −

2π ( r − r ) r 2  4 2π 
r 5
gθ
2
Vz ,0 + Vz ,3
2
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
1 ∂p 0
 ∂V ∂V
= Fz + Vr  r − z
ρ ∂z
∂r
 ∂z
∂rVr ∂rVz
+
=0
∂r
∂z
∂Vθ

 + Vθ
∂z

A turbomachinery approach
Stoke’s stream function
ωθ =
∂ 2ψ
∂r 2
∂Vr ∂Vz
−
∂z
∂r
−
1 ∂ψ ∂ 2ψ
+
= −rωθ
r ∂r ∂z 2
ωθ = Vθ
∂ 2ψ
∂r 2
∂ 2ψ
∂r 2
−
−
d ( rVθ )
dψ
−
r dp 0
ρ dψ
1 ∂ψ ∂ 2ψ
+
=0
r ∂r ∂z 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
)
− p10 = ΩrVθ = Weul
Wu equation
(
0
0

1 ∂ψ ∂ 2ψ  p2 − p1
−
+
= −
∂r 2 r ∂r ∂z 2 
Ω2

∂ 2ψ
The angular momentum
distribution can be assigned
Vθ = k1r n +
k2
r
rVθ = k1r n +1 + k 2
)
ρ

+r
d ( rVθ )
1 dp20
= Ωr 2 − rVθ
 ρ dψ
dψ

2
Free vortex distribution
rVθ = const
(
)
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 − Vz2,hub =
2
(p
ρ
0
)
− pr0,hub − 2 ∫
r
r , hub
Wu hypothesis
∂Vr ,1
∂z
=−
∂Vr ,2
∂z
dVz ,m
1 dp20 Vθ ,2 d ( rVθ ,2 )
=
+ 2Vz ,m
ρ dr
r
dr
dr
Vθ ∂rVθ
r
 ∂V 
dr + 2 ∫
Vz  r  dr − Vr2 − Vr2,hub
r
,
hub
r ∂r
 ∂z ψ
(
)
Wu hypothesis on a streamline
 ∂Vr ,1 
 ∂Vr ,2 

 = −

 ∂z ψ
 ∂z ψ
dVz ,m
dVr ,m
1 dp20 Vθ ,2 d ( rVθ ,2 )
=
+ 2Vz ,m
+ 2Vr ,m
ρ dr
r
dr
dr
dr
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
8 V2
Weul = ⋅ 0
9 2
Conway actuator disk – vortex
theory exact solution for a (almost)
(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
on a streamline
dVs ,m
1 dp20 Vθ ,2 d ( rVθ ,2 )
=
+ 2Vs ,m
ρ dr
r
dr
dr
Vs2,m = Vz2,m + Vr2,m
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
(
∂Vs2,m
∂r
=
)
Coning / yaw effects
Turbulence wake state / stall
)
 1
∂Vs ,m
1 ∂p20
1  1 ∂ 2 2
+ 2Vr ,m
− Vs2,m cos ε 
+
r Vθ
 − 2

∂s
ρ ∂r
 rs ,1 rs ,2  2r ∂q
(
)
Tip effects
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
• 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.
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
Fθ , Z = ρ ⋅ Vz ,m ⋅ s ⋅ (Vθ ,2 − Vθ ,1 ) = ρ ⋅ V 2 z ,m ⋅ s ⋅ ( tan α 2 − tan α1 )
Fz , Z = ( p10 − p20 ) ⋅ 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
 U + Vθ ,1 
 Vz ,1 


β1 = tan −1 
 U + Vθ ,m 
 Vz ,m 


β m = tan −1 
 U + Vθ ,2 
 Vz ,2 


β 2 = tan −1 
Vz ,m Vz ,1 Vz ,2
Cy =
Fy
Fy ,max
= 2⋅
C y = 0.8
Dloc =
Wmax − W2
Wmax
s
( tan β2 − tan β1 ) cos2 β2
cz
Zweifel
Lieblein
cz = c ⋅ cos β m
s
CL = 2 (tan β 2 − tan β1 ) cos β m
c
θ=
π
2
− β m − sen −1 (
CL, ID
2π
)
Inverse Design
Results and discussion
Gaia turbine
Inverse Design
Results and discussion
1
VDz / V0
0.5
0.2
0.4
0.6
0.8
0
1
0
0.2
0.4
40
20
0
0.2
0.4
0.8
1
0.6
0.8
40
20
0
1
0
0.2
0.4
r/R
0.6
0.8
1
0.2
0.1
r/R
1.5
0
0.2
0.4
0.6
0.8
0
0
0.2
0.4
0.6
0.8
0.2
0.4
0.8
60
40
20
0
1
Fn / q0R
0
0.2
0.4
0.6
0.8
0.4
0.4
0.6
0.8
0
0.2
0.4
0.5
0.2
0.4
0.6
r/R
0.6
0.8
1
0.6
0.8
1
0.6
0.8
1
0.2
0
0.2
0.4
r/R
1
0
1
r/R
r/R
0
0.8
0.4
0
1
0.6
r/R
0.6
0.2
0.2
0.2
1
0
1
r/R
0
0
2
0.4
0
80
r/R
0.6
λ=6,Z=3
0.6
1
0
1
r/R
Mt / q0R3
r/R
1
0.5
dCp / d(r/R)
0
Ft / q0R
0.5
0
0
2
1
Psi
p1-p2 / q0
1
Mn / q0R3
0
0.6
r/R
beta1-beta2 [deg]
alpha2 [deg]
r/R
betam [deg]
0
0.5
P / 1/2rhoAV 30
0
Flow characteristics
C/R
W / 1/2V 20
1
0.8
1
0.6
0.4
0.2
0
0
0.2
0.4
r/R
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
Flow characteristics
β = tan −1
Vsenϑ cos δ
( V cosϑ + Ωr ) cos γ
C N = CL cos β + C D sin β
2
W 2 = ( Vcosϑ + Ωr ) cos γ  + ( Vsenϑ cos δ )
Re =
CL =
2
CT = C L sin β − C D cos β
1
dh
dFN = ρ0 W2 c
CN
2
cos δ
cW
ν0
dL
1 ρ W 2 c dh
2 0
CD =
dD
1 ρ W 2 cdh
2 0
dFT =
1
dh
ρ0 W 2 c
CT
2
cos δ
Shaft torque/power
dM = dFT Ω
N BL
1 ϑ
dM Ω
Nϑ ∫ ∫ ∫
MΩ
CP =
=
1 ρ A V3
1 ρ A V3
0
sw
0
2
2 0 sw 0
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
ρ0 V02 dAs
2
dAs = dh r dϑ sinϑ
Momentum theory
α=
V
V0
dFx = 2ρ dA s V(V0 -V)
CTH =
dFx
1
ρ V02 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
CTH =
26
4
(1 − α ) +
15
15
Tip losses
Post stall airfoil performance
correction
Flow curvature
Dynamic stall
Streamtubes expansion
VAWT fluid dynamics
Validation and results
Sandia 5m
Darrieus
NACA0015
Four geometric characteristics
VAWT fluid dynamics
Validation and results
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
H = 1.46m
D = 1.03m
NACA0021
Solidity 0.25
Rotor position
Torque
PDF3
H = 2.5m
D = 1.78m
P = 1.5kW
H(tower) = 3.5m
Rotor position
Torque (electric)
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)
λ = 1.6
In closed wind tunnel there is an
higher velocity due to blockage
effects
VAWT experimental analysis
PDF1 rotor - Aerodynamics
Wind tunnel blockage

1
1
1
1
 

T = AD  p0 + ρV02 − ρVD2  −  p3 + ρV32 − ρVD2  
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
Shed vortex position
Flow characteristics
2
W 2 = ΩR + (U 0 + uC ) cos (θ ) + vC sin (θ )  + (U 0 + uC ) sin (θ ) − vC cos (θ ) 
(U 0 + uC ) sin (θ ) − vC cos (θ )
φ = tan −1 −
ΩR + (U 0 + uC ) cos (θ ) + vC sin (θ )
α =φ − β
2
(
)
xS ,i = xS ,i −1 + U 0 + uS ( xS ,i −1 , yS ,i −1 ) ⋅ dt
y S ,i = yS ,i −1 + vS ( xS ,i −1 , yS ,i −1 ) ⋅ dt
(
)
xS ,i = xS ,i −1 + U 0 + 0.5 ⋅ uS ( x S ,i , y S ,i ) + uS ( xS ,i −1 , yS ,i −1 )  ⋅ dt


(
)
yS ,i = yS ,i −1 + 0.5 ⋅ vS ( xS ,i , y S ,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 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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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.
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