A Prescribed-‐Wake Vortex Line Method for Aerodynamic Analysis and Op=miza=on of Mul=-‐Rotor Wind Turbines A. Rosenberg and A. Sharma Department of Aerospace Engineering Iowa State University NAWEA 2015 Symposium Virginia Tech
Mo=va=on
• Betz limit: – 59.3% of energy capture – Modern HAWTs operate well
below this
• Root loss (≈5%) – Inner 25% of rotor … … designed for structural integrity à Poor root aerodynamics
• Wake loss (≈8−40%) – Not operaWng in isolaWon
NAWEA 2015 Rosenberg & Sharma 2
Motivation
• Betz limit: – 59.3% of energy capture– Modern HAWTs operate well
below this
• Root loss (≈5%)– Inner 25% of rotor … … designed for structural integrity Poor root aerodynamics
• Wake loss (≈8−40%)– Not operating in isolation
NAWEA 2015Rosenberg & Sharma 2
Novel Turbine Concept
• Dual Rotor Wind Turbine (DRWT)* – Add a secondary, co-‐axial rotor – Rotors rotate independently – Co-‐ or counter-‐rotaWng
• Aims: – Reduce root loss – Enhance wake mixing
• Secondary rotor “inversely” designed using BEM
NAWEA 2015 Rosenberg & Sharma 3
Dual rotor wind turbine (DRWT) concept
Lab models (courtesy Dr. Hui Hu, ISU) *A. Rosenberg, S. Selvaraj, A. Sharma, A novel dual-‐rotor turbine for increased wind energy capture, Journal of Physics: Conference Series, vol. 524, 2014.
Vortex Line Method
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• Prescribed-‐ or free-‐wake • Prandtl’s Liding Line Theory
– Blade bound vorWcity – Helmhotlz → trailing vortex helix
• InducWon from Biot-‐Savart law
– Compute local inflow, AoA – A/f polars → secWonal lid & drag à
turbine torque/power
• Goal: Develop computaWonally inexpensive and robust method to analyze DRWTs
dv = ��dl⇥ r
4⇡|r|3
Algorithm • Prescribed wake – No mutual inducWon in trailing vorWces
– Trailing vortex stays intact except: • Helix pitch & Treiz plane locaWon are updated
– Chajot* (2011)
• IteraWve soluWon – Converges when trailing vortex
sheet in equilibrium with blade bound circulaWon
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*J.J. Chajot, Wind turbine aerodynamics: analysis and design, InternaGonal Journal of Aerodynamics, vol. 1, 2011
dv
��i=
X
j
dlj ⇥ rij4⇡|rij|3
Under-‐relax
Algorithm -‐ con=nued
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• Vortex pitch determined by area-‐averaged axial inducWon -‐ 1-‐D momentum theory:
-‐ f, g are mulW-‐valued!
• Buhl’s (2005)*correcWon
makes the funcWon single-‐valued
CT (a) =
8<
:
4aF (1� a) if 0 a 0.4
8
9+
✓4F � 40
9
◆a+
✓50
9� 4F
◆a2 if 0.4 < a 1
*M. Buhl, A new empirical relaWonship between thrust coefficient and inducWon factor for the turbulent windmill state, NREL/TP-‐500-‐36834, 2005
CT = 4a(1� a) :: a = f(CP )
CP = 4a(1� a)2 :: a = g(CT )
VLM – Single Rotor Valida=on
• Tellus T-‐1995 – 95 kW – Stall controlled – 19 m diameter – Chord/twist distribuWon:
• Good agreement with: – Measured data – BEM (blade element method)
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����
����
����
����
���� ���� ���� ��� ��
��
��
�
���
���
��������������
�� �������
�� ������ ���� ��
�� ������
CTCP
� �
Extension to Dual-‐Rotor Wind Turbines
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• MulWple trailing helices – Treat singulariWes in Biot-‐Savart (r > rmin)
• PotenWal flow effects ignored • Both helices convect with
same flow speed – Equiv. CT à inducWon
• Inherently unsteady – Pseudo-‐steady assumpWon – Phase average for final results
(a) c⌧F (b) cT
Figure 7: Comparisons of radial variation of force coe�cients at � = 7.0 against BEMresults for the Tellus T-1995 turbine
2. If the two rotors rotate independently, then the problem becomes in-herently unsteady; approximations need to be made to solve it as asteady problem.
3. If the two rotors are very close, potential flow e↵ects (due to finite bladethickness) also come into play.
To account for mutual induction between the bound and trailing vorticesof the two rotors of a DRWT, we first need to setup the associated vortexlines/lattices. Figure 8 shows the vortex lattice structure of a DRWT. Twosets of helices are now present; one set each for the two rotors. The pitch ofthe two helices is set to be the same since the trailing vorticity is convectedby the same flow speed. Equation 1 is used to determine the area-averageinduction, a, which then determines the pitch of the helices. For DRWTs,an equivalent area-weighted C
T,eq
is used with Eq. 1, where
C
T,eq
= C
T,m
+A
s
A
m
C
T,s
, (2)
and subscripts ‘m’ and ‘s’ refer to the main rotor and the secondary rotorrespectively; ‘A’ is area of rotor disk. Once the vorticity structure is set,computation of induction coe�cients using the Biot-Savart law is straight-forward. It should be emphasized again that induction from bound vorticity
8
RANS/AD Method for DRWT Aero
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RANS/AD: • Actuator disk to model circ. avg. forces • Axisymmetric grids; SIMPLE; k-‐omega RANS model • Previously validated*
Figure 9: Predicted variation in % CP of a DRWT with relative angular position betweenthe primary and the secondary rotors.
Rosenberg et al. [1] suggest that the optimum axial separation for enhancedisolated rotor aerodynamic performance is about 0.2 ⇥ main rotor diameter,which is much greater than the maximum blade chord of the main rotor.Thus, the approximation to neglect potential e↵ects due to blade thicknessshould be valid for such turbines.
3.1. Verification with CFD
The proposed extended vortex line method to analyze dual-rotor windturbines is verified against results obtained using the Reynolds AveragedNavier-Stokes + actuator disk (RANS/AD) method described in Rosenberget al. [1] and Selvaraj [12]. Subtle aspects of this RANS/AD methodologyare summarized here for completeness.
3.1.1. RANS/AD Method
The RANS/AD method [1, 12] solves the incompressible RANS equations(Eq. 3) with the rotor blades modeled as body forces (actuator disk). Thegoverning equations are
@u
i
@x
i
= 0, and,
u
j
@u
i
@x
j
= �1
⇢
@p
@x
i
+ ⌫
@
2u
i
@x
j
2�
@u
0i
u
0j
@x
j
+f
i
⇢
. (3)
10
*A. Rosenberg, S. Selvaraj, A. Sharma, A novel dual-‐rotor turbine for increased wind energy capture, Journal of Physics: Conference Series, vol. 524, 2014.
DRWT: Code-‐to-‐Code Comparisons
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Verify VLM against RANS/AD predicWons: 2 DRWT designs: rWp sec = 0.25 and 0.4
Good overall agreement Discrepancy at r = rWp sec due to circ avg. in AD method
-0.50
-0.25
0.00
0.25
0.50
0.75
1.00
0.00 0.25 0.50 0.75 1.00To
rque
forc
e co
e�.
radius / tip radius of main rotor
CFDVLM
-0.50
-0.25
0.00
0.25
0.50
0.75
1.00
0.00 0.25 0.50 0.75 1.00
Torq
ue fo
rce
coe�
.
radius / tip radius of main rotor
CFDVLM
Poor Results at High Loading
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OpWmizaWon: rudimentary approach – parametric sweeps • Vary secondary rotor radius & TSR • Main rotor geometry & TSR held fixed
RANS/AD
Stability & accuracy issues with VLM @ high loading
VLM
Enhancement to VLM Model
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• Previous approaches assumed: • axial inducWon, a(x) increases linearly from a0 to 2a0 • a(x) à wake helix pitch
• Calibrate wake pitch using RANS results Old model – linear approx.
New Wake Model
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a(x) = a0
✓✓a
xr3
a0� 1
◆tanh(B ⇥ x/x
r3)
tanh(B)+ 1
◆axr3 = 0.47 C
T
+ 0.03
with B=3
solid lines à RANS dashed à new model
Improved Results @ High Loading
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RANS/AD VLM
Repeat parametric sweep
• Stability issues resolved with the proposed wake correcWon • QualitaWve agreement à useful for opWmizaWon • Parameter tuning to improve results ongoing
Conclusion
• Prescribed-‐wake VLM method proposed for DRWTs – Useful for prelim design & opWmizaWon
• Valida=on: experiments & BEMT for SRWTs
• Verifica=on: Code-‐to-‐code agreement -‐ VLM & RANS/AD for DRWTs
• RANS-‐inspired wake model alleviates stability issues at high loading for DRWTs
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QUESTIONS?
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ACKNOWLEDGEMENT • NSF CBET: #1438099 • IEC: #14-‐008-‐OG • ISGC: #475-‐20-‐5 • NSF XSEDE: #TG-‐CTS130004