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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

Rosenberg & Sharma NAWEA 2015 4

•  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


*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


•  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)


��

��

��

��

��

��

��

�

��

��

��

��

��

��

CTCP

� �

Extension to Dual-‐Rotor Wind Turbines


•  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


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


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


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


•  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


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


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


QUESTIONS?


ACKNOWLEDGEMENT •  NSF CBET: #1438099 •  IEC: #14-‐008-‐OG •  ISGC: #475-‐20-‐5 •  NSF XSEDE: #TG-‐CTS130004

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