Vertical-axis wind turbine (VAWT) has received significant attention due to its application in urban environment. Torque produced by VAWT determines its efficiency and power output. In this paper, we develop a reduced-order model of torque VAWT at different tip speed ratios (TSR). We numerically simulate both 2D and 3D flows past a three-bladed Darrieus H-type VAWT and compute overall torque acting on the turbine. We then perform higher-order spectral analysis to identify dominant frequencies and nonlinear couplings. We propose a reduced-order model of torque in the form of modified van der Pol equation with additional quadratic term to allow for even harmonics in addition to odd harmonics present in the system. Using, a perturbation approach of method of multiple scales, we solve the proposed model and compute the coefficients at different TSR. The model not only predicts torque accurately in time domain but also in spectral domain. These reduced-order models provide an accurate and computationally efficient means to predict overall performance and output of the turbine with varying free-stream conditions even in predictive setting.

References

References
1.
Ponta
,
F. L.
,
Seminara
,
J. J.
, and
Otero
,
A. D.
,
2007
, “
On the Aerodynamics of Variable-Geometry Oval-Trajectory Darrieus Wind Turbines
,”
Renewable Energy
,
32
(
1
), pp.
35
56
.10.1016/j.renene.2005.12.007
2.
Sutherland
,
H. J.
,
Berg
,
D. E.
, and
Ashwill
,
T. D.
,
2012
, “
A Retrospective of VAWT Technology
,” Sandia Report No. SAND2012-0304.
3.
Bhutta
,
M. M. A.
,
Hayat
,
N.
,
Farooq
,
A. U.
,
Ali
,
Z.
,
Jamil
,
S. R.
, and
Hussain
,
Z.
,
2012
, “
Vertical Axis Wind Turbine–A Review of Various Configurations and Design Techniques
,”
Renewable Sustainable Energy Rev.
,
16
(
4
), pp.
1926
1939
.10.1016/j.rser.2011.12.004
4.
Debnath
,
B. K.
,
Biswas
,
A.
, and
Gupta
,
R.
,
2009
, “
Computational Fluid Dynamics Analysis of a Combined Three-Bucket Savonius and Three-Bladed Darrieus Rotor at Various Overlap Conditions
,”
J. Renewable Sustainable Energy
,
1
(
3
), p.
033110
.10.1063/1.3152431
5.
Howell
,
R.
,
Qin
,
N.
,
Edwards
,
J.
, and
Durrani
,
N.
,
2010
, “
Wind Tunnel and Numerical Study of a Small Vertical Axis Wind Turbine
,”
Renewable Energy
,
35
(
2
), pp.
412
422
.10.1016/j.renene.2009.07.025
6.
Mohamed
,
M.
,
Janiga
,
G.
,
Pap
,
E.
, and
Thévenin
,
D.
,
2011
, “
Optimal Blade Shape of a Modified Savonius Turbine Using an Obstacle Shielding the Returning Blade
,”
Energy Convers. Manage.
,
52
(
1
), pp.
236
242
.10.1016/j.enconman.2010.06.070
7.
Siddiqui
,
M. S.
,
Durrani
,
N.
, and
Akhtar
,
I.
,
2013
, “
Numerical Study to Quantify the Effects of Struts and Central Hub on the Performance of a Three Dimensional Vertical Axis Wind Turbine Using Sliding Mesh
,”
ASME
Paper No. POWER2013-98300.10.1115/POWER2013-98300
8.
Sirovich
,
L.
,
1987
, “
Turbulence and the Dynamics of Coherent Structures. I-Coherent Structures. II-Symmetries and Transformations. III-Dynamics and Scaling
,”
Q. Appl. Math.
,
45
(3), pp.
561
571
, pp. 573–590.
9.
Deane
,
A. E.
, and
Mavriplis
,
C.
,
1994
, “
Low-Dimensional Description of the Dynamics in Separated Flow Past Thick Airfoils
,”
AIAA J.
,
32
(
6
), pp.
1222
1227
.10.2514/3.12123
10.
Akhtar
,
I.
,
Nayfeh
,
A. H.
, and
Ribbens
,
C. J.
,
2009
, “
On the Stability and Extension of Reduced-Order Galerkin Models in Incompressible Flows
,”
Theor. Comput. Fluid Dyn.
,
23
(
3
), pp.
213
237
.10.1007/s00162-009-0112-y
11.
Graham
,
W. R.
,
Peraire
,
J.
, and
Tang
,
K. Y.
,
1999
, “
Optimal Control of Vortex Shedding Using Low-Order Models. Part I–Open-Loop Model Development
,”
Int. J. Numer. Methods Eng.
,
44
(
7
), pp.
945
972
.10.1002/(SICI)1097-0207(19990310)44:7<945::AID-NME537>3.0.CO;2-F
12.
Akhtar
,
I.
, and
Nayfeh
,
A. H.
,
2010
, “
Model Based Control of Laminar Wake Using Fluidic Actuation
,”
ASME J. Comput. Nonlinear Dyn.
,
5
(
4
), p.
041015
.10.1115/1.4002085
13.
Hay
,
A.
,
Borggaard
,
J.
,
Akhtar
,
I.
, and
Pelletier
,
D.
,
2010
, “
Reduced-Order Models for Parameter Dependent Geometries Based on Shape Sensitivity Analysis
,”
J. Comput. Phys.
,
229
(
4
), pp.
1327
1352
.10.1016/j.jcp.2009.10.033
14.
Akhtar
,
I.
,
Borggaard
,
J.
, and
Hay
,
A.
,
2010
, “
Shape Sensitivity Analysis in Flow Models Using a Finite-Difference Approach
,”
Math. Prob. Eng.
,
2010
, pp.
1
22
.10.1155/2010/209780
15.
Yue
,
Y.
, and
Meerbergen
,
K.
,
2012
, “
Using Krylov-Padé Model Order Reduction for Accelerating Design Optimization of Structures and Vibrations in the Frequency Domain
,”
Int. J. Numer. Methods Eng.
,
90
(
10
), pp.
1207
1232
.10.1002/nme.3357
16.
Wang
,
Z.
,
Akhtar
,
I.
,
Borggaard
,
J.
, and
Iliescu
,
T.
,
2011
, “
Two-Level Discretizations of Nonlinear Closure Models for Proper Orthogonal Decomposition
,”
J. Comput. Phys.
,
230
(
1
), pp.
126
146
.10.1016/j.jcp.2010.09.015
17.
Wang
,
Z.
,
Akhtar
,
I.
,
Borggaard
,
J.
, and
Iliescu
,
T.
,
2012
, “
Proper Orthogonal Decomposition Closure Models for Turbulent Flows: A Numerical Comparison
,”
Comput. Meth. Appl. Mech. Eng.
,
237–240
, pp.
10
26
.10.1016/j.cma.2012.04.015
18.
Hay
,
A.
,
Akhtar
,
I.
, and
Borggaard
,
J.
,
2012
, “
On the Use of Sensitivity Analysis in Model Reduction to Predict Flows for Varying Inflow Conditions
,”
Int. J. Numer. Methods Fluids
,
68
(
1
), pp.
122
134
.10.1002/fld.2512
19.
Ghommem
,
M.
,
Akhtar
,
I.
, and
Hajj
,
M. R.
,
2013
, “
A Low–Dimensional Tool for Predicting Force Decomposition Coefficients for Varying Inflow Conditions
,”
Int. J. Prog. Comput. Fluid Dyn.
,
13
(
6
), pp.
368
381
.10.1504/PCFD.2013.057101
20.
Nayfeh
,
A. H.
,
Owis
,
F.
, and
Hajj
,
M. R.
,
2003
, “
A Model for the Coupled Lift and Drag on a Circular Cylinder
,”
ASME
Paper No. DETC2003/VIB-48455.10.1115/DETC2003/VIB-48455
21.
Nayfeh
,
A. H.
, and
Mook
,
D. T.
,
2008
,
Nonlinear Oscillations
,
Wiley
-VCH. Printed in Birkach, The Federal Republic of Germany.
22.
Nayfeh
,
A. H.
,
2011
,
Introduction to Perturbation Techniques
,
Wiley
-VCH. Printed in Birkach, The Federal Republic of Germany.
23.
Nayfeh
,
A. H.
,
Marzouk
,
O. A.
,
Arafat
,
H. N.
, and
Akhtar
,
I.
,
2005
, “
Modeling the Transient and Steady-State Flow Over a Stationary Cylinder
,”
ASME
Paper No. DETC2005-85376.10.1115/DETC2005-85376
24.
Qin
,
L.
,
2004
, “
Development of Reduced-Order Models for Lift and Drag on Oscillating Cylinders With Higher-Order Spectral Moments
,” Ph.D. thesis Polytechnic Institute and State University, Blacksburg, VA.
25.
Marzouk
,
O.
,
Nayfeh
,
A. H.
,
Akhtar
,
I.
, and
Arafat
,
H. N.
,
2007
, “
Modeling Steady-State and Transient Forces on a Cylinder
,”
J. Vib. Control
,
13
(
7
), pp.
1065
1091
.10.1177/1077546307078737
26.
Akhtar
,
I.
,
Marzouk
,
O. A.
, and
Nayfeh
,
A. H.
,
2009
, “
A van der Pol–Duffing Oscillator Model of Hydrodynamic Forces on Canonical Structures
,”
ASME J. Comput. Nonlinear Dyn.
,
4
(
4
), p.
041006
.10.1115/1.3192127
27.
Janajreh
,
I.
, and
Hajj
,
M.
,
2008
, “
An Analytical Model for the Lift on a Rotationally Oscillating Cylinder
,”
BBAAVI International Colloquium on Bluff Bodies Aerodynamics & Applications
, Milan, Italy, July 20–24, pp.
20
24
.
28.
Blackwell
,
B.
,
1974
, “
Vertical-Axis Wind Turbine: How it Works
,” Sandia Laboratories, Technical Report No. SLA 74 0160, Albuquerque, NM.
29.
Marzouk
,
O. A.
,
2010
, “
A Nonlinear ODE System for the Unsteady Hydrodynamic Force–A New Approach
,”
Int. J. Eng. Math. Sci.
,
6
(
2
), pp.
111
125
.
30.
Claessens
,
M. C.
,
2006
, “
The Design and Testing of Airfoils for Application in Small Vertical Axis Wind Turbines
,” Master of Science Thesis.
31.
Durrani
,
N.
,
Hameed
,
H.
,
Rahman
,
H.
, and
Chaudhry
,
S. R.
,
2011
, “
A Detailed Aerodynamic Design and Analysis of a 2D Vertical Axis Wind Turbine Using Sliding Mesh in CFD
,”
49th AIAA Aerospaces Sciences Meeting and Exhibit
,
Orlando, FL
, Jan. 4–7.
32.
Fluent
,
A.
,
2009
, “
ansys Fluent 12.0 User's Guide
,” Ansys Inc.
33.
Hamada
,
K.
,
Smith
,
T.
,
Durrani
,
N.
,
Qin
,
N.
, and
Howell
,
R.
,
2008
, “
Unsteady Flow Simulation and Dynamic Stall Around Vertical Axis Wind Turbine Blades
,”
46th AIAA Aerospaces Sciences Meeting and Exhibit
, Reno, NV, Jan. 7–10.
34.
Siddiqui
,
M. S.
,
Durrani
,
N.
, and
Akhtar
,
I.
, “
Quantification of the Effects of Geometric Approximations on the Performance of a Vertical Axis Wind Turbine
,”
Renewable Energy
(accepted).
35.
Hajj
,
M. R.
,
Miksad
,
R. W.
, and
Powers
,
E. J.
,
1997
, “
Perspective: Measurements and Analyses of Nonlinear Wave Interactions With Higher-Order Spectral Moments
,”
ASME J. Fluids Eng.
,
119
(
1
), pp.
3
13
.10.1115/1.2819116
36.
Chabalko
,
C. C.
,
2007
, “
Identification of Transient Nonlinear Aeroelastic Phenomena
,” Ph.D. thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA.
You do not currently have access to this content.