0
Research Papers

Nonlinear Reduced-Order Models for Aerodynamic Lift of Oscillating Airfoils2

[+] Author and Article Information
Muhammad Saif Ullah Khalid

Department of Mechanical Engineering,
NUST College of Electrical and
Mechanical Engineering,
National University of Sciences and Technology,
Islamabad 44000, Pakistan
e-mail: m.saifullahkhalid@ceme.nust.edu.pk

Imran Akhtar

Department of Mechanical Engineering,
NUST College of Electrical and
Mechanical Engineering,
National University of Sciences and Technology,
Islamabad 44000, Pakistan
e-mail: akhtar@vt.edu

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received December 5, 2016; final manuscript received February 27, 2017; published online May 17, 2017. Assoc. Editor: Sotirios Natsiavas.

J. Comput. Nonlinear Dynam 12(5), 051019 (May 17, 2017) (8 pages) Paper No: CND-16-1603; doi: 10.1115/1.4036346 History: Received December 05, 2016; Revised February 27, 2017

For the present study, setting Strouhal number as the control parameter, we perform numerical simulations for the flow over oscillating NACA-0012 airfoil at a Reynolds number of 1000. This study reveals that aerodynamic forces produced by oscillating airfoils are independent of the initial kinematic conditions suggesting the existence of limit cycle. Frequencies present in the oscillating lift force are composed of the fundamental harmonics and its odd harmonics. Using these numerical simulations, we analyze the shedding frequencies close to the excitation frequencies. Hence, considering it as a primary resonance case, we model the unsteady lift force with a modified van der Pol oscillator. Using the method of multiple scales and spectral analysis of the steady-state computational fluid dynamics (CFD) solutions, we estimate the frequencies and the damping terms in the reduced-order model (ROM). We show the applicability of this model to all planar motions of the airfoil; heaving, pitching, and flapping. With increasing the Strouhal number, the nonlinear damping terms for all types of motion approach similar magnitudes. Another important aspect in one of the proposed model is its ability to capture the time-averaged value of the aerodynamic lift force. We also notice that increase in the magnitude of the lift force is due to the effect of destabilizing linear damping parameter.

FIGURES IN THIS ARTICLE
<>
Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.

References

Knoller, R. , 1909, “ Gesetze des Luftwiderstands,” Flug Motortechnik (Wien), 3(21), pp. 1–7.
Betz, T. W. , 1912, “ Ein Beitrag zur Erklarung des Segelfluges,” Z. Flugtech. Motorluftschiffahrt, 3(21), pp. 269–272.
Ho, S. , Nassef, H. , Pornsinsirirak, N. , Tai, Y. C. , and Ho, C. M. , 2003, “ Unsteady Aerodynamics and Flow Control for Flapping Wing Flyers,” Prog. Aerosp. Sci., 39(8), pp. 635–681. [CrossRef]
Mittal, R. , Akhtar, I. , Bozkurttas, M. , and Najjar, F. M. , 2003, “ Towards a Conceptual Model of a Bio-Robotic AUV: Pectoral Fin Hydrodynamics,” 13th International Symposium on Unmanned Untethered Submersible Technology (UUST), Durham, NH, Aug. 24–27, p. 61801.
Triantafyllou, M. S. , Techet, A. H. , and Hover, F. S. , 2004, “ Review of Experimental Work in Biomimetic Foils,” IEEE J. Oceanic Eng., 29(3), pp. 31–38. [CrossRef]
Wang, Z. J. , 2005, “ Dissecting Insect Flight,” Annu. Rev. Fluid Mech., 37(1), pp. 183–210. [CrossRef]
Akhtar, I. , and Mittal, R. , 2005, “ A Biologically Inspired Computational Study of Flow Past Tandem Flapping Foils,” AIAA Paper No. 2005-4760.
Lehmann, F. O. , 2004, “ Aerial Locomotion in Flies and Robots: Kinematic Control and Aerodynamics of Oscillating Wings,” Arthropod Struct. Dev., 33(3), pp. 331–345. [CrossRef] [PubMed]
Shyy, W. , Aono, H. , Chimakurthi, S. K. , Trizila, P. , Kang, C. K. , Cesnik, C. E. S. , and Liu, H. , 2010, “ Recent Progress in Flapping Wing Aerodynamics and Aeroelasticity,” Prog. Aerosp. Sci., 46(7), pp. 284–327. [CrossRef]
Khalid, M. S. U. , Akhtar, I. , and Dong, H. , 2016, “ Hydrodynamics of a Tandem Fish School With Asynchronous Undulation of Individuals,” J. Fluids Struct., 66, pp. 19–35. [CrossRef]
Sirovich, L. , 1987, “ Turbulence and the Dynamics of Coherent Structures,” Q. Appl. Math., 45(3), pp. 561–571. [CrossRef]
Akhtar, I. , 2008, “ Parallel Simulation, Reduced-Order Modeling and Feedback Control of Vortex-Shedding Using Fluidic Actuators,” Ph.D. thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA.
Skop, R. , and Griffin, O. , 1973, “ A Model for the Vortex-Excited Resonant Response of Bluff Cylinders,” J. Sound Vib., 27(2), pp. 225–233. [CrossRef]
Skop, R. , and Griffin, O. , 1975, “ On a Theory for the Vortex-Excited Oscillations of Flexible Cylindrical Structures,” J. Sound Vib., 41(3), pp. 263–274. [CrossRef]
Nayfeh, A. H. , Owis, F. , and Hajj, M. R. , 2003, “ Model for the Coupled Lift and Drag on a Circular Cylinder,” ASME Paper No. DETC2003/VIB-48455.
Nayfeh, A. H. , Marzouk, O. A. , Arafat, N. H. , and Akhtar, I. , 2005, “ Modeling the Transient and Steady-State Flow Over a Stationary Cylinder,” ASME Paper No. DETC2005-85376.
Nayfeh, A. H. , 1993, Introduction to Perturbation Techniques, Wiley Classic Library Edition, Wiley, New York.
Fung, J. , 1998, “ Parameter Identification of Nonlinear Systems Using Perturbation Methods and Higher-Order Statistics,” M.S. thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA.
Qin, L. , 2003, “ Development of Reduced-Order Models for Lift and Drag on Oscillating Cylinders With Higher-Order Spectral Moments,” Ph.D. thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA.
Janajreh, I. , and Hajj, M. R. , 2008, “ An Analytical Model for the Lift on a Rotationally Oscillating Cylinder,” BBAA-VI International Colloquium on Bluff Bodies Aerodynamics and Applications, Milan, Italy, July 20–24.
Marzouk, O. A. , 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. [CrossRef]
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.
von Ellenrieder, K. D. , 2006, “ Dynamical Systems Analysis of Flapping Wing Propulsion,” Australian Fluid Mechanics Workshop, Melbourne University, Australia.
Skop, R. A. , and Balasubramanian, S. , 1997, “ A New Twist on an Old Model for Vortex-Induced Vibrations,” J. Fluids Struct., 11(4), pp. 395–412. [CrossRef]
ANSYS, 2014, “ ANSYS Fluent Userguide,” ANSYS, Inc., Canonsburg, PA.
Khalid, M. S. U. , Akhtar, I. , and Durrani, N. I. , 2015, “ Analysis of Strouhal Number Based Equivalence of Pitching and Plunging Airfoils and Wake Deflection,” Proc. Inst. Mech. Eng., Part G, 229(8), pp. 1423–1434.
Moon, F. C. , 1998, Applied Dynamics; With Applications to Multibody and Mechatronic Systems, Wiley, Hoboken, NJ.
von Ellenrieder, K. D. , Parker, K. , and Soria, J. , 2008, “ Fluid Mechanics of Flapping Wings,” Exp. Therm. Fluid Sci., 32(8), pp. 1578–1589. [CrossRef]
Young, J. , and Lai, J. C. S. , 2007, “ Vortex Lock-In Phenomenon in the Wake of a Plunging Airfoil,” AIAA J., 45(2), pp. 485–490. [CrossRef]
Yu, M. L. , Hu, H. , and Wang, Z. J. , 2012, “ Experimental and Numerical Investigations on the Asymmetric Wake Vortex Structures Around an Oscillating Airfoil,” AIAA Paper No. 2012-0299.
Nayfeh, A. H. , and Mook, D. T. , 1995, Nonlinear Oscillations, Wiley Classic Library Edition, Wiley-VCH, Weinheim, Germany.
Asharf, M. A. , Young, J. , and Lai, J. C. S. , 2011, “ Reynolds Number, Thickness and Camber Effects on Flapping Airfoil Propulsion,” J. Fluids Struct., 27(2), pp. 145–160. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Schematic of flow domain around the airfoil

Grahic Jump Location
Fig. 2

Airfoil kinematics (solid lines: upstroke, dotted lines: downstroke), for clarity, only the mean and maximum-amplitude positions are shown here

Grahic Jump Location
Fig. 3

Comparison of CL and CT for different initial positions of heaving airfoil for St = 0.10 in (a) and (c); and for St = 0.30 in (b) and (d)

Grahic Jump Location
Fig. 4

Phase maps using CL and CT their time-derivatives for different initial positions and strokes of heaving airfoil at St = 0.10 in (a) and (c); and for St = 0.30 in (b) and (d)

Grahic Jump Location
Fig. 5

(a) Unsteady lift and (b) amplitude-spectra for St = 0.30

Grahic Jump Location
Fig. 6

Comparison of CFD solution and the proposed model for heaving airfoil at St = 0.45: (a) time histories (solid lines: CFD and circles: ROM) and (b) amplitude-spectrum (solid lines: CFD and dashed lines: ROM)

Grahic Jump Location
Fig. 7

Comparison of CFD solution and the proposed model for pitching airfoil at St = 0.30 (a) time histories (solid lines: CFD and circles: ROM) (b) amplitude-spectrum (solid lines: CFD and dashed lines: ROM)

Grahic Jump Location
Fig. 8

Comparison of CFD solution and the proposed model for flapping airfoil at St = 0.20 (a) time histories (solid lines: CFD and circles: ROM) (b) amplitude-spectrum (solid Lines: CFD and dashed lines: ROM)

Grahic Jump Location
Fig. 9

Comparison of solutions from CFD and ROM for heaving airfoil (a) time histories (solid lines: CFD and circles: ROM) St = 0.20 (b) amplitude-spectrum (solid lines: CFD and dashed lines: ROM) St = 0.40

Grahic Jump Location
Fig. 10

Plots of dynamic parameters for varying St

Grahic Jump Location
Fig. 11

Comparison of CFD solution and the proposed model for pitching airfoil at St = 0.20 (a) time histories (triangles: CFD and solid line: ROM) (b) amplitude-spectrum (dashed line: CFD and solid lines: ROM)

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In