Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-03T00:38:54.790Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear flow-induced instability of an elastically mounted pitching wing

Published online by Cambridge University Press:  27 July 2020

Yuanhang Zhu Open the ORCID record for Yuanhang Zhu [Opens in a new window] ,
Yunxing Su Open the ORCID record for Yunxing Su [Opens in a new window]  and
Kenneth Breuer Open the ORCID record for Kenneth Breuer [Opens in a new window]
Yuanhang Zhu*
Affiliation:
Center for Fluid Mechanics, School of Engineering, Brown University, Providence, RI02912, USA
Yunxing Su
Affiliation:
Center for Fluid Mechanics, School of Engineering, Brown University, Providence, RI02912, USA
Kenneth Breuer
Affiliation:
Center for Fluid Mechanics, School of Engineering, Brown University, Providence, RI02912, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We experimentally study the nonlinear flow-induced instability of an elastically mounted pitching wing in a circulating water tunnel. The structural parameters of the finite-span wing are simulated and regulated using a cyber-physical control system. At a small fixed damping, we systematically vary the stiffness of the wing for different inertia values to test for the stability boundaries of the system. We observe that, for a high-inertia wing, the system dynamics bifurcates from stable fixed points to small-amplitude oscillations followed by large-amplitude limit-cycle oscillations (LCOs) via a subcritical bifurcation, which features hysteretic bistability and an abrupt amplitude jump. Under this condition, the pitching frequency of the wing locks onto its structural frequency and the oscillation is dominated by the inertial force, corresponding to a structural mode. Force and flow field measurements indicate the presence of a secondary leading-edge vortex (LEV). As the wing inertia decreases, the width of the bistable region shrinks. At a sufficiently low inertia, the pitching amplitude changes smoothly with the stiffness without any hysteresis, revealing a supercritical bifurcation. Under this condition, no lock-in phenomenon is observed and the pitching frequency remains relatively constant at a value lower than the structural frequency. Force decomposition shows dominating fluid force, indicating a hydrodynamic mode. The secondary LEV is absent. We show that the onset of large-amplitude LCOs in both the structural mode and the hydrodynamic mode scales with the Cauchy number, and the LCOs in the structural mode collapse with the non-dimensional velocity. We examine the subcritical transition in detail; we find that this transition depends on the static characteristics of the wing, and the secondary LEV starts to emerge at the early stage of the transition. Lastly, we adopt an energy approach to map out the stability of the system and explain the existence of the two distinct types of bifurcations observed for different inertia values.

JFM classification

Instability: Nonlinear instability Vortex Flows: Vortex dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 899 , 25 September 2020 , A35
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Amandolese, X., Michelin, S. & Choquel, M. 2013 Low speed flutter and limit cycle oscillations of a two-degree-of-freedom flat plate in a wind tunnel. J. Fluids Struct. 43, 244–255.CrossRefGoogle Scholar
Baik, Y. S., Bernal, L. P., Granlund, K. & Ol, M. V. 2012 Unsteady force generation and vortex dynamics of pitching and plunging aerofoils. J. Fluid Mech. 709, 37–68.CrossRefGoogle Scholar
Barnes, C. J. & Visbal, M. R. 2018 On the role of flow transition in laminar separation flutter. J. Fluids Struct. 77, 213–230.CrossRefGoogle Scholar
Beatus, T. & Cohen, I. 2015 Wing-pitch modulation in maneuvering fruit flies is explained by an interplay between aerodynamics and a torsional spring. Phys. Rev. E 92 (2), 022712.CrossRefGoogle Scholar
Bergou, A. J., Ristroph, L., Guckenheimer, J., Cohen, I. & Wang, Z. J. 2010 Fruit flies modulate passive wing pitching to generate in-flight turns. Phys. Rev. Lett. 104 (14), 148101.CrossRefGoogle ScholarPubMed
Bergou, A. J., Xu, S. & Wang, Z. J. 2007 Passive wing pitch reversal in insect flight. J. Fluid Mech. 591, 321–337.CrossRefGoogle Scholar
Bhat, S. S. & Govardhan, R. N. 2013 Stall flutter of NACA 0012 airfoil at low Reynolds numbers. J. Fluids Struct. 41, 166–174.CrossRefGoogle Scholar
Boudreau, M., Dumas, G., Rahimpour, M. & Oshkai, P. 2018 Experimental investigation of the energy extraction by a fully-passive flapping-foil hydrokinetic turbine prototype. J. Fluids Struct. 82, 446–472.CrossRefGoogle Scholar
Dimitriadis, G. & Li, J. 2009 Bifurcation behavior of airfoil undergoing stall flutter oscillations in low-speed wind tunnel. AIAA J. 47 (11), 2577–2596.CrossRefGoogle Scholar
Dowell, E., Edwards, J. & Strganac, T. 2003 Nonlinear aeroelasticity. J. Aircraft 40 (5), 857–874.CrossRefGoogle Scholar
Dowell, E. H., Curtiss, H. C., Scanlan, R. H. & Sisto, F. 1989 A Modern Course in Aeroelasticity. Springer.CrossRefGoogle Scholar
Dowell, E. H. & Hall, K. C. 2001 Modeling of fluid-structure interaction. Annu. Rev. Fluid Mech. 33 (1), 445–490.CrossRefGoogle Scholar
Duarte, L., Dellinger, N., Dellinger, G., Ghenaim, A. & Terfous, A. 2019 Experimental investigation of the dynamic behaviour of a fully passive flapping foil hydrokinetic turbine. J. Fluids Struct. 88, 1–12.CrossRefGoogle Scholar
Dugundji, J. 2008 Some aeroelastic and nonlinear vibration problems encountered on the journey to Ithaca. AIAA J. 46 (1), 21–35.CrossRefGoogle Scholar
Eldredge, J. D. & Jones, A. R. 2019 Leading-edge vortices: mechanics and modeling. Annu. Rev. Fluid Mech. 51, 75–104.CrossRefGoogle Scholar
Fagley, C., Seidel, J. & McLaughlin, T. 2016 Cyber-physical flexible wing for aeroelastic investigations of stall and classical flutter. J. Fluids Struct. 67, 34–47.CrossRefGoogle Scholar
Garrick, I. E. 1936 Propulsion of a flapping and oscillating airfoil. NACA Tech. Rep. 567.Google Scholar
Govardhan, R. & Williamson, C. H. K. 2000 Modes of vortex formation and frequency response of a freely vibrating cylinder. J. Fluid Mech. 420, 85–130.CrossRefGoogle Scholar
Govardhan, R. & Williamson, C. H. K. 2002 Resonance forever: existence of a critical mass and an infinite regime of resonance in vortex-induced vibration. J. Fluid Mech. 473, 147–166.CrossRefGoogle Scholar
Granlund, K. O., Ol, M. V. & Bernal, L. P. 2013 Unsteady pitching flat plates. J. Fluid Mech. 733, R5.CrossRefGoogle Scholar
Ho, S., Nassef, H., Pornsinsirirak, N., Tai, Y.-C. & Ho, C.-M. 2003 Unsteady aerodynamics and flow control for flapping wing flyers. Prog. Aerosp. Sci. 39 (8), 635–681.CrossRefGoogle Scholar
Hover, F. S., Miller, S. N. & Triantafyllou, M. S. 1997 Vortex-induced vibration of marine cables: experiments using force feedback. J. Fluids Struct. 11 (3), 307–326.CrossRefGoogle Scholar
Ishihara, D., Yamashita, Y., Horie, T., Yoshida, S. & Niho, T. 2009 Passive maintenance of high angle of attack and its lift generation during flapping translation in crane fly wing. J. Expl Biol. 212 (23), 3882–3891.CrossRefGoogle ScholarPubMed
Jafferis, N. T., Helbling, E. F., Karpelson, M. & Wood, R. J. 2019 Untethered flight of an insect-sized flapping-wing microscale aerial vehicle. Nature 570 (7762), 491–495.CrossRefGoogle ScholarPubMed
Jantzen, R. T., Taira, K., Granlund, K. O. & Ol, M. V. 2014 Vortex dynamics around pitching plates. Phys. Fluids 26 (5), 053606.CrossRefGoogle Scholar
Jin, Y., Kim, J.-T., Fu, S. & Chamorro, L. P. 2019 Flow-induced motions of flexible plates: fluttering, twisting and orbital modes. J. Fluid Mech. 864, 273–285.CrossRefGoogle Scholar
Khalak, A. & Williamson, C. H. K. 1996 Dynamics of a hydroelastic cylinder with very low mass and damping. J. Fluids Struct. 10 (5), 455–472.CrossRefGoogle Scholar
Kim, D., Cossé, J., Cerdeira, C. H. & Gharib, M. 2013 Flapping dynamics of an inverted flag. J. Fluid Mech. 736, R1.CrossRefGoogle Scholar
Lee, J. H., Xiros, N. & Bernitsas, M. M. 2011 Virtual damper-spring system for VIV experiments and hydrokinetic energy conversion. Ocean Engng 38 (5–6), 732–747.CrossRefGoogle Scholar
Mackowski, A. W. & Williamson, C. H. K. 2011 Developing a cyber-physical fluid dynamics facility for fluid-structure interaction studies. J. Fluids Struct. 27 (5–6), 748–757.CrossRefGoogle Scholar
McCroskey, W. J. 1982 Unsteady airfoils. Annu. Rev. Fluid Mech. 14 (1), 285–311.CrossRefGoogle Scholar
Menon, K. & Mittal, R. 2019 Flow physics and dynamics of flow-induced pitch oscillations of an airfoil. J. Fluid Mech. 877, 582–613.CrossRefGoogle Scholar
Morse, T. L. & Williamson, C. H. K. 2009 Prediction of vortex-induced vibration response by employing controlled motion. J. Fluid Mech. 634, 5–39.CrossRefGoogle Scholar
Navrose, & Mittal, S. 2017 The critical mass phenomenon in vortex-induced vibration at low Re. J. Fluid Mech. 820, 159–186.CrossRefGoogle Scholar
Onoue, K. & Breuer, K. S. 2016 Vortex formation and shedding from a cyber-physical pitching plate. J. Fluid Mech. 793, 229–247.CrossRefGoogle Scholar
Onoue, K. & Breuer, K. S. 2017 A scaling for vortex formation on swept and unswept pitching wings. J. Fluid Mech. 832, 697–720.CrossRefGoogle Scholar
Onoue, K., Song, A., Strom, B. & Breuer, K. S. 2015 Large amplitude flow-induced oscillations and energy harvesting using a cyber-physical pitching plate. J. Fluids Struct. 55, 262–275.CrossRefGoogle Scholar
Peng, Z. & Zhu, Q. 2009 Energy harvesting through flow-induced oscillations of a foil. Phys. Fluids 21 (12), 123602.CrossRefGoogle Scholar
Poirel, D., Harris, Y. & Benaissa, A. 2008 Self-sustained aeroelastic oscillations of a NACA0012 airfoil at low-to-moderate Reynolds numbers. J. Fluids Struct. 24 (5), 700–719.CrossRefGoogle Scholar
Poirel, D., Metivier, V. & Dumas, G. 2011 Computational aeroelastic simulations of self-sustained pitch oscillations of a NACA0012 at transitional Reynolds numbers. J. Fluids Struct. 27 (8), 1262–1277.CrossRefGoogle Scholar
Poirel, D. & Yuan, W. 2010 Aerodynamics of laminar separation flutter at a transitional Reynolds number. J. Fluids Struct. 26 (7–8), 1174–1194.CrossRefGoogle Scholar
Rao, S. S. 1995 Mechanical Vibrations. Addison-Wesley.Google Scholar
Razak, N. A., Andrianne, T. & Dimitriadis, G. 2011 Flutter and stall flutter of a rectangular wing in a wind tunnel. AIAA J. 49 (10), 2258–2271.CrossRefGoogle Scholar
Sharma, A. & Visbal, M. 2019 Numerical investigation of the effect of airfoil thickness on onset of dynamic stall. J. Fluid Mech. 870, 870–900.CrossRefGoogle Scholar
Shyy, W., Aono, H., Chimakurthi, S. K., Trizila, P., Kang, C.-K., Cesnik, C. E. S. & Liu, H. 2010 Recent progress in flapping wing aerodynamics and aeroelasticity. Prog. Aerosp. Sci. 46 (7), 284–327.CrossRefGoogle Scholar
Strickland, J. H. & Graham, G. M. 1987 Force coefficients for a NACA-0015 airfoil undergoing constant pitch rate motions. AIAA J. 25 (4), 622–624.CrossRefGoogle Scholar
Strogatz, S. H. 1994 Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Perseus Books.Google Scholar
Su, Y & Breuer, K. S. 2019 Resonant response and optimal energy harvesting of an elastically mounted pitching and heaving hydrofoil. Phys. Rev. Fluids 4 (6), 064701.CrossRefGoogle Scholar
Theodorsen, T. 1935 General theory of aerodynamic instability and the mechanism of flutter. NACA Tech. Rep. 496.Google Scholar
Tzezana, G. A. & Breuer, K. S. 2019 Thrust, drag and wake structure in flapping compliant membrane wings. J. Fluid Mech. 862, 871–888.CrossRefGoogle Scholar
Veilleux, J.-C. & Dumas, G. 2017 Numerical optimization of a fully-passive flapping-airfoil turbine. J. Fluids Struct. 70, 102–130.CrossRefGoogle Scholar
Wang, Z., Du, L., Zhao, J. & Sun, X. 2017 Structural response and energy extraction of a fully passive flapping foil. J. Fluids Struct. 72, 96–113.CrossRefGoogle Scholar
Wang, Z. J. 2005 Dissecting insect flight. Annu. Rev. Fluid Mech. 37, 183–210.CrossRefGoogle Scholar
Williamson, C. H. K. & Govardhan, R. 2004 Vortex-induced vibrations. Annu. Rev. Fluid Mech. 36, 413–455.CrossRefGoogle Scholar
Wu, K. S., Nowak, J. & Breuer, K. S. 2019 Scaling of the performance of insect-inspired passive-pitching flapping wings. J. R. Soc. Interface 16 (161), 20190609.Google Scholar
Xiao, Q. & Zhu, Q. 2014 A review on flow energy harvesters based on flapping foils. J. Fluids Struct. 46, 174–191.CrossRefGoogle Scholar
Young, J., Ashraf, M. A., Lai, J. C. S. & Platzer, M. F. 2013 Numerical simulation of fully passive flapping foil power generation. AIAA J. 51 (11), 2727–2739.CrossRefGoogle Scholar
Young, J., Lai, J. C. S. & Platzer, M. F. 2014 A review of progress and challenges in flapping foil power generation. Prog. Aerosp. Sci. 67, 2–28.CrossRefGoogle Scholar
Zhu, Q. 2011 Optimal frequency for flow energy harvesting of a flapping foil. J. Fluid Mech. 675, 495–517.CrossRefGoogle Scholar
Zhu, Q. 2012 Energy harvesting by a purely passive flapping foil from shear flows. J. Fluids Struct. 34, 157–169.CrossRefGoogle Scholar
Zhu, Q., Haase, M. & Wu, C. H. 2009 Modeling the capacity of a novel flow-energy harvester. Appl. Math. Model. 33 (5), 2207–2217.CrossRefGoogle Scholar
Zhu, Q. & Peng, Z. 2009 Mode coupling and flow energy harvesting by a flapping foil. Phys. Fluids 21 (3), 033601.CrossRefGoogle Scholar

Zhu et al. supplementary movie 1

Vorticity field of flow-induced oscillations for the high-inertia wing.

Download Zhu et al. supplementary movie 1(Video)
Video 5.8 MB

Zhu et al. supplementary movie 2

Vorticity field of flow-induced oscillations for the low-inertia wing.

Download Zhu et al. supplementary movie 2(Video)
Video 2.1 MB