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Flutter instability of a thin flexible plate in a channel

Published online by Cambridge University Press:  24 November 2015

Kourosh Shoele  and
Rajat Mittal
Kourosh Shoele
Affiliation:
Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21218, USA
Rajat Mittal*
Affiliation:
Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21218, USA
*
†Email address for correspondence: [email protected]
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Abstract

The stability of a thin flexible plate confined inside an inviscid two-dimensional channel is examined using a nonlinear eigenvalue analysis method. A new Green’s function for the vortex wake of the flexible plate inside the channel, as well as its rapidly convergent series approximation, is proposed. Comparison with a fully coupled Navier–Stokes fluid–structure interaction model indicates that the current inviscid model successfully predicts the flutter boundary for a confined flexible plate. The analysis also shows that confinement has a destabilizing effect on heavy plates. Furthermore, as the confinement is increased, the oscillating frequency of the plate increases and new peaks appear in its stability curve. Asymmetric placement of the plate within the channel, especially when the plate is very close to one wall, also modifies the stability curve of the system by shifting the mode transition points toward smaller fluid-to-plate inertia ratios. Our study suggests that the degree of confinement and asymmetric placement of the plate in the channel could be used to alter the flutter instability of the plate, and to adjust the frequency of flutter.

JFM classification

Instability: Nonlinear instability Wakes/Jets: Vortex streets
Type
Papers
Information
Journal of Fluid Mechanics , Volume 786 , 10 January 2016 , pp. 29 - 46
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Copyright
© 2015 Cambridge University Press 

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References

Abderrahmane, H. A., Païdoussis, M. P., Fayed, M. & Dick Ng, H. 2012 Nonlinear dynamics of silk and mylar flags flapping in axial flow. J. Wind Engng Ind. Aerodyn. 107, 225–236.CrossRefGoogle Scholar
Alben, S. 2008a The flapping-flag instability as a nonlinear eigenvalue problem. Phys. Fluids 20 (10), 104106.CrossRefGoogle Scholar
Alben, S. 2008b Optimal flexibility of a flapping appendage in an inviscid fluid. J. Fluid Mech. 614, 355–380.CrossRefGoogle Scholar
Alben, S. 2015 Flag flutter in inviscid channel flow. Phys. Fluids 27 (3), 033603.CrossRefGoogle Scholar
Alben, S. & Shelley, M. J. 2008 Flapping states of a flag in an inviscid fluid: bistability and the transition to chaos. Phys. Rev. Lett. 100 (7), 074301.CrossRefGoogle Scholar
Argentina, M. & Mahadevan, L. 2005 Fluid-flow-induced flutter of a flag. Proc. Natl Acad. Sci. USA 102 (6), 1829–1834.CrossRefGoogle ScholarPubMed
Auregan, Y. & Depollier, C. 1995 Snoring: linear stability analysis and in-vitro experiments. J. Sound Vib. 188 (1), 39–53.CrossRefGoogle Scholar
Backus, J. 2005 Small-vibration theory of the clarinet. J. Acoust. Soc. Am. 35 (3), 305–313.CrossRefGoogle Scholar
Balint, T. S. & Lucey, A. D. 2005 Instability of a cantilevered flexible plate in viscous channel flow. J. Fluids Struct. 20 (7), 893–912.CrossRefGoogle Scholar
Boyd, J. P. 2001 Chebyshev and Fourier Spectral Methods. Dover.Google Scholar
Dalmont, J. P. & Frappe, C. 2007 Oscillation and extinction thresholds of the clarinet: comparison of analytical results and experiments. J. Acoust. Soc. Am. 122 (2), 1173–1179.CrossRefGoogle ScholarPubMed
Dessi, D. & Mazzocconi, S. 2015 Aeroelastic behavior of a flag in ground effect. J. Fluids Struct. 55, 303–323.CrossRefGoogle Scholar
Doaré, O., Mano, D. & Ludena, J. C. B. 2011a Effect of spanwise confinement on flag flutter: experimental measurements. Phys. Fluids 23 (11), 111704.CrossRefGoogle Scholar
Doaré, O., Sauzade, M. & Eloy, C. 2011b Flutter of an elastic plate in a channel flow: confinement and finite-size effects. J. Fluids Struct. 27 (1), 76–88.CrossRefGoogle Scholar
Eloy, C., Lagrange, R., Souilliez, C. & Schouveiler, L. 2008 Aeroelastic instability of cantilevered flexible plates in uniform flow. J. Fluid Mech. 611, 97–106.CrossRefGoogle Scholar
Eloy, C., Souilliez, C. & Schouveiler, L. 2007 Flutter of a rectangular plate. J. Fluids Struct. 23 (6), 904–919.CrossRefGoogle Scholar
Gradshteyn, I. S. & Ryzhik, I. M. 2014 Table of Integrals, Series, and Products. Elsevier Science.Google Scholar
Greengard, L. 1990 Potential flow in channels. SIAM J. Sci. Stat. Comput. 11 (4), 603–620.CrossRefGoogle Scholar
Guo, C. Q. & Païdoussis, M. P. 2000a Analysis of hydroelastic instabilities of rectangular parallel-plate assemblies. J. Press. Vessel Technol. 122 (4), 502–508.CrossRefGoogle Scholar
Guo, C. Q. & Païdoussis, M. P. 2000b Stability of rectangular plates with free side-edges in two-dimensional inviscid channel flow. Trans. ASME J. Appl. Mech. 67 (1), 171–176.CrossRefGoogle Scholar
Herrault, F., Hidalgo, P. A., Ji, C. H., Glezer, A. & Allen, M. G. 2012 Cooling performance of micromachined self-oscillating reed actuators in heat transfer channels with integrated diagnostics. In Proceedings of the IEEE 25th International Conference on Micro Electro Mechanical Systems (MEMS), pp. 1217–1220. IEEE.Google Scholar
Howell, R. M., Lucey, A. D., Carpenter, P. W. & Pitman, M. W. 2009 Interaction between a cantilevered-free flexible plate and ideal flow. J. Fluids Struct. 25 (3), 544–566.CrossRefGoogle Scholar
Huang, L. 1995 Flutter of cantilevered plates in axial flow. J. Fluids Struct. 9 (2), 127–147.CrossRefGoogle Scholar
Jaiman, R. K., Parmar, M. K. & Gurugubelli, P. S. 2014 Added mass and aeroelastic stability of a flexible plate interacting with mean flow in a confined channel. Trans. ASME J. Appl. Mech. 81 (4), 041006.CrossRefGoogle Scholar
Kim, G. & Davis, D. C. 1995 Hydrodynamic instabilities in flat-plate-type fuel assemblies. Nucl. Engng Des. 158 (1), 1–17.CrossRefGoogle Scholar
Kornecki, A., Dowell, E. H. & O’Brien, J. 1976 On the aeroelastic instability of two-dimensional panels in uniform incompressible flow. J. Sound Vib. 47 (2), 163–178.CrossRefGoogle Scholar
Lighthill, J. 1975 Mathematical Biofluiddynamics, vol. 17. SIAM.CrossRefGoogle Scholar
Michelin, S. & Llewellyn Smith, S. G. 2009 Resonance and propulsion performance of a heaving flexible wing. Phys. Fluids 21 (7), 071902.CrossRefGoogle Scholar
Michelin, S., Llewellyn Smith, S. G. & Glover, B. J. 2008 Vortex shedding model of a flapping flag. J. Fluid Mech. 617, 1–10.CrossRefGoogle Scholar
Michelin, S. & Smith, S. G. L. 2009 Linear stability analysis of coupled parallel flexible plates in an axial flow. J. Fluids Struct. 25 (7), 1136–1157.CrossRefGoogle Scholar
Miller, D. R. 1960 Critical flow velocities for collapse of reactor parallel-plate fuel assemblies. J. Eng. Power 82 (2), 83–91.CrossRefGoogle Scholar
Muskhelishvili, N. I. & Radok, J. R. M. 2008 Singular Integral Equations: Boundary Problems of Function Theory and Their Application to Mathematical Physics. Dover.Google Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Shelley, M. J. & Zhang, J. 2011 Flapping and bending bodies interacting with fluid flows. Annu. Rev. Fluid Mech. 43, 449–465.CrossRefGoogle Scholar
Shoele, K. & Mittal, R. 2014 Computational study of flow-induced vibration of a reed in a channel and effect on convective heat transfer. Phys. Fluids 26 (12), 127103.CrossRefGoogle Scholar
Shoele, K. & Zhu, Q. 2012 Leading edge strengthening and the propulsion performance of flexible ray fins. J. Fluid Mech. 693, 402–432.CrossRefGoogle Scholar
Shoele, K. & Zhu, Q. 2013 Performance of a wing with nonuniform flexibility in hovering flight. Phys. Fluids 25, 041901.CrossRefGoogle Scholar
Sommerfeldt, S. D. & Strong, W. J. 1988 Simulation of a player clarinet system. J. Acoust. Soc. Am. 83 (5), 1908–1918.CrossRefGoogle Scholar
Tang, D. M., Yamamoto, H. & Dowell, E. H. 2003 Flutter and limit cycle oscillations of two-dimensional panels in three-dimensional axial flow. J. Fluids Struct. 17 (2), 225–242.CrossRefGoogle Scholar
Tetlow, G. A. & Lucey, A. D. 2009 Motions of a cantilevered flexible plate in viscous channel flow driven by a constant pressure drop. Commun. Numer. Meth. Engng 25 (5), 463–482.CrossRefGoogle Scholar
Theodorsen, T.1935 General theory of aerodynamic instability and the mechanism of flutter, NACA Tech. Rep. 496, US National Advisory Committee for Aeronautics, Langley, VA.Google Scholar