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Large-amplitude membrane flutter in inviscid flow

Published online by Cambridge University Press:  26 March 2020

C. Mavroyiakoumou  and
S. Alben Open the ORCID record for S. Alben [Opens in a new window]
C. Mavroyiakoumou*
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI48109, USA
S. Alben*
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI48109, USA
*
†Email addresses for correspondence: chrismav@umich.edu, alben@umich.edu
†Email addresses for correspondence: chrismav@umich.edu, alben@umich.edu
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Abstract

We study the large-amplitude flutter of membranes (of zero bending rigidity) with vortex sheet wakes in two-dimensional inviscid fluid flows. We apply small initial deflections and track their exponential decay or growth and subsequent large-amplitude dynamics in the space of three dimensionless parameters: membrane pretension, mass density and stretching modulus. With both ends fixed, all the membranes converge to steady deflected shapes with single humps that are nearly fore-aft symmetric, except when the deformations are unrealistically large. With leading edges fixed and trailing edges free to move in the transverse direction, the membranes flutter periodically at intermediate values of mass density. As mass density increases, the motions are increasingly aperiodic, and the amplitudes increase and spatial and temporal frequencies decrease. As mass density decreases from the periodic regime, the amplitudes decrease and spatial and temporal frequencies increase until the motions become difficult to resolve numerically. With both edges free to move in the transverse direction, the membranes flutter similarly to the fixed–free case, but also translate vertically with steady, periodic or aperiodic trajectories, and with non-zero slopes that lead to small angles of attack with respect to the oncoming flow.

JFM classification

Instability: Nonlinear instability Vortex Flows: Vortex shedding
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 891 , 25 May 2020 , A23
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Alben, S. 2008a Optimal flexibility of a flapping appendage in an inviscid fluid. J. Fluid Mech. 614, 355–380.CrossRefGoogle Scholar
Alben, S. 2008b The flapping-flag instability as a nonlinear eigenvalue problem. Phys. Fluids 20, 104106.CrossRefGoogle Scholar
Alben, S. 2009 Simulating the dynamics of flexible bodies and vortex sheets. J. Comput. Phys. 228 (7), 2587–2603.CrossRefGoogle Scholar
Alben, S. 2010 Regularizing a vortex sheet near a separation point. J. Comput. Phys. 229 (13), 5280–5298.CrossRefGoogle Scholar
Alben, S. 2012 The attraction between a flexible filament and a point vortex. J. Fluid Mech. 697, 481–503.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
Arbós-Torrent, S., Ganapathisubramani, B. & Palacios, R. 2013 Leading- and trailing-edge effects on the aeromechanics of membrane aerofoils. J. Fluids Struct. 38, 107–126.CrossRefGoogle Scholar
Argentina, M. & Mahadevan, L. 2005 Fluid-flow-induced flutter of a flag. Proc. Natl Acad. Sci. USA 102, 1829–1834.CrossRefGoogle ScholarPubMed
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Brady, M., Leonard, A. & Pullin, D. I. 1998 Regularized vortex sheet evolution in three dimensions. J. Comput. Phys. 146 (2), 520–545.CrossRefGoogle Scholar
Carrier, G. F. 1945 On the non-linear vibration problem of the elastic string. Q. Appl. Maths 3 (2), 157–165.CrossRefGoogle Scholar
Carrier, G. F. 1949 A note on the vibrating string. Q. Appl. Maths 7 (1), 97–101.CrossRefGoogle Scholar
Chatterjee, P. & Bryant, M. 2018 Aeroelastic-photovoltaic ribbons for integrated wind and solar energy harvesting. Smart Mater. Struct. 27 (8), 08LT01.CrossRefGoogle Scholar
Chen, M., Jia, L.-B., Wu, Y.-F., Yin, X.-Z. & Ma, Y.-B. 2014 Bifurcation and chaos of a flag in an inviscid flow. J. Fluids Struct. 45, 124–137.CrossRefGoogle Scholar
Cheney, J. A., Konow, N., Bearnot, A. & Swartz, S. M. 2015 A wrinkle in flight: the role of elastin fibres in the mechanical behaviour of bat wing membranes. J. R. Soc. Interface 12 (106), 20141286.CrossRefGoogle ScholarPubMed
Chorin, A. J. & Bernard, P. S. 1973 Discretization of a vortex sheet, with an example of roll-up. J. Comput. Phys. 13 (3), 423–429.CrossRefGoogle Scholar
Colgate, S. 1996 Fundamentals of Sailing, Cruising, and Racing. WW Norton and Company.Google Scholar
Connell, B. S. H. & Yue, D. K. P. 2007 Flapping dynamics of a flag in a uniform stream. J. Fluid Mech. 581, 33–67.CrossRefGoogle Scholar
Doaré, O. & Michelin, S. 2011 Piezoelectric coupling in energy-harvesting fluttering flexible plates: linear stability analysis and conversion efficiency. J. Fluids Struct. 27 (8), 1357–1375.CrossRefGoogle Scholar
Drachinsky, A. & Raveh, D. E. 2016 Limit-cycle oscillations of a pre-tensed membrane strip. J. Fluids Struct. 60, 1–22.CrossRefGoogle Scholar
Ellis, P. D., Williams, J. E. & Shneerson, J. M. 1993 Surgical relief of snoring due to palatal flutter: a preliminary report. Ann. R. Coll. Surg. Engrs 75 (4), 286–290.Google ScholarPubMed
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, 904–919.CrossRefGoogle Scholar
Erturk, A., Vieira, W. G. R., De Marqui, C. Jr & Inman, D. J. 2010 On the energy harvesting potential of piezoaeroelastic systems. Appl. Phys. Lett. 96 (18), 184103.CrossRefGoogle Scholar
Farlow, S. J. 1993 Partial Differential Equations for Scientists and Engineers. Courier Corporation.Google Scholar
Ghias, R., Mittal, R. & Dong, H. 2007 A sharp interface immersed boundary method for compressible viscous flows. J. Comput. Phys. 225 (1), 528–553.CrossRefGoogle Scholar
Giacomello, A. & Porfiri, M. 2011 Underwater energy harvesting from a heavy flag hosting ionic polymer metal composites. J. Appl. Phys. 109 (8), 084903.CrossRefGoogle Scholar
Gordnier, R. E. 2009 High fidelity computational simulation of a membrane wing airfoil. J. Fluids Struct. 25 (5), 897–917.CrossRefGoogle Scholar
Graff, K. F. 1975 Wave Motion in Elastic Solids. Oxford University Press.Google Scholar
Griffith, B. E., Hornung, R. D., McQueen, D. M. & Peskin, C. S. 2007 An adaptive, formally second order accurate version of the immersed boundary method. J. Comput. Phys. 223 (1), 10–49.CrossRefGoogle Scholar
Griffith, B. E. & Peskin, C. S. 2005 On the order of accuracy of the immersed boundary method: higher order convergence rates for sufficiently smooth problems. J. Comput. Phys. 208 (1), 75–105.CrossRefGoogle Scholar
Hamlet, C., Santhanakrishnan, A. & Miller, L. A. 2011 A numerical study of the effects of bell pulsation dynamics and oral arms on the exchange currents generated by the upside-down jellyfish Cassiopea xamachana. J. Expl Biol. 214 (11), 1911–1921.CrossRefGoogle ScholarPubMed
Haruo, K. 1975 Flutter of hanging roofs and curved membrane roofs. Intl J. Solids Struct. 11 (4), 477–492.CrossRefGoogle 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
Hu, H., Tamai, M. & Murphy, J. T. 2008 Flexible-membrane airfoils at low Reynolds numbers. J. Aircraft 45 (5), 1767–1778.CrossRefGoogle Scholar
Huang, L. 1995a Flutter of cantilevered plates in axial flow. J. Fluids Struct. 9 (2), 127–147.CrossRefGoogle Scholar
Huang, L. 1995b Mechanical modeling of palatal snoring. J. Acoust. Soc. Am. 97 (6), 3642–3648.CrossRefGoogle Scholar
Huang, W. X. & Sung, H. J. 2010 Three-dimensional simulation of a flapping flag in a uniform flow. J. Fluid Mech. 653, 301–336.CrossRefGoogle Scholar
Jaworski, J. W. & Gordnier, R. E. 2012 High-order simulations of low Reynolds number membrane airfoils under prescribed motion. J. Fluids Struct. 31, 49–66.CrossRefGoogle Scholar
Jones, M. A. 2003 The separated flow of an inviscid fluid around a moving flat plate. J. Fluid Mech. 496, 405–441.CrossRefGoogle Scholar
Jones, M. A. & Shelley, M. J. 2005 Falling cards. J. Fluid Mech. 540, 393–425.CrossRefGoogle Scholar
Kashy, E., Johnson, D. A., McIntyre, J. & Wolfe, S. L. 1997 Transverse standing waves in a string with free ends. Am. J. Phys. 65 (4), 310–313.CrossRefGoogle Scholar
Katz, J. & Plotkin, A. 2001 Low-speed Aerodynamics, vol. 13. Cambridge University Press.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
Kimball, J. 2009 Physics of Sailing. CRC Press.CrossRefGoogle Scholar
Knudson, W. C. 1991 Recent advances in the field of long span tension structures. Engng Struct. 13 (2), 164–177.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, 163–178.Google Scholar
Krasny, R. 1986 Desingularization of periodic vortex sheet roll-up. J. Comput. Phys. 65 (2), 292–313.CrossRefGoogle Scholar
Krasny, R. 1987 Computation of vortex sheet roll-up in the Trefftz plane. J. Fluid Mech. 184, 123–155.CrossRefGoogle Scholar
Krasny, R. 1991 Vortex sheet computations: roll-up, wakes, separation. Lectures Appl. Math. 28 (1), 385–401.Google Scholar
Lauder, G. V., Madden, P. G. A., Mittal, R., Dong, H. & Bozkurttas, M. 2006 Locomotion with flexible propulsors: I. Experimental analysis of pectoral fin swimming in sunfish. Bioinspir. Biomim. 1 (4), S25.CrossRefGoogle ScholarPubMed
Le Maître, O., Huberson, S. & De Cursi, E. S. 1999 Unsteady model of sail and flow interaction. J. Fluids Struct. 13 (1), 37–59.CrossRefGoogle Scholar
Lian, Y. & Shyy, W. 2005 Numerical simulations of membrane wing aerodynamics for micro air vehicle applications. J. Aircraft 42 (4), 865–873.CrossRefGoogle Scholar
Lian, Y., Shyy, W., Viieru, D. & Zhang, B. 2003 Membrane wing aerodynamics for micro air vehicles. Prog. Aerosp. Sci. 39 (6–7), 425–465.CrossRefGoogle Scholar
Manela, A. & Weidenfeld, M. 2017 The ‘hanging flag’ problem: on the heaving motion of a thin filament in the limit of small flexural stiffness. J. Fluid Mech. 829, 190–213.CrossRefGoogle Scholar
Michelin, S., Smith, S. G. L. & Glover, B. J. 2008 Vortex shedding model of a flapping flag. J. Fluid Mech. 617, 1–10.CrossRefGoogle Scholar
Narasimha, R. 1968 Non-linear vibration of an elastic string. J. Sound Vib. 8 (1), 134–146.Google Scholar
Nardini, M., Illingworth, S. J. & Sandberg, R. D. 2018 Reduced-order modeling for fluid–structure interaction of membrane wings at low and moderate Reynolds numbers. In 2018 AIAA Aerospace Sciences Meeting, 2018-1544. AIAA.Google Scholar
Nayfeh, A. H. & Pai, P. F. 2008 Linear and Nonlinear Structural Mechanics. John Wiley and Sons.Google Scholar
Newman, B. G. 1987 Aerodynamic theory for membranes and sails. Prog. Aerosp. Sci. 24 (1), 1–27.CrossRefGoogle Scholar
Newman, B. G. & Low, H. T. 1984 Two-dimensional impervious sails: experimental results compared with theory. J. Fluid Mech. 144, 445–462.CrossRefGoogle Scholar
Newman, B. G. & Paidoussis, M. P. 1991 The stability of two-dimensional membranes in streaming flow. J. Fluids Struct. 5 (4), 443–454.CrossRefGoogle Scholar
Nitsche, M. & Krasny, R. 1994 A numerical study of vortex ring formation at the edge of a circular tube. J. Fluid Mech. 276, 139–161.CrossRefGoogle Scholar
Nitsche, M., Taylor, M. A. & Krasny, R. 2003 Comparison of regularizations of vortex sheet motion. In Computational Fluid and Solid Mechanics 2003, pp. 1062–1065. Elsevier.CrossRefGoogle Scholar
Orrego, S., Shoele, K., Ruas, A., Doran, K., Caggiano, B., Mittal, R. & Kang, S. H. 2017 Harvesting ambient wind energy with an inverted piezoelectric flag. Appl. Energy 194, 212–222.CrossRefGoogle Scholar
Peskin, C. S. 2002 The immersed boundary method. Acta Numerica 11, 479–517.CrossRefGoogle Scholar
Piquee, J., López, I., Breitsamter, C., Wüchner, R. & Bletzinger, K.-U. 2018 Aerodynamic characteristics of an elasto-flexible membrane wing based on experimental and numerical investigations. In 2018 Applied Aerodynamics Conference, p. 3338.Google Scholar
Porfiri, M. & Peterson, S. D. 2013 Energy harvesting from fluids using ionic polymer metal composites. In Advances in Energy Harvesting Methods, pp. 221–239. Springer.CrossRefGoogle Scholar
Pullin, D. I. & Wang, Z. J. 2004 Unsteady forces on an accelerating plate and application to hovering insect flight. J. Fluid Mech. 509, 1–21.CrossRefGoogle Scholar
Rojratsirikul, P., Wang, Z. & Gursul, I. 2009 Unsteady fluid–structure interactions of membrane airfoils at low Reynolds numbers. Exp. Fluids 46 (5), 859–872.CrossRefGoogle Scholar
Roma, A. M., Peskin, C. S. & Berger, M. J. 1999 An adaptive version of the immersed boundary method. J. Comput. Phys. 153 (2), 509–534.CrossRefGoogle Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Schomberg, T., Gerland, F., Liese, F., Wünsch, O. & Ruetten, M. 2018 Transition manipulation by the use of an electrorheologically driven membrane. In 2018 Flow Control Conference, p. 3213.Google Scholar
Shelley, M. J., Vandenberghe, N. & Zhang, J. 2005 Heavy flags undergo spontaneous oscillations in flowing water. Phys. Rev. Lett. 94 (9), 094302.CrossRefGoogle ScholarPubMed
Shelley, M. J. & Zhang, J. 2011 Flapping and bending bodies interacting with fluid flows. Annu. Rev. Fluid Mech. 43, 449–465.CrossRefGoogle Scholar
Sheng, J. X., Ysasi, A., Kolomenskiy, D., Kanso, E., Nitsche, M. & Schneider, K. 2012 Simulating vortex wakes of flapping plates. In Natural Locomotion in Fluids and on Surfaces, pp. 255–262. Springer.CrossRefGoogle Scholar
Shukla, R. K. & Eldredge, J. D. 2007 An inviscid model for vortex shedding from a deforming body. Theor. Comput. Fluid Dyn. 21 (5), 343–368.CrossRefGoogle Scholar
Song, A., Tian, X., Israeli, E., Galvao, R., Bishop, K., Swartz, S. & Breuer, K. 2008 Aeromechanics of membrane wings with implications for animal flight. AIAA J. 46 (8), 2096–2106.CrossRefGoogle Scholar
Stanford, B., Ifju, P., Albertani, R. & Shyy, W. 2008 Fixed membrane wings for micro air vehicles: experimental characterization, numerical modeling, and tailoring. Prog. Aerosp. Sci. 44 (4), 258–294.CrossRefGoogle Scholar
Sunny, M. R., Sultan, C. & Kapania, R. K. 2014 Optimal energy harvesting from a membrane attached to a tensegrity structure. AIAA J. 52 (2), 307–319.CrossRefGoogle Scholar
Sygulski, R. 1996 Dynamic stability of pneumatic structures in wind: theory and experiment. J. Fluids Struct. 10 (8), 945–963.CrossRefGoogle Scholar
Sygulski, R. 1997 Numerical analysis of membrane stability in air flow. J. Sound Vib. 201 (3), 281–292.Google Scholar
Sygulski, R. 2007 Stability of membrane in low subsonic flow. Intl J. Non-Linear Mech. 42 (1), 196–202.CrossRefGoogle Scholar
Tadjbakhsh, I. 1966 The variational theory of the plane motion of the extensible elastica. Intl J. Engng Sci. 4 (4), 433–450.Google Scholar
Taira, K. & Colonius, T. 2007 The immersed boundary method: a projection approach. J. Comput. Phys. 225 (2), 2118–2137.CrossRefGoogle Scholar
Tamai, M., Murphy, J. & Hu, H. 2008 An experimental study of flexible membrane airfoils at low Reynolds numbers. In 46th AIAA Aerospace Sciences Meeting and Exhibit, p. 580.Google Scholar
Taneda, S. 1968 Waving motions of flags. J. Phys. Soc. Japan 24, 392–401.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
Tian, F.-B., Luo, H., Zhu, L., Liao, J. C. & Lu, X.-Y. 2011 An efficient immersed boundary-lattice Boltzmann method for the hydrodynamic interaction of elastic filaments. J. Comput. Phys. 230 (19), 7266–7283.CrossRefGoogle ScholarPubMed
Timpe, A., Zhang, Z., Hubner, J. & Ukeiley, L. 2013 Passive flow control by membrane wings for aerodynamic benefit. Exp. Fluids 54 (3), 1471.CrossRefGoogle Scholar
Tiomkin, S. & Raveh, D. E. 2017 On the stability of two-dimensional membrane wings. J. Fluids Struct. 71, 143–163.CrossRefGoogle Scholar
Triantafyllou, M. S. & Howell, C. T. 1994 Dynamic response of cables under negative tension: an ill-posed problem. J. Sound Vib. 173 (4), 433–447.Google Scholar
Tytell, E. D., Hsu, C.-Y., Williams, T. L., Cohen, A. H. & Fauci, L. J. 2010 Interactions between internal forces, body stiffness, and fluid environment in a neuromechanical model of lamprey swimming. Proc. Natl Acad. Sci. USA 107 (46), 19832–19837.CrossRefGoogle 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
Waldman, R. M. & Breuer, K. S. 2013 Shape, lift, and vibrations of highly compliant membrane wings. In 43rd AIAA Fluid Dynamics Conference, p. 3177.Google Scholar
Waldman, R. M. & Breuer, K. S. 2017 Camber and aerodynamic performance of compliant membrane wings. J. Fluids Struct. 68, 390–402.CrossRefGoogle Scholar
Wang, X., Alben, S., Li, C. & Young, Y. L. 2016 Stability and scalability of piezoelectric flags. Phys. Fluids 28 (2), 023601.CrossRefGoogle Scholar
Wang, Z., Rebeiz, E. E. & Shapshay, S. M. 2002 Laser soft palate ‘stiffening’: an alternative to uvulopalatopharyngoplasty. Lasers in Surgery and Medicine: The Official Journal of the American Society for Laser Medicine and Surgery 30 (1), 40–43.CrossRefGoogle ScholarPubMed
Watanabe, Y., Suzuki, S., Sugihara, M. & Sueoka, Y. 2002 An experimental study of paper flutter. J. Fluids Struct. 16 (4), 529–542.CrossRefGoogle Scholar
Welch, P. D. 1967 The use of fast fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms. IEEE Trans. Audio Electroacoust. 15 (2), 70–73.CrossRefGoogle Scholar
Williamson, C. H. K. & Roshko, A. 1988 Vortex formation in the wake of an oscillating cylinder. J. Fluids Struct. 2 (4), 355–381.CrossRefGoogle Scholar
Xu, L., Nitsche, M. & Krasny, R. 2017 Computation of the starting vortex flow past a flat plate. Proc. IUTAM 20, 136–143.CrossRefGoogle Scholar
Yang, S. & Sultan, C. 2016 Modeling of tensegrity-membrane systems. Intl J. Solids Struct. 82, 125–143.CrossRefGoogle Scholar
Zhang, J., Childress, S., Libchaber, A. & Shelley, M. J. 2000 Flexible filaments in a flowing soap film as a model for flags in a two dimensional wind. Nature 408, 835–839.CrossRefGoogle Scholar
Zhu, L. & Peskin, C. S. 2002 Simulation of a flapping flexible filament in a flowing soap film by the immersed boundary method. J. Comput. Phys. 179, 452–468.CrossRefGoogle Scholar

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