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Self-induced flapping dynamics of a flexible inverted foil in a uniform flow

  • P. S. Gurugubelli (a1) and R. K. Jaiman (a1)


We present a numerical study on the self-induced flapping dynamics of an inverted flexible foil in a uniform flow. A high-order coupled fluid–structure solver based on fully coupled Navier–Stokes and nonlinear structural dynamic equations has been employed. Unlike a conventional flexible foil flapping where the leading edge is clamped, the inverted elastic foil is fixed at the trailing edge and the leading edge is allowed to oscillate freely. We investigate the evolution of flapping instability of an inverted foil as a function of the non-dimensional bending rigidity, $K_{B}$ , Reynolds number, $\mathit{Re}$ , and structure-to-fluid mass ratio, $m^{\ast }$ , and identify three distinct stability regimes, namely (i) fixed-point stable, (ii) deformed steady and (iii) unsteady flapping state. With the aid of a simplified analytical model, we show that the fixed-point stable regime loses its stability by static-divergence instability. The transition from the deformed steady state to the unsteady flapping regime is marked by a flow separation at the leading edge. We also show that an inverted foil is more vulnerable to static divergence than a conventional foil. Three distinct unsteady flapping modes have been observed as a function of decreasing $K_{B}$ : (i) inverted limit-cycle oscillations, (ii) deformed flapping and (iii) flipped flapping. We characterize the transition to the deformed-flapping regime through a quasistatic equilibrium analysis between the structural restoring and the fluid forces. We further examine the effects of $m^{\ast }$ on the post-critical flapping dynamics at a fixed $\mathit{Re}=1000$ . Finally, we present the net work done by the fluid and the bending strain energy developed in a flexible foil due to the flapping motion. For small $m^{\ast }$ , we demonstrate that the flapping of an inverted flexible foil can generate $O(10^{3})$ times more strain energy in comparison to a conventional flexible foil flapping, which has a profound impact on energy harvesting devices.


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Akcabay, D. T. & Young, Y. L. 2012 Hydroelastic response and energy harvesting potential of a flexible piezoelectric beam in viscous flow. Phys. Fluids 24, 054106.
Alben, S. 2009 Simulating the dynamics of flexible bodies and vortex sheets. J. Comput. Phys. 228 (7), 25872603.
Allen, J. J. & Smits, A. J. 2001 Energy harvesting eel. J. Fluids Struct. 15, 112.
Antman, S. S. 1995 Nonlinear Problems of Elasticity. Springer.
Argentina, M. & Mahadevan, L. 2004 Fluid-flow-induced flutter of a flag. Proc. Natl Acad. Sci. USA 102, 18291834.
Bisplinghoff, R. L., Ashley, H. & Halfman, R. L. 1957 Aeroelasticity. Addison-Wesley.
Buchak, P., Eloy, C. & Reis, P. M. 2010 The clapping book: wind-driven oscillations in a stack of elastic sheets. Phys. Rev. Lett. 105, 194301.
Connell, B. S. H. & Yue, D. K. P. 2007 Flapping dynamics of a flag in a uniform stream. J. Fluid Mech. 581, 3367.
Crandall, S. H., Dahl, N. C. & Lardner, T. J. 1995 An Introduction to the Mechanics of Solids. McGraw-Hill.
Datta, S. K. & Gottenberg, W. G. 1975 Instability of an elastic strip hanging in an airstream. J. Appl. Mech. 42, 195198.
Dowell, E. H. 1966 Nonlinear oscillations of a fluttering plate. AIAA J. 4 (7), 12671275.
Dowell, E. H. 1970 Panel flutter: a review of the aeroelastic stability of plates and shells. AIAA J. 8 (3), 385399.
Eloy, C., Kofman, N. & Schouveiler, L. 2012 The origin of hysteresis in the flag instability. J. Fluid Mech. 691, 583593.
Eloy, C., Lagrange, R., Souilliez, C. & Schouveiler, L. 2008 Aeroelastic instability of cantilevered flexible plates in uniform flow. J. Fluid Mech. 611, 97106.
Eloy, C., Souilliez, C. & Schouveiler, L. 2007 Flutter of a rectangular plate. J. Fluids Struct. 23 (6), 904919.
Farnell, D. J. J., David, T. & Barton, D. C. 2004 Numerical simulations of a filament in a flowing soap film. Intl J. Numer. Meth. Fluids 44, 313330.
Fung, Y. C. 1969 An Introduction to the Theory of Aeroelasticity. Dover.
Guo, C. Q. & Paidoussis, M. P. 2000 Stability of rectangular plates with free side-edges in two-dimensional inviscid channel flow. J. Appl. Mech. 67 (1), 171176.
Huang, I., Rominger, J. & Nepf, H. 2011 The motion of kelp blades and surface renewal model. Limnol. Oceanogr. 56 (4), 14531462.
Huang, L. 1995 Flutter of cantilevered plates in axial flow. J. Fluids Struct. 9, 127147.
Jaiman, R. K., Loth, E. & Dutton, J. C. 2004 Simulations of normal shock-wave/boundary-layer interaction control using mesoflaps. AIAA J. Propul. Power 20, 344352.
Kim, D., Cossé, J., Huertas Cerdeira, C. & Gharib, M. 2013 Flapping dynamics of an inverted flag. J. Fluid Mech. 736, R1.
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), 163178.
Lighthill, M. J. 1960 Note on the swimming of slender fish. J. Fluid Mech. 9, 305317.
Liu, J., Jaiman, R. K. & Gurugubelli, P. S. 2014 A stable second-order scheme for fluid–structure interaction with strong added-mass effects. J. Comput. Phys. 270, 687710.
Lucey, A. D. 1998 The excitation of waves on a flexible panel in a uniform flow. Phil. Trans. R. Soc. Lond. A 356, 29993039.
Michelin, S. & Doare, O. 2013 Energy harvesting efficiency of piezoelectric flags in axial flows. J. Fluid Mech. 714, 489504.
Michelin, S., Llewellyn Smith, S. G. & Glover, B. J. 2008 Vortex shedding model of a flapping flag. J. Fluid Mech. 617, 110.
Newman, J. N. 1977 Marine Hydrodynamics. MIT Press.
Ol, M. V. & Gharib, M. 2003 Leading-edge vortex structure of nonslender delta wings at low Reynolds numbers. AIAA J. 41 (1), 1626.
Paidoussis, M. P. 2000 Fluid–Structure Interactions: Slender Structures and Axial Flow. Academic.
Schouveiler, L., Eloy, C. & Gal, P. L 2005 Flow-induced vibration of high mass ratio flexible filaments freely hanging in a flow. Phys. Fluids 17, 047104.
Shelley, M., Vandenberghe, N. & Zhang, J. 2005 Heavy flags undergo spontaneous oscillations in flowing water. Phys. Rev. Lett. 94, 094302.
Shelley, M. J. & Zhang, J. 2011 Flapping and bending bodies interacting with fluid flows. Annu. Rev. Fluid Mech. 43 (1), 449465.
Stein, K, Tezduyar, T. & Benney, R. 2003 Mesh moving techniques for fluid–structure interactions with large displacements. J. Appl. Mech. 70, 5863.
Taneda, S. 1968 Waving motions of flags. J. Phys. Soc. Japan 24, 392401.
Tang, L. & Paidoussis, M. P. 2007 On the instability and the post-critical behaviour of two-dimensional cantilevered flexible plates in axial-flow. J. Sound Vib. 305, 97115.
Tang, L., Paidoussis, M. P. & Jiang, J 2009 Cantilevered flexible plates in axial flow: energy transfer and the concept of flutter-mill. J. Sound Vib. 326, 263276.
Temam, R. 2001 Navier–Stokes Equations. Theory and Numerical Analysis. AMS Chelsea.
Wang, Z. J. 2000 Two dimensional mechanism for insect hovering. Phys. Rev. Lett. 85 (10), 22162219.
Watanabe, Y., Isogai, K., Suzuki, S. & Sugihara, M 2002a A theoretical study of paper flutter. J. Fluids Struct. 16, 543560.
Watanabe, Y., Suzuki, S., Sugihara, M. & Sueoka, Y. 2002b An experimental study of paper flutter. J. Fluids Struct. 16, 529542.
Zhang, J., Childress, S., Libchaber, A. & Shelley, M. 2000 Flexible filaments in a flowing soap film as a model for one-dimensional flags in a two-dimensional wind. Nature 408, 835839.
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, 452468.
Zhu, L. & Peskin, C. S. 2003 Interaction of two flapping filaments in a flowing soap film. Phys. Fluids 15, 19541960.
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Journal of Fluid Mechanics
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  • EISSN: 1469-7645
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