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Thrust generation from pitching foils with flexible trailing edge flaps

  • M. Jimreeves David (a1), R. N. Govardhan (a1) and J. H. Arakeri (a1)


In the present experimental study, we investigate thrust production from a pitching flexible foil in a uniform flow. The flexible foils studied comprise a rigid foil in the front (chord length $c_{R}$ ) that is pitched sinusoidally at a frequency $f$ , with a flexible flap of length $c_{F}$ and flexural rigidity $EI$ attached to its trailing edge. We investigate thrust generation for a range of flexural rigidities ( $EI$ ) and flap length to total chord ratio ( $c_{F}/c$ ), with the mean thrust ( $\overline{C_{T}}$ ) and the efficiency of thrust generation ( $\unicode[STIX]{x1D702}$ ) being directly measured in each case. The thrust in the rigid foil cases, as expected, is found to be primarily due to the normal force on the rigid foil ( $\overline{C_{TN}}$ ) with the chordwise or axial thrust contribution ( $\overline{C_{TA}}$ ) being small and negative. In contrast, in the flexible foil cases, the axial contribution to thrust becomes important. We find that using a non-dimensional flexural rigidity parameter ( $R^{\ast }$ ) defined as $R^{\ast }=EI/(0.5\unicode[STIX]{x1D70C}U^{2}c_{F}^{3})$ appears to combine the independent effects of variations in $EI$ and $c_{F}/c$ at a given value of the reduced frequency ( $k=\unicode[STIX]{x03C0}fc/U$ ) for the range of $c_{F}/c$ values studied here ( $U$ is free-stream velocity; $\unicode[STIX]{x1D70C}$ is fluid density). At $k\approx 6$ , the peak mean thrust coefficient is found to be about 100 % higher than the rigid foil thrust, and occurs at $R^{\ast }$ value of approximately 8, while the peak efficiency is found to be approximately 300 % higher than the rigid foil efficiency and occurs at a distinctly different $R^{\ast }$ value of close to 0.01. Corresponding to these two optimal flexural rigidity parameter values, we find two distinct flap deflection shapes; the peak thrust corresponding to a mode 1 type simple bending of the flap with no inflection points, while the peak efficiency corresponds to a distinctly different deflection profile having an inflection point along the flap. The peak thrust condition is found to be close to the ‘resonance’ condition for the first mode natural frequency of the flexible flap in still water. In both these optimal cases, we find that it is the axial contribution to thrust that dominates ( $\overline{C_{TA}}\gg \overline{C_{TN}}$ ), in contrast to the rigid foil case. Particle image velocimetry (PIV) measurements for the flexible cases show significant differences in the strength and arrangement of the wake vortices in these two cases.


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Anderson, J. M., Streitlien, K., Barrett, D. S. & Triantafyllou, M. S. 1998 Oscillating foils of high propulsive efficiency. J. Fluid Mech. 360, 4172.
Bohl, D. G. & Koochesfahani, M. M. 2009 Mtv measurements of the vortical field in the wake of an airfoil oscillating at high reduced frequency. J. Fluid Mech. 620, 6388.
Buchholz, J. H. J. & Smits, A. J. 2008 The wake structure and thrust performance of a rigid low-aspect-ratio pitching panel. J. Fluid Mech. 603, 331365.
Cleaver, D. J., Wang, Z. & Gursul, I. 2012 Bifurcating flows of plunging aerofoils at high Strouhal numbers. J. Fluid Mech. 708, 349376.
Combes, S. A. & Daniel, T. L. 2003 Flexural stiffness in insect wings. J. Expl Biol. 206, 29892997.
Dai, H., Luo, H., Ferreira De Sousa, P. J. S. A. & Doyle, J. F. 2012 Thrust performance of flexible low-aspect-ratio pitching plate. Phys. Fluids 24, 101903.
Daniel, T. L. 1984 Unsteady aspects of aquatic locomotion. Am. Zool. 24 (1), 121134.
Das, P., Govardhan, R. N. & Arakeri, J. H. 2013 Effect of hinged leaflets on vortex pair generation. J. Fluid Mech. 730, 626658.
Dewey, P. A., Boschitsch, B. M., Moored, K. W., Stone, H. A. & Smits, A. J. 2013 Scaling laws for the thrust production of flexible pitching panels. J. Fluid Mech. 732, 2946.
Eldredge, J. D., Toomey, J. & Medina, A. 2010 On the roles of chord-wise flexibility in a flapping wing with hovering kinematics. J. Fluid Mech. 659, 94115.
Godoy-Diana, R., Aider, J.-L. & Wesfreid, J. E. 2008 Transitions in the wake of a flapping foil. Phys. Rev. E 77 (1), 016308.
Govardhan, R. N. & Williamson, C. H. K. 2000 Modes of vortex formation and frequency response of a freely vibrating cylinder. J. Fluid Mech. 420, 85130.
Heathcote, S. & Gursul, I. 2007 Flexible flapping airfoil propulsion at low Reynolds numbers. AIAA J. 45, 10661079.
Kang, C.-K., Aono, H., Cesnik, C. E. S. & Shyy, W. 2011 Effects of flexibility on the aerodynamic performance of flapping wings. J. Fluid Mech. 689, 3274.
Katz, J. & Weihs, D. 1978 Hydrodynamic propulsion by large amplitude oscillation of an airfoil with chordwise flexibility. J. Fluid Mech. 88 (03), 485497.
Kim, D. & Gharib, M. 2011 Flexibility effects on vortex formation of translating plates. J. Fluid Mech. 677, 255271.
Koochesfahani, M. M. 1989 Vortical patterns in wake of an oscillating air foil. AIAA J. 27, 12001205.
Lauder, G. V., Madden, P. G. A., Tangorra, J. L., Anderson, E. & Baker, T. V. 2011 Bioinspiration from fish for smart material design and function. Smart Materials Structures 20 (9), 094014.
Lewin, G. C. & Haj-Hariri, H. 2003 Modelling thrust generation of a two-dimensional heaving airfoil in a viscous flow. J. Fluid Mech. 492, 339362.
Lighthill, M. J. 1970 Aquatic animal propulsion of high hydromechanical efficiency. J. Fluid Mech. 44 (02), 265301.
Low, K. H. 2011 Current and future trends of biologically inspired underwater vehicles. In Defense Science Research Conference and Expo (DSR), 2011, pp. 18. IEEE.
Lu, K., Xie, Y. H. & Zhang, D. 2013 Numerical study of large amplitude, nonsinusoidal motion and camber effects on pitching airfoil propulsion. J. Fluids Struct. 36, 184194.
Mackowski, A. W. & Williamson, C. H. K. 2015 Direct measurement of thrust and efficiency of an airfoil undergoing pure pitching. J. Fluid Mech. 765, 524543.
Marais, C., Thiria, B., Wesfreid, J. E. & Godoy-Diana, R. 2012 Stabilizing effect of flexibility in the wake of a flapping foil. J. Fluid Mech. 710, 659669.
McCroskey, W. J. 1982 Unsteady airfoils. Annu. Rev. Fluid Mech. 14 (1), 285311.
Michelin, S. & Llewellyn Smith, S. G. 2009 Resonance and propulsion performance of a heaving flexible wing. Phys. Fluids 21, 071902.
Paraz, F., Eloy, C. & Schouveiler, L. 2014 Experimental study of the response of a flexible plate to a harmonic forcing in a flow. C. R. Méc. 342 (9), 532538.
Paraz, F., Schouveiler, L. & Eloy, C. 2016 Thrust generation by a heaving flexible foil: resonance, nonlinearities, and optimality. Phys. Fluids 28 (1), 011903.
Platzer, M. F., Jones, K. D., Young, J. & Lai, J. C. S. 2008 Flapping wing aerodynamics: progress and challenges. AIAA J. 46 (9), 21362149.
Prempraneerach, P., Hover, F. S. & Triantafyllou, M. S. 2004 The effect of chordwise flexibility on the thrust and efficiency of a flapping foil. In 13th International Symposium of Unmanned Untethered Submersible Techn.
Quinn, D. B., Lauder, G. V. & Smits, A. J. 2014 Scaling the propulsive performance of heaving flexible panels. J. Fluid Mech. 738, 250267.
Ramananarivo, S., Godoy-Diana, R. & Thiria, B. 2011 Rather than resonance, flapping wing flyers may play on aerodynamics to improve performance. Proc. Natl Acad. Sci. USA 108 (15), 59645969.
Sarkar, S. & Venkatraman, K. 2006 Numerical simulation of thrust generating flow past a pitching foil. Comput. Fluids 35, 1642.
Schnipper, T., Andersen, A. & Bohr, T. 2009 Vortex wakes of a flapping foil. J. Fluid Mech. 633, 411423.
Shinde, S. Y. & Arakeri, J. H. 2014 Flexibility in flapping foil suppresses meandering of induced jet in absence of free stream. J. Fluid Mech. 757, 231250.
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), 284327.
Thiria, B. & Godoy-Diana, R. 2010 How wing compliance drives the efficiency of self-propelled flapping flyers. Phys. Rev. E 82 (1), 015303.
Triantafyllou, M. S., Triantafyllou, G. S. & Yue, D. K. P. 2000 Hydrodynamics of fishlike swimming. Annu. Rev. Fluid Mech. 32 (1), 3353.
Vanella, M., Fitzgerald, T., Preidikman, S., Balaras, E. & Balachandran, B. 2009 Influence of flexibility on the aerodynamic performance of a hovering wing. J. Expl Biol. 212 (1), 95105.
Wang, Z. 2000 Vortex shedding and frequency selection in flapping flight. J. Fluid Mech. 410, 323341.
Williamson, C. H. K. & Roshko, A. 1988 Vortex formation in the wake of an oscillating cylinder. J. Fluids Struct. 2 (4), 355381.
Wu, T. 1971 Hydromechanics of swimming propulsion. Part 1. Swimming of a two-dimensional flexible plate at variable forward speeds in an inviscid fluid. J. Fluid Mech. 46 (02), 337355.
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