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A computational model of the flight dynamics and aerodynamics of a jellyfish-like flying machine

Published online by Cambridge University Press:  27 April 2017

Fang Fang Open the ORCID record for Fang Fang [Opens in a new window] ,
Kenneth L. Ho ,
Leif Ristroph  and
Michael J. Shelley
Fang Fang*
Affiliation:
Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA
Kenneth L. Ho
Affiliation:
Department of Mathematics, Stanford University, Building 380, Stanford, CA 94305, USA
Leif Ristroph
Affiliation:
Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA
Michael J. Shelley
Affiliation:
Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA Center for Computational Biology, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA
*
†Email address for correspondence: [email protected]
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Abstract

We explore theoretically the aerodynamics of a recently fabricated jellyfish-like flying machine (Ristroph & Childress, J. R. Soc. Interface, vol. 11 (92), 2014, 20130992). This experimental device achieves flight and hovering by opening and closing opposing sets of wings. It displays orientational or postural flight stability without additional control surfaces or feedback control. Our model ‘machine’ consists of two mirror-symmetric massless flapping wings connected to a volumeless body with mass and moment of inertia. A vortex sheet shedding and wake model is used for the flow simulation. Use of the fast multipole method allows us to simulate for long times and resolve complex wakes. We use our model to explore the design parameters that maintain body hovering and ascent, and investigate the performance of steady ascent states. We find that ascent speed and efficiency increase as the wings are brought closer, due to a mirror-image ‘ground-effect’ between the wings. Steady ascent is approached exponentially in time, which suggests a linear relationship between the aerodynamic force and ascent speed. We investigate the orientational stability of hovering and ascent states by examining the flyer’s free response to perturbation from a transitory external torque. Our results show that bottom-heavy flyers (centre of mass below the geometric centre) are capable of recovering from large tilts, whereas the orientation of the top-heavy flyers diverges. These results are consistent with the experimental observations in Ristroph & Childress (J. R. Soc. Interface, vol. 11 (92), 2014, 20130992), and shed light upon future designs of flapping-wing micro aerial vehicles that use jet-based mechanisms.

JFM classification

Biological Fluid Dynamics: Propulsion Biological Fluid Dynamics: Swimming/flying
Type
Papers
Information
Journal of Fluid Mechanics , Volume 819 , 25 May 2017 , pp. 621 - 655
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Copyright
© 2017 Cambridge University Press 

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References

Alben, S. 2009 Simulating the dynamics of flexible bodies and vortex sheets. J. Comput. Phys. 228 (7), 2587–2603.Google Scholar
Alben, S. 2010 Regularizing a vortex sheet near a separation point. J. Comput. Phys. 229 (13), 5280–5298.Google Scholar
Alben, S., Miller, L. A. & Peng, J. 2013 Efficient kinematics for jet-propelled swimming. J. Fluid Mech. 733, 100–133.CrossRefGoogle Scholar
Alben, S. & Shelley, M. J. 2005 Coherent locomotion as an attracting state for a free flapping body. Proc. Natl Acad. Sci. USA 102 (32), 11163–11166.CrossRefGoogle ScholarPubMed
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
Anderson, J., Streitlien, K., Barrett, D. & Triantafyllou, M. 1998 Oscillating foils of high propulsive efficiency. J. Fluid Mech. 360, 41–72.CrossRefGoogle Scholar
van Breugel, F., Regan, W. & Lipson, H. 2008 From insects to machines. IEEE Robot. Autom. Mag. 15 (4), 68–74.Google Scholar
Broyden, C. G. 1965 A class of methods for solving nonlinear simultaneous equations. Math. Comput. 19, 577–593.CrossRefGoogle Scholar
Carrier, J., Greengard, L. & Rokhlin, V. 1988 A fast adaptive multipole algorithm for particle simulations. SIAM J. Sci. Stat. Comput. 9 (4), 669–686.CrossRefGoogle Scholar
Cheng, H., Gimbutas, Z., Martinsson, P. G. & Rokhlin, V. 2005 On the compression of low rank matrices. SIAM J. Sci. Comput. 26 (4), 1389–1404.Google Scholar
Childress, S., Vandenberghe, N. & Zhang, J. 2006 Hovering of a passive body in an oscillating airflow. Phys. Fluids 18 (11), 117103.CrossRefGoogle Scholar
Dabiri, J. O., Colin, S. P. & Costello, J. H. 2006 Fast-swimming hydromedusae exploit velar kinematics to form an optimal vortex wake. J. Expl. Biol. 209 (11), 2025–2033.CrossRefGoogle ScholarPubMed
Dabiri, J. O., Colin, S. P., Costello, J. H. & Gharib, M. 2005 Flow patterns generated by oblate medusan jellyfish: field measurements and laboratory analyses. J. Expl. Biol. 208 (7), 1257–1265.CrossRefGoogle ScholarPubMed
Etkin, B. & Reid, L. D. 1995 Dynamics of Flight: Stability and Control. Wiley.Google Scholar
Faruque, I. & Humbert, J. S. 2010 Dipteran insect flight dynamics. Part 1: longitudinal motion about hover. J. Theor. Biol. 264 (2), 538–552.Google Scholar
Gerdes, J. W., Gupta, S. K. & Wilkerson, S. A. 2012 A review of bird-inspired flapping wing miniature air vehicle designs. J. Mech. Robot. 4 (2), 021003.CrossRefGoogle Scholar
Golberg, M. A. 2013 Numerical Solution of Integral Equations. Springer.Google Scholar
Greengard, L. & Rokhlin, V. 1987 A fast algorithm for particle simulations. J. Comput. Phys. 73 (2), 325–348.CrossRefGoogle Scholar
Hoover, A. & Miller, L. 2015 A numerical study of the benefits of driving jellyfish bells at their natural frequency. J. Theor. Biol. 374, 13–25.Google Scholar
How, T. Y., Lowengrub, J. S. & Shelley, M. J. 1994 Removing the stiffness from interfacial flows with surface tension. J. Comput. Phys. 114 (2), 312–338.Google Scholar
Huang, Y., Nitsche, M. & Kanso, E. 2015 Stability versus maneuverability in hovering flight. Phys. Fluids 27 (6), 061706.CrossRefGoogle Scholar
Huang, Y., Nitsche, M. & Kanso, E. 2016 Hovering in oscillatory flows. J. Fluid Mech. 804, 531–549.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.Google Scholar
Krasny, R. 1986 Desingularization of periodic vortex sheet roll-up. J. Comput. Phys. 65, 292–313.Google Scholar
Lentink, D. 2013 Flying like a fly. Nature 498 (7454), 306–307.CrossRefGoogle ScholarPubMed
Lighthill, M. J. 1960 Note on the swimming of slender fish. J. Fluid Mech. 9, 305–317.Google Scholar
Liu, B., Ristroph, L., Weathers, A., Childress, S. & Zhang, J. 2012 Intrinsic stability of a body hovering in an oscillating airflow. Phys. Rev. Lett. 108 (6), 068103.Google Scholar
Lucas, C. H., Pitt, K. A., Purcell, J. E., Lebrato, M. & Condon, R. H. 2011 What’s in a jellyfish? Proximate and elemental composition and biometric relationships for use in biogeochemical studies. ESA Ecol. 92 (8), 1704.Google Scholar
Ma, K. Y., Chirarattananon, P., Fuller, S. B. & Wood, R. J. 2013 Controlled flight of a biologically inspired, insect-scale robot. Science 340 (6132), 603–607.Google Scholar
Martinsson, P. G. & Rokhlin, V. 2007 An accelerated kernel-independent fast multipole method in one dimension. SIAM J. Sci. Comput. 29 (3), 1160–1178.CrossRefGoogle Scholar
Mason, J. C. & Handscomb, D. C. 2002 Chebyshev Polynomials. CRC.Google Scholar
McHenry, M. J. & Jed, J. 2003 The ontogenetic scaling of hydrodynamics and swimming performance in jellyfish (Aurelia aurita). J. Expl. Biol. 206 (22), 4125–4137.CrossRefGoogle ScholarPubMed
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
Miller, L. A. & Peskin, C. S. 2005 A computational fluid dynamics of clap and fling’ in the smallest insects. J. Exp. Biol. 208 (2), 195–212.Google Scholar
Moore, D. W. 1979 The spontaneous appearance of a singularity in the shape of an evolving vortex sheet. Proc. R. Soc. Lond. A 365 (1720), 105–119.Google 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.Google Scholar
Pan, X. & Sheng, X. 2013 Hierarchical interpolative decomposition multilevel fast multipole algorithm for dynamic electromagnetic simulations. Prog. Electromag. Res. 134, 79–94.Google Scholar
Peng, J. & Alben, S. 2012 Effects of shape and stroke parameters on the propulsion performance of an axisymmetric swimmer. Bioinspir. Biomim. 7 (1), 016012.Google Scholar
Peskin, C. S. 2002 The immersed boundary method. Acta Numerica 11, 479–517.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.Google Scholar
Quinn, D. B., Moored, K. W., Dewey, P. A. & Smits, A. J. 2014 Unsteady propulsion near a solid boundary. J. Fluid Mech. 742, 152–170.Google Scholar
Ristroph, L., Bergou, A. J., Ristroph, G., Coumes, K., Berman, G. J., Guckenheimer, J., Wang, Z. J. & Cohen, I. 2010 Discovering the flight autostabilizer of fruit flies by inducing aerial stumbles. Proc. Natl Acad. Sci. USA 107 (11), 4820–4824.Google Scholar
Ristroph, L. & Childress, S. 2014 Stable hovering of a jellyfish-like flying machine. J. R. Soc. Interface 11 (92), 20130992.Google Scholar
Ristroph, L., Ristroph, G., Morozova, S., Bergou, A. J., Chang, S., Guckenheimer, J., Wang, Z. J. & Cohen, I. 2013 Active and passive stabilization of body pitch in insect flight. J. R. Soc. Interface 10 (85), 20130237.Google Scholar
Rosenhead, L. 1931 The formation of vortices from a surface of discontinuity. Proc. R. Soc. Lond. A 134 (823), 170–192.Google Scholar
Saffman, P. G. 1993 Vortex Dynamics. Cambridge University Press.Google Scholar
Shelley, M. J. 1992 A study of singularity formation in vortex-sheet motion by a spectrally accurate vortex method. J. Fluid Mech. 244, 493–526.Google Scholar
Shumway, S. E. & Parsons, G. J. 2011 Scallops: Biology, Ecology and Aquaculture, vol. 40. Elsevier.Google Scholar
Spagnolie, S. & Shelley, M. J. 2009 Shape-changing bodies in fluid: hovering, ratcheting, and bursting. Phys. Fluids 21 (1), 013103.Google Scholar
Stengel, R. F. 2015 Flight Dynamics. Princeton University Press.Google Scholar
Sun, M. & Xiong, Y. 2005 Dynamic flight stability of a hovering bumblebee. J. Exp. Biol. 208 (3), 447–459.CrossRefGoogle ScholarPubMed
Vandenberghe, N., Childress, S. & Zhang, J. 2006 On unidirectional flight of a free flapping wing. Phys. Fluids 18 (1), 014102.Google Scholar
Vandenberghe, N., Zhang, J. & Childress, S. 2004 Symmetry breaking leads to forward flapping flight. J. Fluid Mech. 506, 147–155.Google Scholar
Watkins, S., Milbank, J., Loxton, B. J. & Melbourne, W. H. 2006 Atmospheric winds and their implications for microair vehicles. AIAA J. 44 (11), 2591–2600.Google Scholar
Weathers, A., Folie, B., Liu, B., Childress, S. & Zhang, J. 2010 Hovering of a rigid pyramid in an oscillatory airflow. J. Fluid Mech. 650, 415–425.CrossRefGoogle Scholar
Wu, T. Y. 1961 Swimming of a waving plate. J. Fluid Mech. 10 (03), 321–344.CrossRefGoogle Scholar

Fang et al. supplementary movie

Simulation of a hovering flyer (corresponding to Fig. 2). The black arrows denote instantaneous flow velocities. Red (positive) and blue (negative) vortices, which represent the coarse-grained vortex sheets, show the complex wake. A linearly diminishing downward background flow is added for t≤3, and turned off for t≤3. The flyer’s initial position is shown in gray as a frame reference. The “soup” of vortices stays always around the wings.

Download Fang et al. supplementary movie(Video)
Video 4.4 MB

Fang et al. supplementary movie

Response of a bottom-heavy flyer, with h=−2 (corresponding to Fig. 5 bottom row), to a transitory external torque perturbation applied at tc=3.5 (Eq. 4.2). After the torque impulse, the flyer tilts to a large angle, comes back upright, and then overshoots.

Download Fang et al. supplementary movie(Video)
Video 3.4 MB

Fang et al. supplementary movie

Response of a top-heavy flyer, with h=1 (corresponding to Fig. 5 top row), to a transitory external torque perturbation applied at tc =3.5 (Eq. 4.2). After the torque impulse, the flyer tilts slightly and then the angle keeps increasing slowly.

Download Fang et al. supplementary movie(Video)
Video 3.3 MB

Fang et al. supplementary movie

When θa increases, the flyer’s lift exceeds its weight and the flyer starts to ascend. In each stroke the flyer generates one vortex quadrapole, consisting of two symmetric near-dipoles that move sideways and downwards, carrying downward momentum (corresponding to Fig. 6a).

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Video 2.9 MB

Fang et al. supplementary movie

Free ascending recovery flight of bottom-heavy flyer, with h=−1.5 (corresponding to Fig. 12 left panel). An external torque perturbation (ε=400) is applied at tc=13.5 during the steady ascent of the flyer. The grey dotted line shows the flyer’s trajectory.

Download Fang et al. supplementary movie(Video)
Video 2.2 MB

Fang et al. supplementary movie

Free ascending recovery flight of bottom-heavy flyer, with h=−2 (corresponding to Fig. 12 right panel). An external torque perturbation (ε=200) is applied at tc=13.5 during the steady ascent of the flyer. The grey dotted line shows the flyer’s trajectory.

Download Fang et al. supplementary movie(Video)
Video 2.1 MB