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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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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