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A simple vortex-loop-based model for unsteady rotating wings

Published online by Cambridge University Press:  18 October 2019

Juhi Chowdhury  and
Matthew J. Ringuette Open the ORCID record for Matthew J. Ringuette [Opens in a new window]
Juhi Chowdhury
Affiliation:
Department of Mechanical and Aerospace Engineering, University at Buffalo, The State University of New York, NY 14260, USA
Matthew J. Ringuette*
Affiliation:
Department of Mechanical and Aerospace Engineering, University at Buffalo, The State University of New York, NY 14260, USA
*
†Email address for correspondence: ringum@buffalo.edu
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Abstract

An analytical model is developed for the lift force produced by unsteady rotating wings; this configuration is a simple representation of a flapping wing. Modelling this is important for the aerodynamic and control-system design for bio-inspired drones. Such efforts have often been limited to being two-dimensional, semi-empirical, sometimes computationally expensive, or quasi-steady. The current model is unsteady and three-dimensional, yet simple to implement, requiring knowledge of only the wing kinematics and geometry. Rotating wings produce a vortex loop consisting of the root vortex, leading-edge vortex, tip vortex and trailing-edge vortex, which grows with time. This is modelled as a tilted planar loop, geometrically specified by the wing size, orientation and motion. By equating the angular impulse of the vortex loop to that of the fluid volume driven by the wing, the circulatory lift force is derived. Potential flow theory gives the fluid-inertial lift. Adding these two contributions yields the total lift formula. The model shows good agreement with a range of experimental and computational cases. Also, a steady-state lift model is developed that compares well with previous work for various angles of attack.

JFM classification

Biological Fluid Dynamics: Swimming/flying Vortex Flows: Vortex dynamics Wakes/Jets: Separated flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 880 , 10 December 2019 , pp. 1020 - 1035
Copyright
© 2019 Cambridge University Press 

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References

Ansari, S. A., Żbikowski, R. & Knowles, K. 2006a Non-linear unsteady aerodynamic model for insect-like flapping wings in the hover. Part 1: Methodology and analysis. J. Aerosp. Engng 220 (2), 61–83.Google Scholar
Ansari, S. A., Żbikowski, R. & Knowles, K. 2006b Non-linear unsteady aerodynamic model for insect-like flapping wings in the hover. Part 2: Implementation and validation. J. Aerosp. Engng 220 (3), 169–186.Google Scholar
Babinsky, H., Stevens, R. J., Jones, A. R., Bernal, L. P. & Ol, M. V. 2016 Low order modelling of lift forces for unsteady pitching and surging wings. In 54th AIAA Aerospace Sciences Meeting. AIAA SciTech Forum (2016-0290).Google Scholar
Berman, G. J. & Wang, Z. J. 2007 Energy-minimizing kinematics in hovering insect flight. J. Fluid Mech. 582, 153–168.Google Scholar
Bhat, S. S., Zhao, J., Sheridan, J., Hourigan, K. & Thompson, M. C. 2019 Uncoupling the effects of aspect ratio, Reynolds number and Rossby number on a rotating insect-wing planform. J. Fluid Mech. 859, 921–948.Google Scholar
Carr, Z. R., DeVoria, A. C. & Ringuette, M. J. 2015 Aspect-ratio effects on rotating wings: circulation and forces. J. Fluid Mech. 767, 497–525.Google Scholar
Chen, D., Kolomenskiy, D., Onishi, R. & Liu, H. 2018 Versatile reduced-order model of leading-edge vortices on rotary wings. Phys. Rev. Fluids 3 (114703), 1–13.Google Scholar
Darakananda, D. & Eldredge, J. D. 2019 A versatile taxonomy of low-dimensional models for unsteady aerodynamics. J. Fluid Mech. 858, 917–948.Google Scholar
Dickinson, M. H., Lehmann, F. O. & Sane, P. O. 1999 Wing rotation and the aerodynamic basis of insect flight. Science 284 (5422), 1954–1960.Google Scholar
Eldredge, J. D. & Darakananda, D. 2015 Reduced-order two- and three-dimensional vortex modeling of unsteady separated flows. In 53rd AIAA Aerospace Sciences Meeting. AIAA SciTech Forum (2015-1749).Google Scholar
Eldredge, J. D. & Jones, A. R. 2019 Leading-edge vortices: mechanics and modeling. Annu. Rev. Fluid Mech. 51, 75–104.Google Scholar
Galler, J. N., Weymouth, G. D. & Rival, D. E. 2019 Progress towards modelling unsteady forces using a drift-volume approach. In 57th AIAA Aerospace Sciences Meeting. AIAA SciTech Forum (2019-1147).Google Scholar
Garmann, D. J. & Visbal, M. R. 2014 Dynamics of revolving wings for various aspect ratios. J. Fluid Mech. 748, 932–956.Google Scholar
Gharib, M., Rambod, E. & Shariff, K. 1998 A universal time scale for vortex ring formation. J. Fluid Mech. 360, 121–140.Google Scholar
Harbig, R. R., Sheridan, J. & Thompson, M. C. 2013 Reynolds number and aspect ratio effects on the leading-edge vortex for rotating insect wing planforms. J. Fluid Mech. 717, 166–192.10.1017/jfm.2012.565Google Scholar
Hemati, M. S., Eldredge, J. D. & Speyer, J. L. 2014 Improving vortex models via optimal control theory. J. Fluids Struct. 49, 99–111.Google Scholar
Jardin, T. & Colonius, T. 2018 On the lift-optimal aspect ratio of a revolving wing at low Reynolds number. J. R. Soc. Interface 15 (143), 20170933.Google Scholar
Jardin, T., Farcy, A. & David, L. 2012 Three-dimensional effects in hovering flapping flight. J. Fluid Mech. 702, 102–125.Google Scholar
Jiao, Z., Zhao, L., Shang, Y. & Sun, X. 2018 Generic analytical thrust-force model for flapping wings. AIAA J. 56 (2), 581–593.Google Scholar
Jones, A. R. & Babinsky, H. 2011 Reynolds number effects on leading edge vortex development on a waving wing. Exp. Fluids 51 (1), 197–210.Google Scholar
Jones, A. R., Manar, F., Phillips, N., Nakata, T., Bomphrey, R., Ringuette, M. J., Percin, M., van Oudheusden, B. & Palmer, J. 2016 Leading edge vortex evolution and lift production on rotating wings (invited). In 54th AIAA Aerospace Sciences Meeting. AIAA SciTech Forum (2016-0288).Google Scholar
von Kármán, T. & Sears, W. R. 1938 Airfoil theory for non-uniform motion. J. Aeronaut. Sci. 5 (10), 379–390.Google Scholar
Kim, D. & Gharib, M. 2010 Experimental study of three-dimensional vortex structures in translating and rotating plates. Exp. Fluids 49 (1), 329–339.Google Scholar
Kruyt, J. W., van Heijst, G. F., Altshuler, D. L. & Lentink, D. 2015 Power reduction and the radial limit of stall delay in revolving wings of different aspect ratio. J. R. Soc. Interface 12, 20150051.Google Scholar
Lee, J., Choi, H. & Kim, H. Y. 2015 A scaling law for the lift of hovering insects. J. Fluid Mech. 782, 479–490.Google Scholar
Lee, Y. J., Lua, K. B. & Lim, T. T. 2016a Aspect ratio effects on revolving wings with Rossby number consideration. Bioinspir. Biomim. 11, 056013.Google Scholar
Lee, Y. J., Lua, K. B., Lim, T. T. & Yeo, K. S. 2016b A quasi-steady aerodynamic model for flapping flight with improved adaptability. Bioinspir. Biomim. 11, 036005.Google Scholar
Lentink, D. & Dickinson, M. H. 2009 Rotational accelerations stabilize leading edge vortices on revolving fly wings. J. Expl Biol. 212, 2705–2719.Google Scholar
Li, C., Dong, H. & Cheng, B. 2017 Effects of aspect ratio and angle of attack on tip vortex structures and aerodynamic performance for rotating flat plates. In 47th AIAA Fluid Dynamics Conference. AIAA Aviation Forum (2017-3645).Google Scholar
Li, J. & Wu, Z.-N. 2016 A vortex force study for a flat plate at high angle of attack. J. Fluid Mech. 801, 222–249.Google Scholar
Limacher, E., Morton, C. & Wood, D. 2016 On the trajectory of leading-edge vortices under the influence of coriolis acceleration. J. Fluid Mech. 800, R1.Google Scholar
Manar, F. H., Mancini, P., Mayo, D. & Jones, A. R. 2016 Comparison of rotating and translating wings: force production and vortex characteristics. AIAA J. 54 (2), 519–530.Google Scholar
Mancini, P., Manar, F. H. & Jones, A. R. 2015 A semi-empirical approach to modeling lift production. In 53rd AIAA Aerospace Sciences Meeting. AIAA SciTech Forum (2015-1748).Google Scholar
Nabawy, M. R. A. & Crowther, W. J. 2014 On the quasi-steady aerodynamics of normal hovering flight part II: model implementation and evaluation. J. R. Soc. Interface 11, 20131197.Google Scholar
Nguyen, A. T., Kim, J.-K., Han, J.-S. & Han, J.-H. 2016 Extended unsteady vortex-lattice method for insect flapping wings. J. Aircraft 53 (6), 1709–1718.Google Scholar
Ol, M. & Babinsky, H.2016 Extensions of fundamental flow physics to practical MAV aerodynamics. Tech. Rep. TR-AVT-202. NATO STO.Google Scholar
Ozen, C. A. & Rockwell, D. 2012a Flow structure on a rotating plate. Exp. Fluids 52, 207–223.Google Scholar
Ozen, C. A. & Rockwell, D. 2012b Three-dimensional vortex structure on a rotating wing. J. Fluid Mech. 707, 541–550.Google Scholar
Pedersen, C. B. & Żbikowski, R. 2006 An indicial-Polhamus aerodynamic model of insect-like flapping wings in hover. In Flow Phenomena in Nature (ed. Liebe, R.), Design and Nature, vol. 2, pp. 606–665. WIT Press.Google Scholar
Percin, M. & van Oudheusden, B. W. 2015 Three-dimensional flow structures and unsteady forces on pitching and surging revolving flat plates. Exp. Fluids 56, 47.Google Scholar
Phillips, N., Knowles, K. & Bomphrey, R. J. 2015 The effect of aspect ratio on the leading-edge vortex over an insect-like flapping wing. Bioinspir. Biomim. 10 (5), 056020.Google Scholar
Pitt Ford, C. W. & Babinsky, H. 2013 Lift and the leading-edge vortex. J. Fluid Mech. 720, 280–313.Google Scholar
Polhamus, E. C.1966 A concept of the vortex lift of sharp-edge delta wings based on a leading-edge-suction analogy. NASA Tech. Rep. TN D-3767.Google 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
Ramesh, K., Gopalarathnam, A., Granlund, K., Ol, M. V. & Edwards, J. R. 2014 Discrete-vortex method with novel shedding criterion for unsteady aerofoil flows with intermittent leading-edge vortex shedding. J. Fluid Mech. 751, 500–538.Google Scholar
Roccia, B. A., Preidikman, S., Massa, J. C. & Mook, D. T. 2013 Modified unsteady vortex-lattice method to study flapping wings in hover flight. AIAA J. 51 (11), 2628–2642.Google Scholar
Sane, S. P. & Dickinson, M. H. 2002 The aerodynamic effects of wing rotation and a revised quasi-steady model of flapping flight. J. Expl Biol. 205, 1087–1096.Google Scholar
Sedov, L. 1965 Two-Dimensional Problems in Hydrodynamics and Aerodynamics. Interscience.Google Scholar
Shariff, K. & Leonard, A. 1992 Vortex rings. Annu. Rev. Fluid Mech. 24 (1), 235–279.Google Scholar
Sullivan, I. S., Niemela, J. J., Hershberger, R. E., Bolster, D. & Donnelly, R. J. 2008 Dynamics of thin vortex rings. J. Fluid Mech. 609, 319–347.Google Scholar
Sun, M. & Wu, J. 2004 Large aerodynamic forces on a sweeping wing at low Reynolds number. Acta Mechanica Sin. 20, 24.Google Scholar
Taha, H. E., Hajja, M. R. & Beran, P. S. 2014 State-space representation of the unsteady aerodynamics of flapping flight. Aerosp. Sci. Technol. 34, 1–11.Google Scholar
Theodorsen, T.1935 General theory of aerodynamic instability and the mechanism of flutter. NACA Tech. Rep. 496.Google Scholar
Wagner, H. 1925 Über die entstehung des dynamischen auftriebs von tragflügeln. Z. Angew. Math. Mech. 5, 17–35.Google Scholar
Wang, C. & Eldredge, J. D. 2013 Low-order phenomenological modeling of leading-edge vortex formation. Theor. Comput. Fluid Dyn. 27, 577–598.Google Scholar
Wolfinger, M. & Rockwell, D. 2015 Transformation of flow structure on a rotating wing due to variation of radius of gyration. Exp. Fluids 56 (137), 1–18.Google Scholar
Wong, J. G., Gillespie, G. & Rival, D. E. 2018 Circulation redistribution in leading-edge vortices with spanwise flow. AIAA J. 56 (10), 3857–3862.Google Scholar
Wu, J.-Z., Ma, H. Y. & Zhou, M. D. 2006 Vorticity and Vortex Dynamics. Springer.Google Scholar
Xia, X. & Mohseni, K. 2013 Lift evaluation of a two-dimensional pitching flat plate. Phys. Fluids 25, 091901.Google Scholar
Yan, Z., Taha, H. E. & Hajja, M. R. 2014 Geometrically-exact unsteady model for airfoils undergoing large amplitude maneuvers. Aerosp. Sci. Technol. 39, 293–306.Google Scholar
Żbikowski, R. 2002 On aerodynamic modelling of an insect-like flapping wing in hover for micro air vehicles. Phil. Trans. R. Soc. Lond. A 360 (1791), 273–290.Google Scholar