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Reduced coupled flapping wing-fluid computational model with unsteady vortex wake

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Abstract

Insect flight research is propelled by their unmatched flight capabilities. However, complex underlying aerodynamic phenomena make computational modeling of insect-type flapping flight a challenging task, limiting our ability in understanding insect flight and producing aerial vehicles exploiting same aerodynamic phenomena. To this end, novel mid-fidelity approach to modeling insect-type flapping vehicles is proposed. The approach is computationally efficient enough to be used within optimal design and optimal control loops, while not requiring experimental data for fitting model parameters, as opposed to widely used quasi-steady aerodynamic models. The proposed algorithm is based on Helmholtz–Hodge decomposition of fluid velocity into curl-free and divergence-free parts. Curl-free flow is used to accurately model added inertia effects (in almost exact manner), while expressing system dynamics by using wing variables only, after employing symplectic reduction of the coupled wing-fluid system at zero level of vorticity (thus reducing out fluid variables in the process). To this end, all terms in the coupled body-fluid system equations of motion are taken into account, including often neglected terms related to the changing nature of the added inertia matrix (opposed to the constant nature of rigid body mass and inertia matrix). On the other hand—in order to model flapping wing system vorticity effects—divergence-free part of the flow is modeled by a wake of point vortices shed from both leading (characteristic for insect flight) and trailing wing edges. The approach is evaluated for a numerical case involving fruit fly hovering, while quasi-steady aerodynamic model is used as benchmark tool with experimentally validated parameters for the selected test case. The results indicate that the proposed approach is capable of mid-fidelity accurate calculation of aerodynamic loads on the insect-type flapping wings.

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References

  1. Banazadeh, A., Taymourtash, N.: Adaptive attitude and position control of an insect-like flapping wing air vehicle. Nonlinear Dyn. 85(1), 47–66 (2016). https://doi.org/10.1007/s11071-016-2666-8

    Article  MathSciNet  Google Scholar 

  2. Berman, G.J., Wang, Z.J.: Energy-minimizing kinematics in hovering insect flight. J. Fluid Mech. 582, 153–168 (2007). https://doi.org/10.1017/S0022112007006209

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, D., Kolomenskiy, D., Liu, H.: Closed-form solution for the edge vortex of a revolving plate. J. Fluid Mech. 821, 200–218 (2017). https://doi.org/10.1017/jfm.2017.257

    Article  MathSciNet  MATH  Google Scholar 

  4. Cho, H., Lee, N., Kwak, J.Y., Shin, S.J., Lee, S.: Three-dimensional fluid-structure interaction analysis of a flexible flapping wing under the simultaneous pitching and plunging motion. Nonlinear Dyn. 86(3), 1951–1966 (2016). https://doi.org/10.1007/s11071-016-3007-7

    Article  Google Scholar 

  5. Deng, S., Percin, M., van Oudheusden, B.W., Bijl, H., Remes, B., Xiao, T.: Numerical simulation of a flexible X-wing flapping-wing micro air vehicle. AIAA J. 55(7), 2295–2306 (2017). https://doi.org/10.2514/1.J054816

    Article  Google Scholar 

  6. Dickinson, M.H., Lehmann, F.O., Sane, S.P.: Wing rotation and the aerodynamic basis of insect flight. Science 284(5422), 1954–1960 (1999). https://doi.org/10.1126/science.284.5422.1954

    Article  Google Scholar 

  7. Ellington, C.P., van den Berg, C., Willmott, A.P., Thomas, A.L.R.: Leading-edge vortices in insect flight. Nature 384(6610), 626–630 (1996). https://doi.org/10.1038/384626a0

    Article  Google Scholar 

  8. Fry, S.N., Sayaman, R., Dickinson, M.H.: The aerodynamics of hovering flight in Drosophila. J. Exp. Biol. 208(12), 2303–2318 (2005). https://doi.org/10.1242/jeb.01612

    Article  Google Scholar 

  9. Garcia, D., Ghommem, M., Collier, N., Varga, B.O.N., Calo, V.M.: PyFly: a fast, portable aerodynamics simulator. J. Comput. Appl. Math. 344, 875–903 (2018). https://doi.org/10.1016/j.cam.2018.03.003

    Article  MathSciNet  MATH  Google Scholar 

  10. García-Naranjo, L.C., Vankerschaver, J.: Nonholonomic LL systems on central extensions and the hydrodynamic Chaplygin sleigh with circulation. J. Geom. Phys. 73, 56–69 (2013). https://doi.org/10.1016/j.geomphys.2013.05.002

    Article  MathSciNet  MATH  Google Scholar 

  11. Hammer, P., Altman, A., Eastep, F.: Validation of a discrete vortex method for low Reynolds number unsteady flows. AIAA J. 52(3), 643–649 (2014). https://doi.org/10.2514/1.J052510

    Article  Google Scholar 

  12. Kanso, E., Marsden, J.E., Rowley, C.W., Melli-Huber, J.B.: Locomotion of articulated bodies in a perfect fluid. J. Nonlinear Sci. 15(4), 255–289 (2005). https://doi.org/10.1007/s00332-004-0650-9

  13. Katz, J., Plotkin, A.: Low-Speed Aerodynamics, 2nd edn. Cambridge University Press, Cambridge (2001). https://doi.org/10.1017/CBO9780511810329

    Book  MATH  Google Scholar 

  14. Lambert, T., Abdul Razak, N., Dimitriadis, G.: Vortex lattice simulations of attached and separated flows around flapping wings. Aerospace 4(2), 22 (2017). https://doi.org/10.3390/aerospace4020022

    Article  Google Scholar 

  15. Leonard, N.E.: Stability of a bottom-heavy underwater vehicle. Automatica 33(3), 331–346 (1997). https://doi.org/10.1016/S0005-1098(96)00176-8

    Article  MathSciNet  MATH  Google Scholar 

  16. Liang, Zy., Wei, L., Lu, Jy.: Numerical simulation of a two-dimensional flapping wing in advanced mode. J. Hydrodyn. Ser. B 29(6), 1076–1080 (2017). https://doi.org/10.1016/S1001-6058(16)60801-6

    Article  Google Scholar 

  17. Liu, L., Li, H., Ang, H., Xiao, T.: Numerical investigation of flexible flapping wings using computational fluid dynamics/computational structural dynamics method.Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering, vol. 232, no. 1, pp. 85–95 (2018). https://doi.org/10.1177/0954410016671343. (Publisher: IMECHE)

  18. Liu, L., Sun, M.: The added mass forces in insect flapping wings. J. Theor. Biol. 437, 45–50 (2018). https://doi.org/10.1016/j.jtbi.2017.10.014

    Article  MATH  Google Scholar 

  19. Marsden, J., Misiolek, G., Ortega, J.P., Perlmutter, M., Ratiu, T.: Hamiltonian Reduction by Stages. Springer-Verlag, Berlin, Heidelberg (2007)

    MATH  Google Scholar 

  20. Nguyen, A.T., Kim, J.K., Han, J.S., Han, J.H.: Extended unsteady vortex-lattice method for insect flapping wings. J. Aircr. 53(6), 1709–1718 (2016). https://doi.org/10.2514/1.C033456

    Article  Google Scholar 

  21. Persson, P.O., Willis, D.J., Peraire, J.: Numerical simulation of flapping wings using a panel method and a high-order Navier-Stokes solver. Int. J. Numer. Methods Eng. 89(10), 1296–1316 (2012). https://doi.org/10.1002/nme.3288

  22. Roccia, B.A., Preidikman, S., Balachandran, B.: Computational dynamics of flapping wings in hover flight: a co-simulation strategy. AIAA J. 55(6), 1806–1822 (2017). https://doi.org/10.2514/1.J055137

  23. Sane, S.P.: The aerodynamics of insect flight. J. Exp. Biol. 206(23), 4191–4208 (2003). https://doi.org/10.1242/jeb.00663

    Article  Google Scholar 

  24. Sane, S.P., Dickinson, M.H.: The control of flight force by a flapping wing: lift and drag production. J. Exp. Biol. 204(15), 2607–2626 (2001). https://doi.org/10.1242/jeb.204.15.2607

  25. Sane, S.P., Dickinson, M.H.: The aerodynamic effects of wing rotation and a revised quasi-steady model of flapping flight. J. Exp. Biol. 205(8), 1087–1096 (2002). https://doi.org/10.1242/jeb.205.8.1087

    Article  Google Scholar 

  26. Sutradhar, A., Paulino, G., Gray, L.J.: Symmetric Galerkin Boundary Element Method. Springer-Verlag, Berlin, Heidelberg (2008)

    MATH  Google Scholar 

  27. Suzuki, K., Yoshino, M.: Numerical simulations for aerodynamic performance of a butterfly-like flapping wing-body model with various wing Planforms. Commun. Comput. Phys. (2018). https://doi.org/10.4208/cicp.OA-2016-0238

    Article  MathSciNet  MATH  Google Scholar 

  28. Taha, H.E., Hajj, M.R., Nayfeh, A.H.: Flight dynamics and control of flapping-wing MAVs: a review. Nonlinear Dyn. 70(2), 907–939 (2012). https://doi.org/10.1007/s11071-012-0529-5

    Article  MathSciNet  Google Scholar 

  29. Terze, Z., Müller, A., Zlatar, D.: Lie-group integration method for constrained multibody systems in state space. Multibody Syst. Dyn. 34(3), 275–305 (2015). https://doi.org/10.1007/s11044-014-9439-2

    Article  MathSciNet  MATH  Google Scholar 

  30. Terze, Z., Pandža, V., Kasalo, M., Zlatar, D.: Discrete mechanics and optimal control optimization of flapping wing dynamics for Mars exploration. Aerosp. Sci. Technol. 106, 106131 (2020). https://doi.org/10.1016/j.ast.2020.106131

    Article  Google Scholar 

  31. Terze, Z., Pandža, V., Kasalo, M., Zlatar, D.: Optimized flapping wing dynamics via DMOC approach. Nonlinear Dyn. 103(1), 399–417 (2021). https://doi.org/10.1007/s11071-020-06119-y

    Article  Google Scholar 

  32. Vankerschaver, J., Kanso, E., Marsden, J.E.: The geometry and dynamics of interacting rigid bodies and point vortices. J. Geom. Mech. (2009). https://doi.org/10.3934/jgm.2009.1.223

    Article  MathSciNet  MATH  Google Scholar 

  33. Vankerschaver, J., Kanso, E., Marsden, J.E.: The dynamics of a rigid body in potential flow with circulation. Regul. Chaotic Dyn. 15, 606–629 (2010). https://doi.org/10.1134/S1560354710040143

    Article  MathSciNet  MATH  Google Scholar 

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Funding

This work has been fully supported by Croatian Science Foundation under the project IP-2016-06-6696.

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Authors and Affiliations

  1. Department of Aeronautical Engineering, Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Zagreb, Croatia

    Zdravko Terze, Viktor Pandža, Marijan Andrić & Dario Zlatar

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  1. Zdravko Terze
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  2. Viktor Pandža
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Correspondence to Viktor Pandža.

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Terze, Z., Pandža, V., Andrić, M. et al. Reduced coupled flapping wing-fluid computational model with unsteady vortex wake. Nonlinear Dyn 109, 975–987 (2022). https://doi.org/10.1007/s11071-022-07482-8

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