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On the static aeroelastic instability of an inverted plate in wall effect: a continuum model and its solution approximation

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Abstract

This paper aims at the static instability of a thin, flexible plate loaded by low-speed airflow in the wall effect. The present plate model has the flow impinging on its leading and free edge and is called an inverted cantilevered plate. A theoretical continuum model is first established, which mathematically presents such an instability problem as a mathematical function approximation problem within the framework of differential operators. The mirror image method considers the wall’s effect on fluids. The fluid force is presented as a Possio integral equation composed of Hilbert and Tricomi operators. We bring no approximation at the first equation level, and the derived instability equation is on the continuum. A convergent numerical scheme based on the Weierstrass theorem and the least square method is proposed to solve the instability equation. The results show that the current analysis strategy successfully predicts the plate instability in the wall effect compared with other theoretical and experimental studies. The confinement of the wall plays a destabilizing effect, and the critical flow velocity significantly decreases as the confinement is increased; however, the plate instability modes are not sensitive to wall confinement. The plate instability modes are close to the plate’s first natural ones and not sensitive to the channel character parameters. This conclusion allows further theoretical exploration of an approximation of the instability boundary from the obtained instability equation, a five-second scaling relation. The present plate aeroelasticity model on the continuum and its solution approximation may provide a new idea and essential reference for other instability problems.

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References

  1. Dowell, E.H.: Aeroelasticity of Plates and Shells. Noordhoff International Publishing, Leyden (1975)

    MATH  Google Scholar 

  2. Paidoussis, M.P.: Fluid-Structure Interactions. Slender Structures and Axial Flow, vol. 2, 1st edn. Elsevier Academic Press, London (2004)

    Google Scholar 

  3. Dowell, E.H.: Nonlinear oscillations of a fluttering plate II. AIAA J. 5, 1856–1862 (1967). https://doi.org/10.2514/6.1967-13

    Article  Google Scholar 

  4. Algazin, S.D., Kijko, I.A.: Aeroelastic vibrations and stability of plates and shells, vol. 25. Walter de Gruyter GmbH and Co KG (2014). https://doi.org/10.1515/9783110338379.20

  5. Li, P., Yang, Y.R., Xu, W.: Nonlinear dynamics analysis of a two-dimensional thin panel with an external forcing in incompressible subsonic flow. Nonlinear Dyn. 67, 1251–1267 (2012). https://doi.org/10.1007/s11071-011-0162-8

    Article  MathSciNet  Google Scholar 

  6. Li, P., Zhang, D.C., Li, Z.W., et al.: Bifurcations and post-critical behaviors of a nonlinear curved plate in subsonic airflow. Arch. Appl. Mech. 89(2), 343–362 (2019). https://doi.org/10.1007/s00419-018-1471-x

    Article  Google Scholar 

  7. Li, P., Li, Z.W., Liu, S., et al.: Non-linear limit cycle flutter of a plate with Hertzian contact in axial flow. J. Fluids Struct. 81, 131–160 (2018). https://doi.org/10.1016/j.jfluidstructs.2018.04.014

    Article  Google Scholar 

  8. Li, P., Li, Z.W., Dai, C.D., et al.: On the non-linear dynmics of a forced plate with boundary conditions correction in subsonic flow. Appl. Math. Model. 64, 15–46 (2018). https://doi.org/10.1016/j.apm.2018.07.012

    Article  MathSciNet  MATH  Google Scholar 

  9. Zhang, D.C., Li, P., Zhu, Y.Z., et al.: Aeroleastic instability of an inverted cantilevered plate with cracks in axial subsonic airflow. Appl. Math. Model. 107, 782–801 (2022). https://doi.org/10.1016/j.apm.2022.03.019

    Article  MathSciNet  MATH  Google Scholar 

  10. Guo, C.Q., Païdoussis, M.P.: Stability of rectangular plates with free side-edges in two-dimensional inviscid channel flow. J. Appl. Mech. 67(1), 171–176 (2000). https://doi.org/10.1115/1.321143

    Article  MATH  Google Scholar 

  11. Watanabe, Y., Suzuki, S., Sugihara, M., et al.: A experimental study of paper flutter. J. Fluids Struct. 16, 529–542 (2002). https://doi.org/10.1006/jfls.2001.0435

    Article  Google Scholar 

  12. Watanabe, Y., Isogai, K., Suzuki, S., et al.: A theoretical study of paper flutter. J. Fluids Struct. 16, 543–560 (2002). https://doi.org/10.1006/jfls.2001.0436

    Article  Google Scholar 

  13. Doare, O., Michelin, S.: Piezoelectric coupling in energy-harvesting fluttering flexible plates: linear stability analysis and conversion efficiency. J. Fluids Struct. 27(8), 1357–1375 (2011). https://doi.org/10.1016/j.jfluidstructs.2011.04.008

    Article  Google Scholar 

  14. Tang, D.M., Dowell, E.H.: Aeroelastic response and energy harvesting from a cantilevered piezoelectric laminated plate. J. Fluids Struct. 76, 14–36 (2018). https://doi.org/10.1016/j.jfluidstructs.2017.09.007

    Article  Google Scholar 

  15. Lucey, A.D.: The excitation of waves on a flexible panel in a uniform flow. Philos. Trans. R. Soc. Lond, A 356, 2999–3039 (1998). https://doi.org/10.1098/rsta.1998.0306

    Article  MathSciNet  MATH  Google Scholar 

  16. de Breuker, R., Abdalla, M.M., Gürdal, Z.: Flutter of partially rigid cantilevered plates in axial flow. AIAA J. 46, 936–946 (2008). https://doi.org/10.2514/1.31887

    Article  Google Scholar 

  17. Zhang, J., Liu, N.S., Lu, X.Y.: Locomotion of a passively flapping flat plate. J. Fluid Mech. 659, 43–68 (2010). https://doi.org/10.1017/S0022112010002387

    Article  MathSciNet  MATH  Google Scholar 

  18. Dugundji, J., Dowell, E.H., Perkin, B.: Subsonic flutter of panels on continuous elastic foundations. AIAA J. 5, 1146–1154 (1963). https://doi.org/10.2514/3.1738

    Article  Google Scholar 

  19. Kornecki, A., Dowell, E.H., O’brien, J.: On the aeroelastic instability of two-dimensional panels in uniform incompressible flow. J. Sound Vib. 47(2), 163–178 (1976). https://doi.org/10.1016/0022-460X(76)90715-X

    Article  MATH  Google Scholar 

  20. Huang, L.: Flutter of cantilevered plates in axial flow. J. Fluids Struct. 9(2), 127–147 (1995). https://doi.org/10.1006/jfls.1995.1007

    Article  Google Scholar 

  21. Shelley, M.J., Zhang, J.: Flapping and bending bodies interacting with fluid flows. Annu. Rev. Fluid Mech. 43, 449–465 (2011). https://doi.org/10.1146/annurev-fluid-121108-145456

    Article  MathSciNet  MATH  Google Scholar 

  22. Yu, Y., Liu, Y., Amandolese, X.: A review on fluid-induced flag vibrations. Appl. Mech. Rev. 71(1), 010801 (2019). https://doi.org/10.1115/1.4042446

    Article  Google Scholar 

  23. Michelin, S., Doaré, O.: Energy harvesting efficiency of piezoelectric flags in axial flows. J. Fluid Mech. 714, 489–504 (2013). https://doi.org/10.1017/jfm.2012.494

    Article  MathSciNet  MATH  Google Scholar 

  24. Shoele, K., Mittal, R.: Energy harvesting by flow-induced flutter in a simple model of an inverted piezoelectric flag. J. Fluid Mech. 790, 582–606 (2016). https://doi.org/10.1017/jfm.2016.40

    Article  MathSciNet  MATH  Google Scholar 

  25. Silva-Leon, J., Cioncolini, A., Nabawy, M.R., et al.: Simultaneous wind and solar energy harvesting with inverted flags. Appl. Energy 239, 846–858 (2019). https://doi.org/10.1016/j.apenergy.2019.01.246

    Article  Google Scholar 

  26. Mazharmanesh, S., Young, J., Tian, F.B., et al.: Energy harvesting of two inverted piezoelectric flags in tandem, side-by-side and staggered arrangements. Int. J. Heat Fluid Flow 83, 108589 (2020). https://doi.org/10.1016/j.ijheatfluidflow.2020.108589

    Article  Google Scholar 

  27. Park, S.G., Kim, B., Chang, C.B., et al.: Enhancement of heat transfer by a self-oscillating inverted flag in a Poiseuille channel flow. Int. J. Heat Mass Transf. 96, 362–370 (2016). https://doi.org/10.1063/1.5037747

    Article  Google Scholar 

  28. Fan, B.: Fluid-structure Interactions of Inverted Leaves and Flags. Doctoral dissertation, California Institute of Technology (2015)

  29. Buchak, P., Eloy, C., Reis, P.M.: The clapping book: wind-driven oscillations in a stack of elastic sheets. Phys. Rev. Lett. 105(19), 194301 (2010). https://doi.org/10.1103/PhysRevLett.105.194301

    Article  Google Scholar 

  30. Tavallaeinejad, M., Païdoussis, M.P., Legrand, M.: Nonlinear static response of low-aspect-ratio inverted flags subjected to a steady flow. J. Fluids Struct. 83, 413–428 (2018). https://doi.org/10.1016/j.jfluidstructs.2018.09.003

    Article  Google Scholar 

  31. Tavallaeinejad, M., Païdoussis, M.P., Legrand, M., et al.: Instability and the post-critical behaviour of two-dimensional inverted flags in axial flow. J. Fluid Mech. 890A14, 1–14 (2020). https://doi.org/10.1017/jfm.2020.111

    Article  MathSciNet  MATH  Google Scholar 

  32. Serry, M., Tuffaha, A.: Static stability analysis of a thin plate with a fixed trailing edge in axial subsonic flow: Possion integral equation approach. Appl. Math. Model. 63, 644–659 (2018). https://doi.org/10.1016/j.apm.2018.07.005

    Article  MathSciNet  MATH  Google Scholar 

  33. Kim, D., Cossé, J., Cerdeira, C.H., et al.: Flapping dynamics of an inverted flag. J. Fluid Mech. 736R1, 1–12 (2013). https://doi.org/10.1017/jfm.2013.555

    Article  MATH  Google Scholar 

  34. Goza, A., Colonius, T., Sader, J.E.: Global modes and nonlinear analysis of inverted-flag flapping. J. Fluid Mech. 857, 312–344 (2016). https://doi.org/10.1017/jfm.2018.728

    Article  MathSciNet  MATH  Google Scholar 

  35. Zhang, D., Liang, S., Li, P., et al.: A numerical and experimental study on the divergence instability of an inverted cantilevered plate in wall effect. Arch. Appl. Mech. 90, 1509–1528 (2020). https://doi.org/10.1007/s00419-020-01681-8

    Article  Google Scholar 

  36. Ryu, J., Park, S.G., Kim, B., et al.: Flapping dynamics of an inverted flag in a uniform flow. J. Fluids Struct. 57, 159–169 (2015). https://doi.org/10.1016/j.jfluidstructs.2015.06.006

    Article  Google Scholar 

  37. Tang, C., Liu, N.S., Lu, X.Y.: Dynamics of an inverted flexible plate in a uniform flow. Phys. Fluids 27(7), 073601 (2015). https://doi.org/10.1063/1.4923281

    Article  Google Scholar 

  38. Sader, J.E., Cossé, J., Kim, D., Fan, B., et al.: Large-amplitude flapping of an inverted flag in a uniform steady flow-a vortex-induced vibration. J. Fluid Mech. 793, 524–555 (2016). https://doi.org/10.1017/jfm.2016.139

    Article  MathSciNet  MATH  Google Scholar 

  39. Balakrishnan, A.V., IIiff, K.W.: Continuum aeroelastic model for inviscid subsonic bending-torsion wing flutter. J. Aerosp. Eng. 20, 152–164 (2007). https://doi.org/10.1061/(ASCE)0893-1321(2007)20:(3152)

    Article  Google Scholar 

  40. Balakrishnan, A.V.: Aeroelasticity: The Continuum Theory. Springer (2012)

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant Nos.: 12072298; 11772273).

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Authors and Affiliations

  1. School of Mechanics and Aerospace Engineering, Applied Mechanics and Structure Safety Key Laboratory of Sichuan Province, Southwest Jiaotong University, Chengdu, 610031, People’s Republic of China

    Peng Li, Dechun Zhang, Hong Yin, Junzhe Cui & Yiren Yang

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Li, P., Zhang, D., Yin, H. et al. On the static aeroelastic instability of an inverted plate in wall effect: a continuum model and its solution approximation. Arch Appl Mech 93, 1825–1840 (2023). https://doi.org/10.1007/s00419-022-02358-0

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