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Length and stiffness effects of the attached flexible plate on the flow over a traveling wavy foil

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Abstract

Flow over a traveling wavy foil attached with a flexible plate has been numerically investigated using the lattice Boltzmann method combined with the immersed boundary method. The influence of the flexibility and length of the caudal fin on the locomotion of swimming fish through this simplified model, whereas the fish body is modeled by the undulating foil and the caudal fin by the plate passively flapping as a consequence of fluid-structure interaction. It is found that the plate flexibility denoted by the bending stiffness, as well as the length ratio of tail and body, plays an important role in improving thrust generation and propulsive efficiency. It is also revealed that there exists a parameter region of the plate length and stiffness, in which positive propeller efficiency can be achieved. The effect of the passively flapping flexible plate on the pressure field and the vortex production on the wake is further discussed. It is found that when the length ratio of caudal fin and body is greater than 0.2, a reverse von Kármán vortex street occurs when the bending stiffness is about greater than 1.0, and a great thrust is generated as a result of a large pressure difference occurring across the flexible plate. This work provides physical insight into the role of the caudal fin in fish swimming and may inspire the design of robotic fish.

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References

  1. Fish, F.E.: Biomechanics and energetics in aquatic and semiaquatic mammals: platypus to whale. Physiol. Biochem. Zool. 73, 683–698 (2000)

    Article  Google Scholar 

  2. Lauder, G.V., Drucker, E.G.: Morphology and experimental hydrodynamics of fish fin control surfaces. IEEE J. Ocean Eng. 29, 556–571 (2004)

    Article  Google Scholar 

  3. Gray, J.: Studies in animal locomotion VI. The propulsive powers of the dolphin. J. Exp. Biol. 13, 192–199 (1936)

    Article  Google Scholar 

  4. Barrett, D.S., Triantafyllou, M.S., Yue, D.K.P., et al.: Drag reduction in fish-like locomotion. J. Fluid Mech. 392, 183–212 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. Webb, P.W.: The Biology of Fish Swimming. Mechanics and Physiology of Animal Swimming. Cambridge University Press, Cambridge (1994)

    Google Scholar 

  6. Sfakiotakis, M., Lane, D.M., Davies, J.B.C.: Review of fish swimming modes for aquatic locomotion. IEEE J. Ocean Eng. 24, 237–252 (1999)

    Article  Google Scholar 

  7. Ayancik, F., Zhong, Q., Quinn, D.B., et al.: Scaling laws for the propulsive performance of three-dimensional pitching propulsors. J. Fluid Mech. 871, 1117–1138 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lin, X.J., He, G.Y., He, X.Y., et al.: Hydrodynamic studies on two wiggling hydrofoils in an oblique arrangement. Acta Mech. Sin. 34, 446–451 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gopalkrishnan, R., Triantafyllou, M.S., Triantafyllou, G.S., et al.: Active vorticity control in a shear-flow using a flapping foil. J. Fluid Mech. 274, 1–21 (1994)

    Article  Google Scholar 

  10. Maertens, A.P., Gao, A., Triantafyllou, M.S.: Optimal undulatory swimming for a single fish-like body and for a pair of interacting swimmers. J. Fluid Mech. 813, 301–345 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hemmati, A., Smits, A.J.: The effect of pitching frequency on the hydrodynamics of oscillating foils. J. Appl. Mech. 86, 101010 (2019)

  12. Katz, J., Weihs, D.: Hydrodynamic propulsion by large-amplitude oscillation of an airfoil with chordwise flexibility. J. Fluid Mech. 88, 485–497 (1978)

    Article  MATH  Google Scholar 

  13. Thekkethil, N., Sharma, A., Agrawal, A.: Unified hydrodynamics study for various types of fishes-like undulating rigid hydrofoil in a free stream flow. Phys. Fluids 30, 077107 (2018)

    Article  Google Scholar 

  14. Tuncer, I.H., Kaya, M.: Optimization of flapping airfoils for maximum thrust and propulsive efficiency. AIAA J. 43, 2329–2336 (2005)

    Article  Google Scholar 

  15. Zhang, D., Pan, G., Chao, L.M., et al.: Effects of Reynolds number and thickness on an undulatory self-propelled foil. Phys. Fluids 30, 071902 (2018)

    Article  Google Scholar 

  16. Zhao, Z.J., Dou, L.: Effects of the structural relationships between the fish body and caudal fin on the propulsive performance of fish. Ocean Eng. 186, 106117 (2019)

    Article  Google Scholar 

  17. Muller, U.K., VandenHeuvel, B.L.E., Stamhuis, E.J., et al.: Fish foot prints: morphology and energetics of the wake behind a continuously swimming mullet (Chelon labrosus risso). J. Exp. Biol. 200, 2893–2906 (1997)

    Article  Google Scholar 

  18. Thekkethil, N., Sharma, A., Agrawal, A.: Self-propulsion of fishes-like undulating hydrofoil: a unified kinematics based unsteady hydrodynamics study. J. Fluids Struct. 93, 102875 (2020)

    Article  Google Scholar 

  19. Raspa, V., Ramananarivo, S., Thiria, B., et al.: Vortex-induced drag and the role of aspect ratio in undulatory swimmers. Phys. Fluids 26, 041701 (2014)

    Article  Google Scholar 

  20. Liu, N.S., Peng, Y., Liang, Y.W., et al.: Flow over a traveling wavy foil with a passively flapping flat plate. Phys. Rev. E 85, 056316 (2012)

    Article  Google Scholar 

  21. Liu, N.S., Peng, Y., Lu, X.Y.: Length effects of a built-in flapping flat plate on the flow over a traveling wavy foil. Phys. Rev. E 89, 063019 (2014)

  22. Bandyopadhyay, P.R., Castano, J.M., Nedderman, W.H., et al.: Experimental simulation of fish-inspired unsteady vortex dynamics on a rigid cylinder. J. Fluids Eng. Trans. ASME 122, 219–238 (2000)

    Article  Google Scholar 

  23. Yeh, P.D., Alexeev, A.: Biomimetic flexible plate actuators are faster and more efficient with a passive attachment. Acta Mech. Sin. 32, 1001–1011 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Eldredge, J.D., Toomey, J., Medina, A.: On the roles of chord-wise flexibility in a flapping wing with hovering kinematics. J. Fluid Mech. 659, 94–115 (2010)

    Article  MATH  Google Scholar 

  25. Lauder, G.V., Drucker, E.G.: Forces, fishes, and fluids: Hydrodynamic mechanisms of aquatic locomotion. News Physiol. Sci. 17, 235–240 (2002)

    Google Scholar 

  26. Hua, R.N., Zhu, L.D., Lu, X.Y.: Locomotion of a flapping flexible plate. Phys. Fluids 25, 121901 (2013)

    Article  Google Scholar 

  27. Bergmann, M., Iollo, A., Mittal, R.: Effect of caudal fin flexibility on the propulsive efficiency of a fish-like swimmer. Mech. Mach. Theory 9, 4 (2014)

    Google Scholar 

  28. Kancharala, A.K., Philen, M.K.: Study of flexible fin and compliant joint stiffness on propulsive performance: theory and experiments. Bioinspir. Biomim. 9, 036011 (2014)

    Article  Google Scholar 

  29. Krishnadas, A., Ravichandran, S., Rajagopal, P.: Analysis of biomimetic caudal fin shapes for optimal propulsive efficiency. Ocean Eng. 153, 132–142 (2018)

    Article  Google Scholar 

  30. Gaolt, A., Triantafyllou, M.S.: Independent caudal fin actuation enables high energy extraction and control in two-dimensional fish-like group swimming. J. Fluid Mech. 850, 304–335 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  31. Reddy, N.S., Sen, S., Har, C.: Effect of flexural stiffness distribution of a fin on propulsion performance. Mech. Mach. Theory 129, 218–231 (2018)

    Article  Google Scholar 

  32. Ferry, L.A., Lauder, G.V.: Heterocercal tail function in leopard sharks: a three-dimensional kinematic analysis of two models. J. Exp. Biol. 199, 2253–2268 (1996)

    Article  Google Scholar 

  33. Wilga, C.D., Lauder, G.V.: Function of the heterocercal tail in sharks: quantitative wake dynamics during steady horizontal swimming and vertical maneuvering. Am. Zool. 40, 1259 (2000)

    Google Scholar 

  34. Han, P., Lauder, G.V., Dong, H.B.: Hydrodynamics of median-fin interactions in fish-like locomotion: effects of fin shape and movement. Phys. Fluids 32, 011902 (2020)

    Article  Google Scholar 

  35. Dagenais, P., Aegerter, C.M.: How shape and flapping rate affect the distribution of fluid forces on flexible hydrofoils. J. Fluid Mech. 901 (2020)

  36. Luo, Y., Xiao, Q., Shi, G.Y., et al.: A fluid-structure interaction solver for the study on a passively deformed fish fin with non-uniformly distributed stiffness. J. Fluid Struct. 92, 102778 (2020)

    Article  Google Scholar 

  37. Heathcote, S., Gursul, I.: Flexible flapping airfoil propulsion at low Reynolds numbers. AIAA J. 45, 1066–1079 (2007)

    Article  Google Scholar 

  38. David, M.J., Govardhan, R.N., Arakeri, J.H.: Thrust generation from pitching foils with flexible trailing edge flaps. J. Fluid Mech. 828, 70–103 (2017)

    Article  Google Scholar 

  39. Dong, G.J., Lu, X.Y.: Characteristics of flow over traveling wavy foils in a side-by-side arrangement. Phys. Fluids 19, 057107 (2007)

    Article  MATH  Google Scholar 

  40. Wu, J.Z., Pan, Z.L., Lu, X.Y.: Unsteady fluid-dynamic force solely in terms of control-surface integral. Phys. Fluids 17, 098102 (2005)

    Article  MATH  Google Scholar 

  41. Lu, X.Y., Yin, X.Z.: Propulsive performance of a fish-like travelling wavy wall. Acta Mech. 175, 197–215 (2005)

    Article  MATH  Google Scholar 

  42. Borazjani, I., Sotiropoulos, F.: Numerical investigation of the hydrodynamics of anguilliform swimming in the transitional and inertial flow regimes. J. Exp. Biol. 212, 576–592 (2009)

    Article  Google Scholar 

  43. Bainbridge, R., Brown, R.H.J.: An apparatus for the study of the locomotion of fish. J. Exp. Biol. 35, 134–137 (1958)

    Article  Google Scholar 

  44. Hutchins, M., Thoney, D.A., Loiselle, P.V., et al.: Grzimek’s Animal Life Encyclopedia, edited by M. Hutchins. Fishes I-II. Gale Group, Farmington Hills, MI (2003)

    Google Scholar 

  45. Wang, W.J., Huang, H.B., Lu, X.Y.: Self-propelled plate in wakes behind tandem cylinders. Phys. Rev. E 100, 033114 (2019)

    Article  Google Scholar 

  46. Wang, W.J., Huang, H.B., Lu, X.Y.: Optimal chordwise stiffness distribution for self-propelled heaving flexible plates. Phys. Fluids 32, 111905 (2020)

    Article  Google Scholar 

  47. Bedon, C., Amadio, C.: Analytical and numerical assessment of the strengthening effect of structural sealant joints for the prediction of the ltb critical moment in laterally restrained glass beams. Mater. Struct. 49, 2471–2492 (2016)

    Article  Google Scholar 

  48. Chen, S., Doolen, G.D.: Lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech. 30, 329–364 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  49. Xia, Z.H., Connington, K.W., Rapaka, S., et al.: Flow patterns in the sedimentation of an elliptical particle. J. Fluid Mech. 625, 249–272 (2009)

    Article  MATH  Google Scholar 

  50. Guo, Z.L., Zheng, C.G., Shi, B.C.: Discrete lattice effects on the forcing term in the lattice Boltzmann method. Phys. Rev. E 65, 046308 (2002)

  51. Peskin, C.S.: The immersed boundary method. Acta Numer. 11, 479–517 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  52. Tezduyar, T.E., Behr, M., Mittal, S., et al.: A new strategy for finite-element computations involving moving boundaries and interfaces—the deforming-spatial-domain space-time procedure. 2. computation of free-surface flows, 2-liquid flows, and flows with drifting cylinders. Comput. Meth. Appl. Mech. Eng. 94, 353–371 (1992)

  53. Huang, W.X., Shin, S.J., Sung, H.J.: Simulation of flexible filaments in a uniform flow by the immersed boundary method. J. Comput. Phys. 226, 2206–2228 (2007)

  54. Tian, F.B., Luo, H.X., Zhu, L.D., et al.: Interaction between a flexible filament and a downstream rigid body. Phys. Rev. E 82, 026301 (2010)

    Article  Google Scholar 

  55. Shen, L., Zhang, X., Yue, D.K.P., et al.: Turbulent flow over a flexible wall undergoing a streamwise travelling wave motion. J. Fluid Mech. 484, 197–221 (2003)

    Article  MATH  Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grants 92052301, 91752110, 11621202, and 1572312) and Science Challenge Project (Grant TZ2016001).

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Authors and Affiliations

  1. Department of Modern Mechanics, University of Science and Technology of China, Hefei, 230026, China

    Lin Tian, Zhiye Zhao, Wenjiang Wang & Nansheng Liu

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  1. Lin Tian
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Correspondence to Nansheng Liu.

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Executive Editor: Shizhao Wang.

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Tian, L., Zhao, Z., Wang, W. et al. Length and stiffness effects of the attached flexible plate on the flow over a traveling wavy foil. Acta Mech. Sin. 37, 1404–1415 (2021). https://doi.org/10.1007/s10409-021-01110-1

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  • DOI: https://doi.org/10.1007/s10409-021-01110-1

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