Hostname: page-component-7479d7b7d-8zxtt Total loading time: 0 Render date: 2024-07-15T12:21:43.078Z Has data issue: false hasContentIssue false

Intermittent swimming of two self-propelled flexible fins with laterally constrained heaving motions in a side-by-side configuration

Published online by Cambridge University Press:  11 April 2023

Young Dal Jeong
Department of Mechanical Engineering, UNIST, 50 UNIST-gil, Eonyang-eup, Ulsan 44919, Korea
Min Je Kim
Department of Mechanical Engineering, UNIST, 50 UNIST-gil, Eonyang-eup, Ulsan 44919, Korea
Jae Hwa Lee*
Department of Mechanical Engineering, UNIST, 50 UNIST-gil, Eonyang-eup, Ulsan 44919, Korea
 Email address for correspondence:


Inspired by the intermittent locomotion of fish schools, numerical simulations are performed with two self-propelled flexible fins in a side-by-side configuration with anti-phase oscillation actuated by laterally constrained heaving motions. For an intermittent swimming gait, one type of the half-tail-beating mode (HT mode) and two types of multiple-tail-beating modes coasting at the smallest (MTS mode) and largest (MTL mode) lateral gap distances are applied. Similar to the continuous-tail-beating mode (CT mode), equilibrium lateral gap distances between two fins with HT and MTL modes exist, whereas two fins with MTS mode do not maintain a lateral equilibrium state. Although the cycle-averaged lateral force acting on two fins with CT and MTL modes is mostly determined by an outward deflected jet and enhanced positive pressure between two fins, an added-mass lateral force related to an asymmetric flapping kinematics by passive flexibility also plays an important role in MTL mode to achieve a stable state with a lateral gap distance smaller than that in CT mode. When the cruising speed or the cycle-averaged input power is identical in a stable state, the cost of transport (COT) for two fins with MTL mode is smaller than that with CT mode due to not only a benefit from the intermittent swimming gait but also an enhanced schooling benefit with a small equilibrium lateral gap distance. The COT for two fins with CT mode is reduced further when the bending rigidity increases, whereas it is opposite with MTL mode.

JFM Papers
© The Author(s), 2023. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


Akoz, E., Han, P., Liu, G., Dong, H. & Moored, K.W. 2019 Large-amplitude intermittent swimming in viscous and inviscid flows. AIAA J. 57 (9), 36783685.CrossRefGoogle Scholar
Akoz, E. & Moored, K.W. 2018 Unsteady propulsion by an intermittent swimming gait. J. Fluid Mech. 834, 149172.CrossRefGoogle Scholar
Ashraf, I., Godoy-Diana, R., Halloy, J., Collignon, B. & Thiria, B. 2016 Synchronization and collective swimming patterns in fish (Hemigrammus bleheri). J. R. Soc. Interface 13 (123), 20160734.CrossRefGoogle ScholarPubMed
Baddoo, P.J., Kurt, M., Ayton, L.J. & Moored, K.W. 2020 Exact solutions for ground effect. J. Fluid Mech. 891, R2.CrossRefGoogle Scholar
Baddoo, P.J., Moore, N.J., Oza, A.U. & Crowdy, D.G. 2021 Generalization of waving-plate theory to multiple interacting swimmers. Preprint, arXiv:2106.09167.Google Scholar
Blake, R.W. 1983 Functional design and burst-and-coast swimming in fishes. Can. J. Zool. 61, 24912494.CrossRefGoogle Scholar
Boschitsch, B.M., Dewey, P.D. & Smits, A.J. 2014 Propulsive performance of unsteady tandem hydrofoils in an in-line configuration. Phys. Fluids 26, 051901.CrossRefGoogle Scholar
Chung, M.-H. 2009 On burst-and-coast swimming performance in fish-like locomotion. Bioinspir. Biomim. 4 (3), 036001.CrossRefGoogle ScholarPubMed
Cong, L., Teng, B. & Cheng, L. 2020 Hydrodynamic behavior of two-dimensional tandem-arranged flapping flexible foils in uniform flow. Phys. Fluids 32, 021903.Google Scholar
Dai, L., He, G. & Zhang, X. 2016 Self-propelled swimming of a flexible plunging foil near a solid wall. Bioinspir. Biomim. 11, 046005.CrossRefGoogle Scholar
Dai, L., He, G., Zhang, X. & Zhang, X. 2018 a Intermittent locomotion of a fish-like swimmer driven by passive elastic mechanism. Bioinspir. Biomim. 13, 056011.CrossRefGoogle ScholarPubMed
Dai, L., He, G., Zhang, X. & Zhang, X. 2018 b Stable formations of self-propelled fish-like swimmers induced by hydrodynamic interactions. J. R. Soc. Interface 15, 20180490.CrossRefGoogle ScholarPubMed
Dewey, P.A., Quinn, D.B., Boschitsch, B.M. & Smits, A.J. 2014 Propulsive performance of unsteady tandem hydrofoils in a side-by-side configuration. Phys. Fluids 26, 041903.CrossRefGoogle Scholar
Dong, G.-J. & Lu, X.-Y. 2007 Characteristics of flow over traveling wavy foils in a side-by-side arrangement. Phys. Fluids 19, 057107.CrossRefGoogle Scholar
Dong, H., Mittal, R. & Najjar, F.M. 2006 Wake topology and hydrodynamic performance of low-aspect-ratio flapping foils. J. Fluid Mech. 566, 309343.CrossRefGoogle Scholar
Fish, F.E. 2010 Swimming strategies for energy economy. In Fish Locomotion: An Etho-ecological Perspective (ed. P. Domenici & B.G. Kapoor), pp. 90–122. Science Publishers.CrossRefGoogle Scholar
Fish, F.E., Fegely, J.F. & Xanthopoulos, C.J. 1991 Burst-and-coast swimming in schooling fish (Notemigonus crysoleucas) with implications for energy economy. Comput. Biochem. Physiol. 100, 633637.CrossRefGoogle Scholar
Floryan, D., Buren, T.V. & Smits, A.J. 2017 Forces and energetics of intermittent swimming. Acta Mechanica Sin. 33 (4), 725732.CrossRefGoogle Scholar
Fuiman, L.A. & Webb, P.W. 1988 Ontogeny of routine swimming activity and performance in zebra danios (Teleostei: Cyprinidae). Animal Behav. 36, 250261.CrossRefGoogle Scholar
Godoy-Diana, R., Marais, C., Aider, J.-L. & Wesfreid, J.E. 2009 A model for the symmetry breaking of the reverse Bénard-von Kármán vortex street produced by a flapping foil. J. Fluid Mech. 622, 2332.CrossRefGoogle Scholar
Hemelrijk, C.K., Reid, D.A.P., Hildenbrandt, H. & Paddling, J.T. 2015 The increased efficiency of fish swimming in a school. Fish Fish. 16, 511521.CrossRefGoogle Scholar
Hua, R.-N., Zhu, L. & Lu, X.-Y. 2013 Locomotion of a flapping flexible plate. Phys. Fluids 25, 121901.CrossRefGoogle Scholar
Huang, W.-X., Shin, S.J. & Sung, H.J. 2007 Simulation of flexible filaments in a uniform flow by the immersed boundary method. J. Comput. Phys. 226, 22062228.CrossRefGoogle Scholar
Jardin, T., David, L. & Farcy, A. 2009 Characterization of vortical structures and loads based on time-resolved PIV for asymmetric hovering flapping flight. Exp. Fluids 46, 847857.CrossRefGoogle Scholar
Jeong, Y.D. & Lee, J.H. 2017 Passive control of a single flexible flag using two side-by-side flags. Intl J. Heat Fluid Flow 65, 90104.CrossRefGoogle Scholar
Jeong, Y.D., Lee, J.H. & Park, S.G. 2021 Flow-mediated interactions between two self-propelled flexible fins near sidewalls. J. Fluid Mech. 913, A39.CrossRefGoogle Scholar
Kim, K., Baek, S.J. & Sung, H.J. 2002 An implicit velocity decoupling procedure for incompressible Navier–Stokes equations. Intl J. Numer. Meth. Fluids 38, 125138.CrossRefGoogle Scholar
Kim, M.J. & Lee, J.H. 2019 Wake transitions of flexible foils in a viscous uniform flow. Phys. Fluids 31, 111906.Google Scholar
Kim, S., Huang, W.-X. & Sung, H.J. 2010 Constructive and destructive interaction modes between two tandem flexible flags in viscous flow. J. Fluid Mech. 661, 511521.CrossRefGoogle Scholar
Kramer, D.L. & McLaughlin, R.L. 2001 The behavioral ecology of intermittent locomotion. Am. Zool. 41, 137153.Google Scholar
Kurt, M., Cochran-Carney, J., Zhong, Q., Mivehchi, A., Quinn, D. & Moored, K.W. 2019 Swimming freely near the ground leads to flow-mediated equilibrium altitudes. J. Fluid Mech. 875, R1.CrossRefGoogle Scholar
Kurt, M. & Moored, K.W. 2018 Flow interactions of two- and three-dimensional networked bio-inspired control elements in an in-line arrangement. Bioinspir. Biomim. 13, 045002.CrossRefGoogle Scholar
Kurt, M., Ormonde, P.C., Mivehchi, A. & Moored, K.W. 2021 Two dimensionally stable self-organization arises in simple schooling swimmers through hydrodynamic interactions. Preprint, arXiv:2102.03571.Google Scholar
Kurtulus, D.F., Scarano, F. & David, L. 2007 Unsteady aerodynamic forces estimation on a square cylinder by TR-PIV. Exp. Fluids 42, 185196.CrossRefGoogle Scholar
Lewin, G.C. & Haj-Hariri, H. 2003 Modelling thrust generation of a two-dimensional heaving airfoil in a viscous flow. J. Fluid Mech. 492, 339362.CrossRefGoogle Scholar
Lighthill, M.J. 1971 Large-amplitude elongated-body theory of fish locomotion. Proc. R. Soc. Lond. B 179 (1055), 125138.Google Scholar
Lin, X., Wu, J., Zhang, T. & Yang, L. 2021 Flow-mediated organization of two freely flapping swimmers. J. Fluid Mech. 912, A37.CrossRefGoogle Scholar
Liu, K., Huang, H. & Lu, X.-Y. 2020 Hydrodynamic benefits of intermittent locomotion of a self-propelled flapping plate. Phys. Rev. E 102, 053106.CrossRefGoogle ScholarPubMed
Lua, K.B., Lim, T.T., Yeo, K.S. & Oo, G.Y. 2007 Wake-structure formation of a heaving two-dimensional elliptic airfoil. AIAA J. 45 (7), 15711583.CrossRefGoogle Scholar
Lua, K.B., Zhang, X.H., Lim, T.T. & Yeo, K.S. 2016 Aerodynamics of two-dimensional flapping wings in tandem configuration. Phys. Fluids 28, 121901.CrossRefGoogle Scholar
McHenry, M.J. & Lauder, G.V. 2005 The mechanical scaling of coasting in zebrafish (Danio rerio). J. Expl Biol. 208, 22892301.CrossRefGoogle ScholarPubMed
Müller, U.K., Stamhuis, E.J. & Videler, J.J. 2000 Hydrodynamics of unsteady fish swimming and the effects of body size: comparing the flow fields of fish larvae and adults. J. Expl Biol. 203 (2), 193206.CrossRefGoogle ScholarPubMed
Muscutt, L.E., Weymouth, G.D. & Ganapathisubramani, B. 2017 Performance augmentation mechanism of in-line tandem flapping foils. J. Fluid Mech. 827, 484505.CrossRefGoogle Scholar
Oza, A.U., Ristroph, L. & Shelley, M.J. 2019 Lattices of hydrodynamically interacting flapping swimmers. Phys. Rev. X 9, 041024.Google Scholar
Park, S.G., Kim, B. & Sung, H.J. 2017 Hydrodynamics of a self-propelled flexible fin near the ground. Phys. Fluids 29, 051902.CrossRefGoogle Scholar
Park, S.G. & Sung, H.J. 2016 Vortex interaction between two tandem flexible propulsor with a paddling-based locomotion. J. Fluid Mech. 793, 612632.CrossRefGoogle Scholar
Park, S.G. & Sung, H.J. 2018 Hydrodynamics of flexible fins propelled in tandem, diagonal, triangular and diamond configurations. J. Fluid Mech. 840, 154189.CrossRefGoogle Scholar
Peng, Z.-R., Huang, H. & Lu, X.-Y. 2018 Collective locomotion of two closely spaced self-propelled flapping plates. J. Fluid Mech. 849, 10681095.CrossRefGoogle Scholar
Quinn, D.B., Moored, K.W., Dewey, P.A. & Smits, A.J. 2014 Unsteady propulsion near a soild boundary. J. Fluid Mech. 742, 152170.CrossRefGoogle Scholar
Raj, K.M. & Arumuru, V. 2020 Jet deflection by two side-by-side arranged hydrofoils pitching in a quiescent fluid. AIP Adv. 10, 105128.CrossRefGoogle Scholar
Ramananarivo, S., Fang, F., Oza, A., Zhang, J. & Ristroph, L. 2016 Flow interactions lead to orderly formations of flapping wings in forward flight. Phys. Rev. Fluids 1 (7), 071201.CrossRefGoogle Scholar
Ryu, J. & Sung, H.J. 2019 Intermittent locomotion of a self-propelled plate. Phys. Fluids 31, 111902.Google Scholar
Shen, L., Chan, E.S. & Lin, P. 2009 Calculation of hydrodynamic forces acting on a submerged moving object using immersed boundary method. Comput. Fluids 38, 691702.CrossRefGoogle Scholar
Son, Y. & Lee, J.H. 2017 Flapping dynamics of coupled flexible flags in a uniform viscous flow. J. Fluids Struct. 68, 339355.CrossRefGoogle Scholar
Uddin, E., Huang, W.-X. & Sung, H.J. 2013 Interaction modes of multiple flexible flags in a uniform flow. J. Fluid Mech. 729, 563583.CrossRefGoogle Scholar
Uddin, E., Huang, W.-X. & Sung, H.J. 2015 Actively flapping tandem flexible flags in a viscous flow. J. Fluid Mech. 780, 120142.CrossRefGoogle Scholar
Videler, J.J. 1981 Swimming movements, body structure and propulsion in Cod Gadus morhua. Symp. Zool. Soc. Lond. 48, 127.Google Scholar
Videler, J.J. & Weihs, D. 1982 Energetic advantages of burst-and-coast swimming of fish at high speeds. J. Expl Biol. 97 (1), 169178.CrossRefGoogle ScholarPubMed
Weihs, D. 1973 Hydromechanics of fish schooling. Nature 241, 290291.CrossRefGoogle Scholar
Weihs, D. 1974 Energetic advantages of burst swimming of fish. J. Theor. Biol. 48 (1), 215229.CrossRefGoogle ScholarPubMed
Weihs, D. 1980 Energetic significance of changes in swimming modes during growth of larval anchovy, Engraulis mordax. Fish. Bull. 77, 597604.Google Scholar
Wu, B., Shu, C., Wan, M., Wang, Y. & Chen, S. 2022 Hydrodynamic performance of an unconstrained flapping swimmer with flexible fin: a numerical study. Phys. Fluids 34, 011901.CrossRefGoogle Scholar
Wu, G., Yang, Y. & Zeng, L. 2007 Kinematics, hydrodynamics and energetic advantages of burst-and-coast swimming of koi carps (Cyprinus carpio koi). J. Expl Biol. 210 (12), 21812191.CrossRefGoogle ScholarPubMed
Zhang, C., Huang, H. & Lu, X.-Y. 2017 Free locomotion of a flexible plate near the ground. Phys. Fluids 29, 041903.CrossRefGoogle Scholar
Zhao, J., Mao, Q., Pan, G., Huang, Q.G. & Sung, H.J. 2021 Hydrodynamic benefit of impulsive bursting in a self-propelled flexible plate. Phys. Fluids 33, 111904.CrossRefGoogle Scholar
Zheng, Z.C. & Wei, Z. 2012 Study of mechanisms and factors that influence the formation of vortical wake of a heaving airfoil. Phys. Fluids 24, 103601.CrossRefGoogle Scholar
Zhong, Q., Han, T., Moored, K.W. & Quinn, D.B. 2021 Aspect ratio affects the equilibrium altitude of near-ground swimmers. J. Fluid Mech. 917, A36.CrossRefGoogle Scholar
Zhu, X., He, G. & Zhang, X. 2014 a Flow-mediated interactions between two self-propelled flapping filaments in tandem configuration. Phys. Rev. Lett. 113, 238105.CrossRefGoogle ScholarPubMed
Zhu, X., He, G. & Zhang, X. 2014 b How flexibility affects the wake symmetry properties of a self-propelled plunging foil. J. Fluid Mech. 751, 164183.CrossRefGoogle Scholar