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Optimal undulatory swimming for a single fish-like body and for a pair of interacting swimmers

  • Audrey P. Maertens (a1), Amy Gao (a1) and Michael S. Triantafyllou (a1)

Abstract

We establish through numerical simulation conditions for optimal undulatory propulsion for a single fish, and for a pair of hydrodynamically interacting fish, accounting for linear and angular recoil. We first employ systematic two-dimensional (2-D) simulations to identify conditions for minimal propulsive power of a self-propelled fish, and continue with targeted 3-D simulations for a danio-like fish; all at Reynolds number 5000. We find that the Strouhal number, phase angle between heave and pitch at the trailing edge, and angle of attack are principal parameters. For 2-D simulations, imposing a deformation based on measured displacement for carangiform swimming provides, at best, efficiency of 35 %, which increases to 50 % for an optimized motion; for a 3-D fish, the efficiency increases from 22 % to 34 %. Indeed, angular recoil has significant impact on efficiency, and optimized body bending requires maximum bending amplitude upstream of the trailing edge. Next, we turn to 2-D simulation of two hydrodynamically interacting fish. We find that the upstream fish benefits energetically only for small distances. In contrast, the downstream fish can benefit at any position that allows interaction with the upstream wake, provided its body motion is timed appropriately with respect to the oncoming vortices. For an in-line configuration, one body length apart, the efficiency of the downstream fish can increase from 50 % to 60 %; for an offset arrangement it can reach 80 %. This proves that in groups of fish, energy savings can be achieved for downstream fish through interaction with oncoming vortices, even when the downstream fish lies directly inside the jet-like flow of an upstream fish.

Copyright

Corresponding author

Present address: EPFL, LMH, Avenue de Cour 33 bis, 1007 Lausanne, Switzerland. Email address for correspondence: audrey.maertens@epfl.ch

References

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Abrahams, M. V. & Colgan, P. W. 1987 Fish schools and their hydrodynamic function: a reanalysis. Environ. Biol. Fish. 20 (1), 7980.
Akanyeti, O. & Liao, J. C. 2013 A kinematic model of Karman gaiting in rainbow trout. J. Expl Biol. jeb.093245.
Alben, S. 2009 Wake-mediated synchronization and drafting in coupled flags. J. Fluid Mech. 641, 489496.
Anderson, J. M., Streitlien, K., Barrett, D. S. & Triantafyllou, M. S. 1998 Oscillating foils of high propulsive efficiency. J. Fluid Mech. 360, 4172.
Bainbridge, R. 1961 Problems of fish locomotion. In Symp. Zool. Soc. Lond., vol. 5, pp. 1332.
Bale, R., Hao, M., Bhalla, A. P. S. & Patankar, N. A. 2014 Energy efficiency and allometry of movement of swimming and flying animals. Proc. Natl Acad. Sci. 111 (21), 75177521.
Beal, D. N., Hover, F. S., Triantafyllou, M. S., Liao, J. C. & Lauder, G. V. 2006 Passive propulsion in vortex wakes. J. Fluid Mech. 549, 385402.
Bergmann, M., Iollo, A. & Mittal, R. 2014 Effect of caudal fin flexibility on the propulsive efficiency of a fish-like swimmer. Bioinspir. Biomim. 9 (4), 046001.
Blondeaux, P., Fornarelli, F., Guglielmini, L., Triantafyllou, M. S. & Verzicco, R. 2005 Numerical experiments on flapping foils mimicking fish-like locomotion. Phys. Fluids 17 (11), 113601.
Borazjani, I. & Sotiropoulos, F. 2008 Numerical investigation of the hydrodynamics of carangiform swimming in the transitional and inertial flow regimes. J. Expl Biol. 211 (10), 15411558.
Borazjani, I. & Sotiropoulos, F. 2010 On the role of form and kinematics on the hydrodynamics of self-propelled body/caudal fin swimming. J. Expl Biol. 213 (1), 89107.
Boschitsch, B. M., Dewey, P. A. & Smits, A. J. 2014 Propulsive performance of unsteady tandem hydrofoils in an in-line configuration. Phys. Fluids 26 (5), 051901.
Breder, C. M. 1926 The locomotion of fishes. Zoologica 4, 159297.
Carling, J., Williams, T. L. & Bowtell, G. 1998 Self-propelled anguilliform swimming: simultaneous solution of the two-dimensional Navier–Stokes equations and Newton’s laws of motion. J. Expl Biol. 201 (23), 31433166.
Connell, B. S. H. & Yue, D. K. P. 2007 Flapping dynamics of a flag in a uniform stream. J. Fluid Mech. 581, 3367.
Daghooghi, M. & Borazjani, I. 2015 The hydrodynamic advantages of synchronized swimming in a rectangular pattern. Bioinspir. Biomim. 10 (5), 056018.
Deng, H.-B., Xu, Y.-Q., Chen, D.-D., Dai, H., Wu, J. & Tian, F.-B. 2013 On numerical modeling of animal swimming and flight. Comput. Mech. 52 (6), 12211242.
Deng, J. & Shao, X.-m. 2006 Hydrodynamics in a diamond-shaped fish school Project supported by the National Lab of Hydrodynamics of China. J. Hydrodyn. B 18 (3), 438442.
Dong, G.-J. & Lu, X.-Y. 2007 Characteristics of flow over traveling wavy foils in a side-by-side arrangement. Phys. Fluids 19 (5), 057107.
Dong, H., Mittal, R. & Najjar, F. M. 2006 Wake topology and hydrodynamic performance of low-aspect-ratio flapping foils. J. Fluid Mech. 566, 309343.
Drucker, E. G. & Lauder, G. V. 2001 Locomotor function of the dorsal fin in teleost fishes: experimental analysis of wake forces in sunfish. J. Expl Biol. 204 (17), 29432958.
Eldredge, J. D. 2006 Numerical simulations of undulatory swimming at moderate Reynolds number. Bioinspir. Biomim. 1 (4), S19.
Eloy, C. 2013 On the best design for undulatory swimming. J. Fluid Mech. 717, 4889.
Förster, C., Wall, W. A. & Ramm, E. 2007 Artificial added mass instabilities in sequential staggered coupling of nonlinear structures and incompressible viscous flows. Comput. Meth. Appl. Mech. Engng 196 (7), 12781293.
Gazzola, M., Argentina, M. & Mahadevan, L. 2014 Scaling macroscopic aquatic locomotion. Nat. Phys. 10 (10), 758761.
Gero, D. R. 1952 The hydrodynamic aspects of fish propulsion. Fish propulsion 1601, 132.
Ginneken, V. v., Antonissen, E., Miller, U. K., Booms, R., Eding, E., Verreth, J. & Thillart, G. v. d. 2005 Eel migration to the Sargasso: remarkably high swimming efficiency and low energy costs. J. Expl Biol. 208 (7), 13291335.
Gopalkrishnan, R., Triantafyllou, M. S., Triantafyllou, G. S. & Barrett, D. 1994 Active vorticity control in a shear flow using a flapping foil. J. Fluid Mech. 274, 121.
Gray, J. 1933 Studies in animal locomotion. I. The movement of fish with special reference to the eel. J. Expl Biol. 10 (1), 88104.
Harper, D. G. & Blake, R. W. 1990 Fast-Start Performance of Rainbow Trout Salmo Gairdneri and Northern Pike Esox Lucius. J. Expl Biol. 150 (1), 321342.
Hemelrijk, C., Reid, D., Hildenbrandt, H. & Padding, J. 2015 The increased efficiency of fish swimming in a school. Fish and Fisheries 16 (3), 511521.
Ijspeert, A. J. 2014 Biorobotics: using robots to emulate and investigate agile locomotion. Science 346 (6206), 196203.
Johnson, S. G.2013 The NLopt nonlinear-optimization package, http://ab-initio.mit.edu/nlopt.
Kern, S. & Koumoutsakos, P. 2006 Simulations of optimized anguilliform swimming. J. Expl Biol. 209 (24), 48414857.
Killen, S. S., Marras, S., Steffensen, J. F. & McKenzie, D. J. 2012 Aerobic capacity influences the spatial position of individuals within fish schools. Proc. Biol. Sci. 279 (1727), 357364.
Lauder, G. V. & Madden, P. G. A. 2007 Fish locomotion: kinematics and hydrodynamics of flexible foil-like fins. Exp. Fluids 43 (5), 641653.
Liao, J. C. 2007 A review of fish swimming mechanics and behaviour in altered flows. Phil. Trans. R. Soc. B: Biol. Sci. 362 (1487), 19731993.
Liao, J. C., Beal, D. N., Lauder, G. V. & Triantafyllou, M. S. 2003a Fish exploiting vortices decrease muscle activity. Science 302 (5650), 15661569.
Liao, J. C., Beal, D. N., Lauder, G. V. & Triantafyllou, M. S. 2003b The Karman gait: novel body kinematics of rainbow trout swimming in a vortex street. J Expl Biol. 206 (6), 10591073.
Lighthill, M. J. 1960 Note on the swimming of slender fish. J. Fluid Mech. 9 (02), 305317.
Liu, G., Yu, Y.-L. & Tong, B.-G. 2011 Flow control by means of a traveling curvature wave in fishlike escape responses. Phys. Rev. E 84 (5), 056312.
Maertens, A. P.2015 Fish swimming optimization and exploiting multi-body hydrodynamic interactions for underwater navigation. PhD thesis, Department of Mechanical Engineering, Massachusetts Institute of Technology.
Maertens, A. P., Triantafyllou, M. S. & Yue, D. K. P. 2015 Efficiency of fish propulsion. Bioinspir. Biomim.; (submitted) (under review).
Maertens, A. P. & Weymouth, G. D. 2015 Accurate Cartesian-grid simulations of near-body flows at intermediate Reynolds numbers. Comput. Meth. Appl. Mech. Engng 283, 106129.
Marras, S., Killen, S. S., Lindstrom, J., Mckenzie, D. J., Steffensen, J. F. & Domenici, P. 2014 Fish swimming in schools save energy regardless of their spatial position. Behav. Ecol. Sociobiol. 18.
Partridge, B. L. & Pitcher, T. J. 1979 Evidence against a hydrodynamic function for fish schools. Nature 279 (5712), 418419.
Peng, Z. & Zhu, Q. 2009 Energy harvesting through flow-induced oscillations of a foil. Phys. Fluids 21 (12), 123602.
Pitcher, T. J. 1986 Functions of shoaling behaviour in teleosts. In The Behaviour of Teleost Fishes (ed. Pitcher, T. J.), pp. 294337. Springer.
Portugal, S. J., Hubel, T. Y., Fritz, J., Heese, S., Trobe, D., Voelkl, B., Hailes, S., Wilson, A. M. & Usherwood, J. R. 2014 Upwash exploitation and downwash avoidance by flap phasing in ibis formation flight. Nature 505 (7483), 399402.
Powell, M. J. D.2009 The BOBYQA algorithm for bound constrained optimization without derivatives. Cambridge NA Report NA2009/06, University of Cambridge, Cambridge.
Read, D. A., Hover, F. S. & Triantafyllou, M. S. 2003 Forces on oscillating foils for propulsion and maneuvering. J. Fluids Struct. 17 (1), 163183.
van Rees, W. M., Gazzola, M. & Koumoutsakos, P. 2013 Optimal shapes for anguilliform swimmers at intermediate Reynolds numbers. J. Fluid Mech. 722, R3.
Reid, Daniel A. P., Hildenbrandt, H., Padding, J. T. & Hemelrijk, C. K. 2009 Flow around fishlike shapes studied using multiparticle collision dynamics. Phys. Rev. E 79 (4), 046313.
Reid, D. A. P., Hildenbrandt, H., Padding, J. T. & Hemelrijk, C. K. 2012 Fluid dynamics of moving fish in a two-dimensional multiparticle collision dynamics model. Phys. Rev. E 85 (2), 021901.
Rios, L. M. & Sahinidis, N. V. 2013 Derivative-free optimization: a review of algorithms and comparison of software implementations. J. Glob. Optim. 56 (3), 12471293.
Roberts, T. J. & Azizi, E. 2011 Flexible mechanisms: the diverse roles of biological springs in vertebrate movement. J. Expl Biol. 214 (3), 353361.
Sefati, S., Neveln, I. D., Roth, E., Mitchell, T. R. T., Snyder, J. B., Maciver, M. A., Fortune, E. S. & Cowan, N. J. 2013 Mutually opposing forces during locomotion can eliminate the tradeoff between maneuverability and stability. PNAS 110 (47), 1879818803.
Sfakiotakis, M., Lane, D. M. & Davies, J. B. C. 1999 Review of fish swimming modes for aquatic locomotion. IEEE J. Ocean. Engng 24 (2), 237252.
Shen, L., Zhang, X., Yue, D. K. P. & Triantafyllou, M. S. 2003 Turbulent flow over a flexible wall undergoing a streamwise travelling wave motion. J. Fluid Mech. 484, 197221.
Shirgaonkar, A. A., MacIver, M. A. & Patankar, N. A. 2009 A new mathematical formulation and fast algorithm for fully resolved simulation of self-propulsion. J. Comput. Phys. 228 (7), 23662390.
Stefanini, C., Orofino, S., Manfredi, L., Mintchev, S., Marrazza, S., Assaf, T., Capantini, L., Sinibaldi, E., Grillner, S., Walln, P. et al. 2012 A novel autonomous, bioinspired swimming robot developed by neuroscientists and bioengineers. Bioinspir. Biomim. 7 (2), 025001.
Streitlien, K., Triantafyllou, G. S. & Triantafyllou, M. S. 1996 Efficient foil propulsion through vortex control. AIAA J. 34 (11), 23152319.
Toki, G. & Yue, D. K. P. 2012 Optimal shape and motion of undulatory swimming organisms. Proc. R. Soc. Lond. B 279 (1740), 30653074.
Triantafyllou, G. S., Triantafyllou, M. S. & Grosenbaugh, M. A. 1993 Optimal thrust development in oscillating foils with application to fish propulsion. J. Fluids Struct. 7 (2), 205224.
Triantafyllou, M. S. & Triantafyllou, G. S. 1995 An efficient swimming machine. Sci. Am. 272, 6470.
Triantafyllou, M. S., Triantafyllou, G. S. & Gopalkrishnan, R. 1991 Wake mechanics for thrust generation in oscillating foils. Phys. Fluids A 3 (12), 28352837.
Tytell, E. D. 2004 The hydrodynamics of eel swimming II. Effect of swimming speed. J. Expl Biol. 207 (19), 32653279.
Tytell, E. D. & Lauder, G. V. 2004 The hydrodynamics of eel swimming I. Wake structure. J. Expl Biol. 207 (11), 18251841.
Videler, J. J. 1993 Fish Swimming. Springer.
Videler, J. J. & Hess, F. 1984 Fast continuous swimming of two pelagic predators, Saithe (Pollachius virens) and Mackerel (Scomber scombrus): a kinematic analysis. J. Expl Biol. 109 (1), 209228.
Webb, P. W. 1971 The swimming energetics of trout II. Oxygen consumption and swimming efficiency. J. Expl Biol. 55 (2), 521540.
van Weerden, J. F., Reid, D. A. P. & Hemelrijk, C. K. 2014 A meta-analysis of steady undulatory swimming. Fish Fish 15 (3), 397409.
Weihs, D. 1973 Hydromechanics of fish schooling. Nature 241 (5387), 290291.
Weymouth, G. D., Dommermuth, D. G., Hendrickson, K. & Yue, D. K.-P. 2006 Advancements in Cartesian-grid methods for computational ship hydrodynamics. In 26th Symposium on Naval Hydrodynamics, Rome, Italy, 17–22 September 2006.
Weymouth, G. D. & Triantafyllou, M. S. 2013 Ultra-fast escape of a deformable jet-propelled body. J. Fluid Mech. 721, 367385.
Wibawa, M. S., Steele, S. C., Dahl, J. M., Rival, D. E., Weymouth, G. D. & Triantafyllou, M. S. 2012 Global vorticity shedding for a vanishing wing. J. Fluid Mech. 695, 112134.
Wolfgang, M. J., Anderson, J. M., Grosenbaugh, M. A., Yue, D. K. & Triantafyllou, M. S. 1999 Near-body flow dynamics in swimming fish. J. Expl Biol. 202 (17), 23032327.
Zhu, L. & Peskin, C. S. 2003 Interaction of two flapping filaments in a flowing soap film. Phys. Fluids 15 (7), 19541960.
Zhu, Q. & Shoele, K. 2008 Propulsion performance of a skeleton-strengthened fin. J. Expl Biol. 211 (13), 20872100.
Zhu, Q., Wolfgang, M. J., Yue, D. K. P. & Triantafyllou, M. S. 2002 Three-dimensional flow structures and vorticity control in fish-like swimming. J. Fluid Mech. 468, 128.
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Optimal undulatory swimming for a single fish-like body and for a pair of interacting swimmers

  • Audrey P. Maertens (a1), Amy Gao (a1) and Michael S. Triantafyllou (a1)

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