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      Optimal specific wavelength for maximum thrust production in undulatory propulsion

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          Abstract

          What wavelengths do undulatory swimmers use during propulsion? In this work we find that a wide range of body/caudal fin (BCF) swimmers, from larval zebrafish and herring to fully–grown eels, use specific wavelength (ratio of wavelength to tail amplitude of undulation) values that fall within a relatively narrow range. The possible emergence of this constraint is interrogated using numerical simulations of fluid–structure interaction. Based on these, it was found that there is an optimal specific wavelength (OSW) that maximizes the swimming speed and thrust generated by an undulatory swimmer. The observed values of specific wavelength for BCF animals are relatively close to this OSW. The mechanisms underlying the maximum propulsive thrust for BCF swimmers are quantified and are found to be consistent with the mechanisms hypothesized in prior work. The adherence to an optimal value of specific wavelength in most natural hydrodynamic propulsors gives rise to empirical design criteria for man–made propulsors.

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          Most cited references40

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          Review of fish swimming modes for aquatic locomotion

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            Flying and swimming animals cruise at a Strouhal number tuned for high power efficiency.

            Dimensionless numbers are important in biomechanics because their constancy can imply dynamic similarity between systems, despite possible differences in medium or scale. A dimensionless parameter that describes the tail or wing kinematics of swimming and flying animals is the Strouhal number, St = fA/U, which divides stroke frequency (f) and amplitude (A) by forward speed (U). St is known to govern a well-defined series of vortex growth and shedding regimes for airfoils undergoing pitching and heaving motions. Propulsive efficiency is high over a narrow range of St and usually peaks within the interval 0.2 < St < 0.4 (refs 3-8). Because natural selection is likely to tune animals for high propulsive efficiency, we expect it to constrain the range of St that animals use. This seems to be true for dolphins, sharks and bony fish, which swim at 0.2 < St < 0.4. Here we show that birds, bats and insects also converge on the same narrow range of St, but only when cruising. Tuning cruise kinematics to optimize St therefore seems to be a general principle of oscillatory lift-based propulsion.
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              Oscillating foils of high propulsive efficiency

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                Author and article information

                Contributors
                Role: Editor
                Journal
                PLoS One
                PLoS ONE
                plos
                plosone
                PLoS ONE
                Public Library of Science (San Francisco, CA USA )
                1932-6203
                2017
                27 June 2017
                : 12
                : 6
                : e0179727
                Affiliations
                [1 ]Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL, United States of America
                [2 ]Department of Mechanical Engineering, Northwestern University, Evanston, IL, United States of America
                University at Buffalo - The State University of New York, UNITED STATES
                Author notes

                Competing Interests: The authors have declared that no competing interests exist.

                • Conceptualization: NN RB NAP.

                • Formal analysis: NN RB NAP.

                • Funding acquisition: NN NAP.

                • Investigation: NN RB NC YH.

                • Software: NN RB NAP.

                • Writing – original draft: NN NAP.

                • Writing – review & editing: NN RB NC YH NAP.

                [¤a]

                Current address: Advanced Institute for Computational Science (AICS), RIKEN, Kobe, Japan

                [¤b]

                Current address: Department of Mechanical Engineering, University of California, Berkeley, CA, United States of America

                [¤c]

                Current address: Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA, United States of America

                Article
                PONE-D-16-41077
                10.1371/journal.pone.0179727
                5487070
                28654649
                e1579aea-902c-4e5b-b474-db97561690cb
                © 2017 Nangia et al

                This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

                History
                : 14 October 2016
                : 2 June 2017
                Page count
                Figures: 13, Tables: 1, Pages: 23
                Funding
                Funded by: funder-id http://dx.doi.org/10.13039/100000001, National Science Foundation;
                Award ID: CBET–0828749
                Award Recipient :
                Funded by: funder-id http://dx.doi.org/10.13039/100000001, National Science Foundation;
                Award ID: CMMI-0941674
                Award Recipient :
                Funded by: funder-id http://dx.doi.org/10.13039/100000001, National Science Foundation;
                Award ID: CBET–1066575
                Award Recipient :
                Funded by: funder-id http://dx.doi.org/10.13039/100000001, National Science Foundation;
                Award ID: DGE–1324585
                Award Recipient :
                This work was supported by National Science Foundation grants CBET-0828749, CMMI-0941674 and CBET-1066575 to NAP. Computational resources were provided by Northwestern University High Performance Computing System-Quest. This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number ACI-1053575. This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1324585 to NN. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
                Categories
                Research Article
                Biology and Life Sciences
                Physiology
                Biological Locomotion
                Swimming
                Medicine and Health Sciences
                Physiology
                Biological Locomotion
                Swimming
                Physical Sciences
                Physics
                Classical Mechanics
                Continuum Mechanics
                Fluid Mechanics
                Fluid Dynamics
                Reynolds Number
                Physical Sciences
                Mathematics
                Geometry
                Aspect Ratio
                Physical Sciences
                Physics
                Classical Mechanics
                Thrust
                Biology and Life Sciences
                Organisms
                Animals
                Vertebrates
                Fishes
                Osteichthyes
                Eels
                Physical Sciences
                Physics
                Classical Mechanics
                Kinematics
                Biology and Life Sciences
                Organisms
                Animals
                Vertebrates
                Fishes
                Engineering and Technology
                Mechanical Engineering
                Robotics
                Custom metadata
                All relevant data are within the paper and its Supporting Information files.

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