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arxiv: 2503.23652 · v2 · submitted 2025-03-31 · ⚛️ physics.flu-dyn · physics.bio-ph

Flow-induced dorso-ventral deformation enhances propulsive efficiency in flexible caudal fins

Sushrut Kumar , Matthew J. McHenry , Jung-Hee Seo , Rajat Mittal This is my paper

Pith reviewed 2026-05-22 22:57 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn physics.bio-ph
keywords caudal findorso-ventral deformationpropulsive efficiencyflow-structure interactionfish swimmingflexible propulsorpassive deformationnumerical simulation
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The pith

Flow-induced dorso-ventral deformation in flexible caudal fins yields 70% higher efficiency than rigid fins at equal thrust.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Numerical simulations of flow-structure interaction reveal that flexible fish tails bend dorso-ventrally under fluid loads, cutting the power needed to produce thrust by 70% compared with rigid tails. The bending reorients pressure forces forward and vertically instead of letting them create large sideways loads that waste energy. These shape changes align with the moments of highest sideward speed in the tail-beat cycle, lowering total energy cost without any active control. Readers care because the result shows how passive flexibility alone can deliver efficient propulsion that engineered vehicles usually achieve only with sensors and computers.

Core claim

Using numerical simulations of the dynamics of flow-structure interaction, we have found that dorso-ventral deformation in flexible caudal fins results in a 70% increase in efficiency of caudal fin swimmers compared to a rigid fin generating the same amount of thrust. By correlating fin deformation to the flow physics, we find that the greater power requirements of rigid fins can be largely attributed to their propensity to generate high-magnitude lateral forces. In contrast, flexible fins achieve high efficiency local-redirection of force where deformations orient pressure forces on the fin in fore-aft and dorso-ventral directions to reduce the power demand of generating thrust forces. In a

What carries the argument

Flow-induced dorso-ventral deformation of the flexible caudal fin, which passively reorients pressure forces fore-aft and vertically to lower lateral force magnitude and net power during the tail-beat cycle.

Load-bearing premise

The numerical flow-structure interaction model accurately reproduces the passive material response and three-dimensional deformation of real biological fins driven only by fluid loading.

What would settle it

A controlled experiment that measures power input and thrust output for a physical flexible caudal fin model versus a rigid counterpart at identical swimming speeds and frequencies would confirm or refute the reported 70% efficiency gain.

Figures

Figures reproduced from arXiv: 2503.23652 by Jung-Hee Seo, Matthew J. McHenry, Rajat Mittal, Sushrut Kumar.

Figure 1
Figure 1. Figure 1: Modeling the flow-structure interactions of caudal fins. (A) Video recording of a batfish (Platax orbicularis) from a posterior view shows how the mid-chord of the fin bends (right arrow) relative to the dorsal (down arrow) and ventral (up arrow) margins of the caudal fin as it beats toward the right. (B) Our model of the caudal fin operates in a Cartesian coordinate system (with axes x1, x2, and x3) with … view at source ↗
Figure 2
Figure 2. Figure 2: The effects the contribution of lateral forces on the power requirements for propulsion in a rigid fin (Fin-R, A–C) and the most flexible fin (Fin-HF, D–F). (A) The predicted flow field (isosurfaces of the spanwise vorticity, which identifies coherent vortices by comparing the local rotation, here Q = 5 colored by ω3C/U∞) for the rigid fin, as it is actuated with (B) oscillatory changes in lateral position… view at source ↗
Figure 3
Figure 3. Figure 3: Phase relationship between pressure and velocity. (A) All fins were actuated with oscillations in lateral position (black) and pitch angle (dark gray), with a leftward half tail-beat (gray field) highlighted. The resulting integrated (B) pressure differences (1/Sfin R ∆pdS) and (C) velocity (1/Sfin R VfindS) of elements of the fin mesh (Fig. 1E) for fins that are rigid (Fin-R, dark blue), and of intermedia… view at source ↗
Figure 4
Figure 4. Figure 4: Force, power, and deflection among fins are varying flexibility. (A) All fins were actuated with oscillations in lateral position (black) and pitch angle (dark gray), with a leftward half tail-beat (gray field) highlighted. The resulting coefficients of (B) thrust, (C) lateral force, and (D) power are shown for a rigid fin (Fin-R, dark blue), and fins of intermediate (Fin-IR, medium blue), and high (Fin-HR… view at source ↗
read the original abstract

Fish swim with flexible fins that stand in stark contrast to the rigid propulsors of engineered vehicles. Using numerical simulations of the dynamics of flow-structure interaction, we have found that dorso-ventral deformation in flexible caudal fins results in a 70% increase in efficiency of caudal fin swimmers compared to a rigid fin generating the same amount of thrust. By correlating fin deformation to the flow physics, we find that the greater power requirements of rigid fins can be largely attributed to their propensity to generate high-magnitude lateral forces. In contrast, flexible fins achieve high efficiency local-redirection of force where deformations orient pressure forces on the fin in fore-aft and dorso-ventral directions to reduce the power demand of generating thrust forces. These deformations occur at phases in the tail-beat cycle where the fin experiences large lateral velocities and pressure differentials and this reduces the net power expended by the flexible fins. In this way, the flexibility of a caudal fin offers a simple and elegant solution for efficient locomotion which does not require sensing, computation and control that might otherwise be provided by the nervous system of a fish or a computer within a underwater vehicle. These flow-induced dorso-ventral fin deformations therefore imbue a mechanical intelligence in these fins that provides propulsive advantages to caudal fin swimmers and they also offer solutions for efficient propulsion in engineered systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript reports numerical simulations of flow-structure interactions in caudal fins, claiming that flow-induced dorso-ventral deformations in flexible fins yield a 70% increase in propulsive efficiency relative to rigid fins that produce the same thrust. The authors link this to reduced lateral forces and phase-specific force redirection enabled by passive deformations during the tail-beat cycle.

Significance. Should the simulation results prove robust, the work identifies a passive mechanical mechanism that confers propulsive advantages to flexible fins, potentially explaining aspects of fish swimming efficiency and informing the design of flexible propulsors for underwater vehicles. The analysis correlating deformation timing with flow features provides a clear mechanistic account of the efficiency benefit.

major comments (2)
  1. [Numerical Methods] Numerical Methods section: No information is supplied on mesh resolution, time-step convergence, or validation of the fluid-structure interaction model against experimental data for fin deformation or forces. This directly undermines assessment of the central 70% efficiency claim.
  2. [Results] Results section: The rigid-fin baseline is not described (identical planform and kinematics? infinite stiffness? same mean thrust generation protocol?). Without this, the efficiency comparison cannot be evaluated.
minor comments (2)
  1. [Abstract] Abstract: 'local-redirection of force' is awkward; rephrase for clarity.
  2. [Abstract] Abstract: The 70% figure is given without error bars or uncertainty quantification.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments, which highlight important areas for improving the clarity and rigor of our manuscript. We address each major comment below and will make the requested revisions to the Numerical Methods and Results sections. These changes will better support evaluation of the reported efficiency gains without altering the core findings.

read point-by-point responses

  1. Referee: [Numerical Methods] Numerical Methods section: No information is supplied on mesh resolution, time-step convergence, or validation of the fluid-structure interaction model against experimental data for fin deformation or forces. This directly undermines assessment of the central 70% efficiency claim.

    Authors: We agree that additional details on the numerical methods are necessary for assessing the robustness of the results. In the revised manuscript, we will add a new subsection to Numerical Methods that reports mesh resolution (including grid convergence studies for both fluid and structural domains), time-step independence tests, and references to prior validation of the FSI solver against experimental benchmarks for flexible propulsors in flow. While direct experimental validation data for this exact fin geometry and deformation mode are not available in the literature, we will discuss the model's fidelity using established benchmarks for similar fluid-structure interaction problems. This revision will directly address the concern regarding the 70% efficiency claim. revision: yes

  2. Referee: [Results] Results section: The rigid-fin baseline is not described (identical planform and kinematics? infinite stiffness? same mean thrust generation protocol?). Without this, the efficiency comparison cannot be evaluated.

    Authors: The rigid-fin baseline employs the identical planform geometry and prescribed kinematics as the flexible case, but with infinite stiffness enforced (no dorso-ventral deformation permitted). The comparison is performed at matched mean thrust by using the same kinematic parameters for both cases, as the flexibility modulates the resulting forces while the motion is prescribed. We will revise the Results section to explicitly state these details, including how the rigid case is implemented numerically and confirmation that mean thrust is matched between cases. This will enable clear evaluation of the efficiency comparison. revision: yes

Circularity Check

0 steps flagged

No circularity: efficiency gain is direct output of FSI simulations

full rationale

The paper's central result (70% efficiency increase) is obtained from numerical flow-structure interaction simulations comparing flexible and rigid fins under equivalent thrust. No equations, fitted parameters, or self-citations are shown that reduce the reported efficiency or deformation effects to inputs by construction. The deformation arises from passive fluid loading in the model, and the efficiency comparison is a computed output rather than a self-definitional or renamed known result. The derivation chain is self-contained against external simulation benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated in the provided text.

pith-pipeline@v0.9.0 · 5781 in / 1152 out tokens · 57908 ms · 2026-05-22T22:57:27.465627+00:00 · methodology

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Reference graph

Works this paper leans on

53 extracted references · 53 canonical work pages

  1. [1]

    and we present a detailed description along with the validation of the model in our earlier work (52). While the fin is geometrically represented as a zero-thickness membrane in the flow solver, the dynamical model of the membrane employs a finite thickness given in the above equations by h∗= h/C. This parameter appears in both the elastic and bending mod...

    work page 2035

  2. [2]

    SA Combes, TL Daniel, Flexural stiffness in insect wings: Scaling and the influence of wing venation. J. Exp. Biol. 206, 2979–2987 (2003)

    work page 2003

  3. [3]

    Science 319, 1250–1253 (2008)

    F Muijres, et al., Leading-edge vortex improves lift in slow-flying bats. Science 319, 1250–1253 (2008)

    work page 2008

  4. [4]

    RM Alexander, Energy-saving mechanisms in walking and running. J. Exp. Biol. 160, 55–69 (1991)

    work page 1991

  5. [5]

    DM Dudek, RJ Full, An isolated insect leg’s passive recovery from dorso-ventral perturbations. J. Exp. Biol. 210, 3209–3217 (2007)

    work page 2007

  6. [6]

    DL Jindrich, RJ Full, Dynamic stabilization of rapid hexapedal locomotion. J. Exp. Biol. 205, 2803–2823 (2002)

    work page 2002

  7. [7]

    MA Daley, AA Biewener, Running over rough terrain reveals limb control for intrinsic stability. Proc. Natl. Acad. Sci. 103, 15681–15686 (2006)

    work page 2006

  8. [8]

    TJ Roberts, E Azizi, Flexible mechanisms: the diverse roles of biological springs in vertebrate movement. J. experimental biology 214, 353–361 (2011)

    work page 2011

  9. [9]

    A Gupta, S Savarese, S Ganguli, L Fei-Fei, Embodied intelligence via learning and evolution. Nat. communications 12, 5721 (2021)

    work page 2021

  10. [10]

    Biomimetics 9, 248 (2024)

    Z Zhao, et al., Exploring Embodied Intelligence in Soft Robotics: A Review. Biomimetics 9, 248 (2024)

    work page 2024

  11. [11]

    R Pfeifer, How the body shapes the way we think: A new view of intelligence (2006)

    work page 2006

  12. [12]

    D Howard, et al., Evolving embodied intelligence from materials to machines. Nat. Mach. Intell. 1, 12–19 (2019)

    work page 2019

  13. [13]

    F Fish, GV Lauder, Passive and active flow control by swimming fishes and mammals. Annu. Rev. Fluid Mech. 38, 193–224 (2006)

    work page 2006

  14. [14]

    J Long, K Nipper, The importance of body stiffness in undulatory propulsion. Am. Zool. 36, 678–694 (1996)

    work page 1996

  15. [15]

    M Taguchi, JC Liao, Rainbow trout consume less oxygen in turbulence: the energetics of swimming behaviors at different speeds. J. Exp. Biol. 214, 1428–1436 (2011)

    work page 2011

  16. [16]

    JC Liao, Neuromuscular control of trout swimming in a vortex street: implications for energy economy during the Karman gait. J. Exp. Biol. 207, 3495–3506 (2004)

    work page 2004

  17. [17]

    DN Beal, FS Hover, MS Triantafyllou, JC Liao, GV Lauder, Passive propulsion in vortex wakes. J. fluid mechanics 549, 385–402 (2006)

    work page 2006

  18. [18]

    Science 302, 1566–1569 (2003)

    JC Liao, D Beal, GV Lauder, M Triantafyllou, Fish exploiting vortices decrease muscle activity. Science 302, 1566–1569 (2003)

    work page 2003

  19. [19]

    (SIAM), (1975)

    SJ Lighthill, Mathematical biofluiddynamics. (SIAM), (1975)

    work page 1975

  20. [20]

    Bioinspiration & Biomimetics 17, 066020 (2022)

    JH Seo, R Mittal, Improved swimming performance in schooling fish via leading-edge vortex enhancement. Bioinspiration & Biomimetics 17, 066020 (2022)

    work page 2022

  21. [21]

    R Bainbridge, Caudal fin and body movement in the propulsion of some fish. J. Exp. Biol. 40, 23–56 (1963)

    work page 1963

  22. [22]

    GV Lauder, Function of the caudal fin during locomotion in fishes: kinematics, flow visualization, and evolutionary patterns. Am. Zool. 40, 101–122 (2000)

    work page 2000

  23. [23]

    BE Flammang, GV Lauder, Speed-dependent intrinsic caudal fin muscle recruitment during steady swimming in bluegill sunfish, lepomis macrochirus. J. Exp. Biol. 211, 587–598 (2008)

    work page 2008

  24. [24]

    AC Gibb, KA Dickson, GV Lauder, Tail kinematics of the chub mackerel scomber japonicus: testing the homocercal tail model of fish propulsion. J. experimental biology 202, 2433–2447 (1999)

    work page 1999

  25. [25]

    Bioinspiration & biomimetics 9, 046001 (2014)

    M Bergmann, A Iollo, R Mittal, Effect of caudal fin flexibility on the propulsive efficiency of a fish-like swimmer. Bioinspiration & biomimetics 9, 046001 (2014)

    work page 2014

  26. [26]

    D Fern ´andez-Guti´errez, WM Van Rees, Effect of leading-edge curvature actuation on flapping fin performance. J. Fluid Mech. 921, A22 (2021)

    work page 2021

  27. [27]

    MS Triantafyllou, G Triantafyllou, DK Yue, Hydrodynamics of fishlike swimming. Annu. review fluid mechanics 32, 33–53 (2000)

    work page 2000

  28. [28]

    Space Nav

    J Rohr, et al., Observations of dolphin swimming speed and strouhal number. Space Nav. Warf. Syst. Cent. Tech. Rep. 1769(1998)

    work page 1998

  29. [29]

    Nature 425, 707–711 (2003)

    GK Taylor, RL Nudds, ALR Thomas, Flying and swimming animals cruise at a Strouhal number tuned for high power efficiency. Nature 425, 707–711 (2003)

    work page 2003

  30. [30]

    GS Triantafyllou, MS Triantafyllou, MA Grosenbaugh, Optimal thrust development in oscillating foils with application to fish propulsion. J. Fluids Struct. 7, 205–224 (1993)

    work page 1993

  31. [31]

    PLOS ONE 8, 1–10 (2013)

    L Zheng, TL Hedrick, R Mittal, Time-varying wing-twist improves aerodynamic efficiency of forward flight in butterflies. PLOS ONE 8, 1–10 (2013)

    work page 2013

  32. [32]

    SJ Hsu, H Deng, J Wang, H Dong, B Cheng, Wing deformation improves aerodynamic performance of forward flight of bluebottle flies flying in a flight mill. J. Royal Soc. Interface 21, 20240076 (2024)

    work page 2024

  33. [33]

    L Johansson, P Henningsson, Butterflies fly using efficient propulsive clap mechanism owing to flexible wings. J. Royal Soc. Interface 18, 20200854 (2021)

    work page 2021

  34. [34]

    FE Fish, et al., Conceptual design for the construction of a biorobotic auv based on biological hydrodynamics in 13th International Symposium on Unmanned Untethered Submersible Technology, Durham, NH, Aug. pp. 24–27 (2003)

    work page 2003

  35. [35]

    MS Triantafyllou, AH Techet, FS Hover, Review of experimental work in biomimetic foils. IEEE J. Ocean. Eng. 29, 585–594 (2004)

    work page 2004

  36. [36]

    HS Raut, JH Seo, R Mittal, Hydrodynamic performance and scaling laws for a modelled wave-induced flapping-foil propulsor. J. Fluid Mech. 999, A1 (2024)

    work page 2024

  37. [37]

    HS Raut, JH Seo, R Mittal, Dynamics and thrust performance of a modeled multifoil wave-induced flapping foil propulsor. Ocean. Eng. 317, 119930 (2025)

    work page 2025

  38. [38]

    J Y oung, JC Lai, MF Platzer, A review of progress and challenges in flapping foil power generation. Prog. Aerosp. Sci. 67, 2–28 (2014)

    work page 2014

  39. [39]

    Q Xiao, Q Zhu, A review on flow energy harvesters based on flapping foils. J. Fluids Struct. 46, 174–191 (2014)

    work page 2014

  40. [40]

    H Dong, R Mittal, FM Najjar, Wake topology and hydrodynamic performance of low-aspect-ratio flapping foils. J. Fluid Mech. 566, 309–343 (2006)

    work page 2006

  41. [41]

    JM Anderson, K Streitlien, D Barrett, MS Triantafyllou, Oscillating foils of high propulsive efficiency. J. Fluid mechanics 360, 41–72 (1998)

    work page 1998

  42. [42]

    PLoS One 14, e0218672 (2019)

    S Sekhar, P Windes, X Fan, DK Tafti, Canonical description of wing kinematics and dynamics for a straight flying insectivorous bat (hipposideros pratti). PLoS One 14, e0218672 (2019)

    work page 2019

  43. [43]

    R Mittal, et al., A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries. J. computational physics 227, 4825–4852 (2008)

    work page 2008

  44. [44]

    K Shoele, R Mittal, Computational study of flow-induced vibration of a reed in a channel and effect on convective heat transfer. Phys. Fluids 26 (2014)

    work page 2014

  45. [45]

    K Shoele, R Mittal, Flutter instability of a thin flexible plate in a channel. J. Fluid Mech. 786, 29–46 (2016)

    work page 2016

  46. [46]

    S Bailoor, JH Seo, LP Dasi, S Schena, R Mittal, A computational study of the hemodynamics of bioprosthetic aortic valves with reduced leaflet motion. J. Biomech. 120, 110350 (2021)

    work page 2021

  47. [47]

    H Dong, M Bozkurttas, R Mittal, P Madden, GV lauder, Computational modelling and analysis of the hydrodynamics of a highly deformable fish pectoral fin. J. Fluid Mech. 645, 345–373 (2010)

    work page 2010

  48. [48]

    arXiv preprint arXiv:2501.09299 (2025)

    J Zhou, JH Seo, R Mittal, Hydrodynamically beneficial school configurations in carangiform swimmers: Insights from a flow-physics informed model. arXiv preprint arXiv:2501.09299 (2025)

    work page arXiv 2025

  49. [49]

    L Zheng, TL Hedrick, R Mittal, A multi-fidelity modelling approach for evaluation and optimization of wing stroke aerodynamics in flapping flight. J. Fluid Mech. 721, 118–154 (2013)

    work page 2013

  50. [50]

    MD de Tullio, G Pascazio, A moving-least-squares immersed boundary method for simulating the fluid–structure interaction of elastic bodies with arbitrary thickness. J. Comput. Phys. 325, 201–225 (2016)

    work page 2016

  51. [51]

    DA Fedosov, B Caswell, GE Karniadakis, Systematic coarse-graining of spectrin-level red blood cell models. Comput. Methods Appl. Mech. Eng. 199, 1937–1948 (2010)

    work page 1937

  52. [52]

    V Spandan, et al., A parallel interaction potential approach coupled with the immersed boundary method for fully resolved simulations of deformable interfaces and membranes. J. computational physics 348, 567–590 (2017)

    work page 2017

  53. [53]

    arXiv preprint arXiv:2501.02034 (2025)

    S Kumar, JH Seo, R Mittal, Computational modeling and analysis of the coupled aero structural dynamics in bat inspired wings. arXiv preprint arXiv:2501.02034 (2025). Kumar et al

    work page arXiv 2025