pith. sign in

arxiv: 2604.21054 · v1 · submitted 2026-04-22 · ⚛️ physics.flu-dyn

Hydrodynamic loads and vortex evolution from a bio-inspired pectoral fin near a solid body

Xiaowei He , Kenneth Breuer This is my paper

Pith reviewed 2026-05-09 22:43 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords hydrodynamic loadsvortex evolutionbio-inspired finStrouhal numberreduced frequencyhysteresisparticle image velocimetrythrust generation
0
0 comments X

The pith

Hydrodynamic loads on a bio-inspired pectoral fin near a body show significant hysteresis tied to Strouhal number and reduced frequency.

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

A rigid fin attached to a body is flapped in a water tunnel to mimic fish pectoral fins across a range of angles and speeds. The resulting forces display marked differences between the rising and falling parts of each flap cycle. Flow visualization captures how vortices form and orbit around the fin tip at higher speeds, altering the forces produced. A statistical method then identifies that the force magnitudes are best predicted by quadratic expressions involving the Strouhal number together with the reduced frequency.

Core claim

Quasi-steady hydrodynamic loads exhibit significant hysteresis during the upstroke and downstroke phases of the fin flapping. Particle image velocimetry measurements show the details of the shear layer and vortex development in dynamic flapping cases, including orbiting behaviors of the fin tip vortices in larger Strouhal number cases. The strong dependency on the reduced frequency and Strouhal number leads to scalings of the hydrodynamic loads using a data-driven method, where the most significant terms selected are quadratic terms of the Strouhal number and its nonlinear combinations with the reduced frequency. PIV results reveal the influence of vortices on hydrodynamic loads in terms of

What carries the argument

Data-driven selection of scaling terms for hydrodynamic loads based on quadratic Strouhal number and nonlinear combinations with reduced frequency, observed via PIV vortex orbiting and shear layer development.

If this is right

  • Hysteresis causes substantially different force production between the upstroke and downstroke phases.
  • Orbiting fin tip vortices appear at larger Strouhal numbers and contribute to thrust generation.
  • Vortex development directly drives observed lift fluctuations.
  • Hydrodynamic loads can be predicted from motion parameters using quadratic Strouhal terms and their nonlinear interactions with reduced frequency.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The reported scalings could allow simplified load estimates for robotic fins operating near hulls without full fluid simulations.
  • If hysteresis persists for flexible fins, it may alter how fish select flapping frequencies for efficient propulsion close to their bodies.
  • Vortex orbiting near the body may represent an exploitable mechanism for enhanced thrust in bio-inspired designs.

Load-bearing premise

The idealized rigid-fin flapping motion and water-tunnel conditions sufficiently represent the flexible, three-dimensional motions and flow environments of actual fish pectoral fins for the observed hysteresis, vortex orbiting, and scalings to generalize.

What would settle it

A follow-up experiment using a flexible three-dimensional fin in realistic flow that shows no hysteresis between upstroke and downstroke or different dominant scaling terms would falsify the reported load behaviors.

Figures

Figures reproduced from arXiv: 2604.21054 by Kenneth Breuer, Xiaowei He.

Figure 1
Figure 1. Figure 1: Fin-body experimental setup: (a) physical model; (b) coordinates and geometry [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Baseline hydrodynamic loads vs. fin flap angle [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: PIV results for the baseline case (𝑘 = 0.006) at 𝜃 = 10◦ : (a) the fin moves away from the body (𝜃 increasing); (b) the fin moves toward the body (𝜃 decreasing); the color map shows the normalized vorticity field. flap-up (𝜃 increasing) and flap-down (𝜃 decreasing) phases of the fin flapping are marked by blue and green colors, respectively. The baseline lift coefficient varies from zero to −0.7 as the fla… view at source ↗
Figure 4
Figure 4. Figure 4: Dynamic lift coefficients fluctuations Δ𝐶𝐿: (a) 𝑓 = 0.25 Hz, 𝑘 = 0.16; (b) 𝑓 = 0.5 Hz, 𝑘 = 0.31; (c) 𝑓 = 1 Hz, 𝑘 = 0.63; (d) 𝑓 = 2 Hz, 𝑘 = 1.26; blue data points: 𝐴𝜃 = 7.5 ◦ ; red data points: 𝐴𝜃 = 15◦ ; green data points: 𝐴𝜃 = 30◦ ; error bars: 1𝜎 standard deviation intervals; white solid line: normalized fin angle 𝜃/(2𝐴𝜃 ); white dotted line: Δ𝐶𝐿 = 0 reference. observed from the positive (red) color map … view at source ↗
Figure 5
Figure 5. Figure 5: PIV results at 𝑘 = 1.257 ( 𝑓 = 2 Hz) and 𝑆𝑡 = 0.105 (𝐴𝜃 = 7.5 ◦ ); (a) to (d) corresponds to snapshots of flow fields at 𝑡/𝑇 = 0, 1/4, 1/2, and 3/4, respectively; color map: nondimensional vorticity 𝜔 ∗ 𝑧 ; green arrow: instantaneous flapping direction [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: PIV results at 𝑘 = 1.257 ( 𝑓 = 2 Hz) and 𝑆𝑡 = 0.209 (𝐴𝜃 = 15◦ ); (a) to (d) corresponds to snapshots of flow fields at 𝑡/𝑇 = 0, 1/4, 1/2, and 3/4, respectively; color map: nondimensional vorticity 𝜔 ∗ 𝑧 ; green arrow: instantaneous flapping direction. through wall shear, as mentioned above. Intuitively, and as the PIV results confirm, this suction effect is more apparent at high 𝑆𝑡 since the fin accelerati… view at source ↗
Figure 7
Figure 7. Figure 7: PIV results at 𝑘 = 1.257 ( 𝑓 = 2 Hz) and 𝑆𝑡 = 0.419 (𝐴𝜃 = 30◦ ); (a) to (d) corresponds to snapshots of flow fields at 𝑡/𝑇 = 0, 1/4, 1/2, and 3/4, respectively; color map: nondimensional vorticity 𝜔 ∗ 𝑧 ; green arrow: instantaneous flapping direction [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Dynamic drag coefficients fluctuations Δ𝐶𝐷: (a) 𝑓 = 0.25 Hz, 𝑘 = 0.16; (b) 𝑓 = 0.5 Hz, 𝑘 = 0.31; (c) 𝑓 = 1 Hz, 𝑘 = 0.63; (d) 𝑓 = 2 Hz, 𝑘 = 1.26; blue data points: 𝐴𝜃 = 7.5 ◦ ; red data points: 𝐴 = 15◦ 𝜃 ; green data points: 𝐴𝜃 = 30◦ ; error bars: 1𝜎 standard deviation intervals; white solid line: normalized fin angle 𝜃/(2𝐴𝜃 ); white dotted line: Δ𝐶𝐷 = 0 reference. geometry change and the added mass effect … view at source ↗
Figure 9
Figure 9. Figure 9: Drag coefficient variation at 𝑘 = 1.26 ( 𝑓 = 2 Hz) and 𝑆𝑡 = 0.209 (𝐴𝜃 = 15◦ ): the dotted white line is the Δ𝐶𝐷 = 0 reference; numbers mark out the data points of interest, where the fin flaps down, and thrust is generated (negative Δ𝐶𝐷) [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Flow fields of interested data points in drag fluctuation; color map: [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Flowchart of SINDy algorithm (He et al. 2019). play significant roles, as indicated by the selection from SINDy iterations. The identification of drag scaling is interpreted in the same way. In addition, the candidate selection results are more sensitive to the sparsity parameter as shown in figure 12b. The number of identified scaling terms in the 𝜣𝑪𝑫 vector decreases from 4 to 1 as 𝜆 changes from 0.1 to… view at source ↗
Figure 12
Figure 12. Figure 12: SINDy coefficients identification results: (a) lift scaling; (b) drag scaling; the [PITH_FULL_IMAGE:figures/full_fig_p013_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Amplitude scaling results: (a) lift amplitudes; (b) drag amplitudes; vertical [PITH_FULL_IMAGE:figures/full_fig_p013_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Scaling errors: ratio of absolute scaling errors to their corresponding [PITH_FULL_IMAGE:figures/full_fig_p013_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Scaled load fluctuations: (a) and (b) are scaled lift fluctuations, [PITH_FULL_IMAGE:figures/full_fig_p014_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Phase lag between the scaled hydrodynamic loads and the fin flapping angle: [PITH_FULL_IMAGE:figures/full_fig_p015_16.png] view at source ↗
read the original abstract

A fin-body configuration is tested in a water tunnel to study the hydrodynamic loads and vortex evolution under dynamic fin-flapping motions, which is an idealized approximation of the pectoral fins of fish. The fin flaps about its leading edge, which is attached to the side of the body, at a range of combinations of amplitudes ($0^\circ-30^\circ$) and frequencies ($0.25\,\mathrm{Hz}-2\,\mathrm{Hz}$ or $k=0.16-1.26$), so the Strouhal number ($St=0.013-0.419$). The quasi-steady hydrodynamic loads exhibit significant hysteresis during the upstroke and downstroke phases of the fin flapping. Particle image velocimetry (PIV) measurements show the details of the shear layer and vortex development in dynamic flapping cases. Orbiting behaviors of the fin tip vortices are observed in larger Strouhal number cases. PIV results also reveal the influence of vortices on hydrodynamic loads in terms of lift fluctuations and thrust generation. The strong dependency on the reduced frequency and Strouhal number leads to scalings of the hydrodynamic loads using a data-driven method to select highly correlated terms. The most significant terms selected by the scaling process are quadratic terms of the Strouhal number and its nonlinear combinations with the reduced frequency.

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 water-tunnel experiments on a rigid bio-inspired pectoral fin attached to a body, flapping about its leading edge over amplitudes 0°–30° and frequencies 0.25–2 Hz (corresponding to reduced frequencies k = 0.16–1.26 and Strouhal numbers St = 0.013–0.419). It documents significant hysteresis in the quasi-steady hydrodynamic loads between upstroke and downstroke phases, presents PIV visualizations of shear-layer and vortex evolution including orbiting of fin-tip vortices at higher St, and applies a data-driven method to identify scaling terms for the loads, concluding that quadratic terms in St and their nonlinear combinations with k are the most significant.

Significance. The experimental observations of load hysteresis and vortex dynamics provide concrete data on unsteady fin-body interactions that could inform bio-inspired underwater propulsion. The standard water-tunnel load-cell and PIV measurements lend support to the qualitative claims of hysteresis and vortex orbiting. If the data-driven scalings can be shown to be robust and reproducible, they would supply useful empirical relations for predicting forces in similar configurations.

major comments (2)
  1. [scaling analysis following PIV results] The data-driven scaling process for the hydrodynamic loads (described after the PIV results) lacks any specification of the term-selection algorithm, regularization approach, multicollinearity diagnostics, or validation procedure (e.g., cross-validation, out-of-sample testing, or sensitivity to the tested ranges St = 0.013–0.419 and k = 0.16–1.26). Without these, it is impossible to assess whether the reported dominance of quadratic St terms and St–k nonlinear combinations is unique or an artifact of the particular dataset and noise level in the load measurements.
  2. [scaling analysis following PIV results] The central empirical claim that the selected quadratic-St and nonlinear St–k terms provide the scaling for the loads is presented without error bars on the fit coefficients, goodness-of-fit metrics, or comparison against simpler models (e.g., linear St or k-only terms). This leaves the quantitative utility of the scaling unverified even within the experimental parameter space.
minor comments (2)
  1. [methods] Notation for the reduced frequency k and Strouhal number St should be defined explicitly at first use in the methods section rather than only in the abstract.
  2. [PIV results figures] Figure captions for the PIV velocity fields would benefit from explicit indication of the phase within the flapping cycle and the instantaneous angle of attack.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments on our manuscript. We address each major comment below and have revised the scaling analysis section to incorporate the requested methodological details, quantitative metrics, and comparisons.

read point-by-point responses

  1. Referee: The data-driven scaling process for the hydrodynamic loads (described after the PIV results) lacks any specification of the term-selection algorithm, regularization approach, multicollinearity diagnostics, or validation procedure (e.g., cross-validation, out-of-sample testing, or sensitivity to the tested ranges St = 0.013–0.419 and k = 0.16–1.26). Without these, it is impossible to assess whether the reported dominance of quadratic St terms and St–k nonlinear combinations is unique or an artifact of the particular dataset and noise level in the load measurements.

    Authors: We agree that the original manuscript provided insufficient detail on the data-driven scaling procedure. In the revised version, we have expanded this section to specify that term selection uses forward stepwise regression based on Pearson correlation with the load targets, with no regularization applied given the small candidate pool (polynomials up to quadratic order in St and k). Multicollinearity is diagnosed via variance inflation factors (all selected terms have VIF < 5). Validation consists of 5-fold cross-validation plus sensitivity checks over the full experimental ranges of St and k, which confirm that the quadratic-St and St–k interaction terms remain the dominant contributors with stable coefficients. revision: yes

  2. Referee: The central empirical claim that the selected quadratic-St and nonlinear St–k terms provide the scaling for the loads is presented without error bars on the fit coefficients, goodness-of-fit metrics, or comparison against simpler models (e.g., linear St or k-only terms). This leaves the quantitative utility of the scaling unverified even within the experimental parameter space.

    Authors: We acknowledge this shortcoming in the original presentation. The revised manuscript now includes standard errors on all regression coefficients, reports R² and adjusted R² values (0.88–0.93 for the selected model), and provides explicit comparisons against baseline linear models in St alone, k alone, and St + k. These comparisons show that the quadratic and nonlinear terms yield statistically significant improvements in fit and reduce residual variance by 25–40 % within the tested parameter space, thereby supporting the reported dominance. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives scalings for hydrodynamic loads via explicit data-driven term selection on measured loads, using physically independent inputs (St and k ranges from flapping parameters). This is an empirical correlation process on the experimental dataset rather than any self-definitional loop, fitted parameter renamed as prediction, or load-bearing self-citation. Vortex evolution and hysteresis claims rest directly on PIV and force measurements without reduction to prior author results or ansatz smuggling. The derivation chain is self-contained against the reported water-tunnel data.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard incompressible-flow assumptions and an empirical data-driven scaling whose coefficients are fitted to the present dataset; no new physical entities are postulated.

free parameters (1)
  • coefficients of selected scaling terms
    The data-driven method selects and fits quadratic and nonlinear combinations of Strouhal number and reduced frequency to the measured loads.
axioms (1)
  • domain assumption Quasi-steady approximation remains useful for describing time-varying loads during flapping
    The abstract explicitly refers to quasi-steady hydrodynamic loads that nevertheless exhibit hysteresis.

pith-pipeline@v0.9.0 · 5535 in / 1330 out tokens · 26846 ms · 2026-05-09T22:43:17.652338+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

  1. [1]

    Integrative and Comparative Biology42(1), 102–117

    Bandyopadhyay, Promode R.2002 Maneuvering hydrodynamics of fish and small underwater vehicles. Integrative and Comparative Biology42(1), 102–117

    work page 2002

  2. [2]

    & Bandyopadhyay, Promode R.2007 A harmonic model of hydrodynamic forces produced by a flapping fin.Experiments in Fluids43(5), 675–682

    Beal, David N. & Bandyopadhyay, Promode R.2007 A harmonic model of hydrodynamic forces produced by a flapping fin.Experiments in Fluids43(5), 675–682

    work page 2007

  3. [3]

    W.1979 The Mechanics of Labriform Locomotion: I

    Blake, R. W.1979 The Mechanics of Labriform Locomotion: I. Labriform Locomotion in the Angelfish (Pterophyllum Eimekei): an Analysis of the Power Stroke.Journal of Experimental Biology82(1), 255–271

    work page 1979

  4. [4]

    Bozkurttas, M., Mittal, R., Dong, H., Lauder, G. V. & Madden, P.2009 Low-dimensional models and performance scaling of a highly deformable fish pectoral fin.Journal of Fluid Mechanics631, 311–342. 16

    work page 2009

  5. [5]

    & Smits, Alexander J.2013 Scaling laws for the thrust production of flexible pitching panels.Journal of Fluid Mechanics732, 29–46

    Dewey, Peter A., Boschitsch, Birgitt M., Moored, Keith W., Stone, Howard A. & Smits, Alexander J.2013 Scaling laws for the thrust production of flexible pitching panels.Journal of Fluid Mechanics732, 29–46

    work page 2013

  6. [6]

    Journal of Experimental Biology200(8), 1165–1178

    Domenici, Paolo & Blake, Robert W.1997 The kinematics and performance of fish fast-start swimming. Journal of Experimental Biology200(8), 1165–1178

    work page 1997

  7. [7]

    & Lauder, G

    Dong, H., Bozkurttas, M., Mittal, R., Mandden, P. & Lauder, G. V.2010 Computational modelling and analysis of the hydrodynamics of a highly deformable fish pectoral fin.Journal of Fluid Mechanics 645, 345–373

    work page 2010

  8. [8]

    & Najjar, F

    Dong, H., Mittal, R. & Najjar, F. M.2006 Wake topology and hydrodynamic performance of low-aspect- ratio flapping foils.Journal of Fluid Mechanics566, 309–343

    work page 2006

  9. [9]

    Drucker, Eliot G. & Lauder, George V.1999 Locomotor forces on a swimming fish: three-dimensional vortex wake dynamics quantified using digital particle image velocimetry.Journal of Experimental Biology202(18), 2393–2412

    work page 1999

  10. [10]

    & Smits, Alexander J.2017 Scaling the propulsive performance of heaving and pitching foils.Journal of Fluid Mechanics822, 386–397

    Floryan, Daniel, Van Buren, Tyler, Rowley, Clarence W. & Smits, Alexander J.2017 Scaling the propulsive performance of heaving and pitching foils.Journal of Fluid Mechanics822, 386–397

    work page 2017

  11. [11]

    Floryan, Daniel, Van Buren, Tyler & Smits, Alexander J.2019 Large-amplitude oscillations of foils for efficient propulsion.Physical Review Fluids4(9), 093102

    work page 2019

  12. [12]

    & Smits, Alexander J.2011 The unsteady three-dimensional wake produced by a trapezoidal pitching panel.Journal of Fluid Mechanics685, 117–145

    Green, Melissa A., Rowley, Clarence W. & Smits, Alexander J.2011 The unsteady three-dimensional wake produced by a trapezoidal pitching panel.Journal of Fluid Mechanics685, 117–145

    work page 2011

  13. [13]

    InActive Flow and Combustion Control 2018, pp

    He, Xiaowei, Le Provost, Mathieu, An, Xuanhong & Williams, David R.2019 Unsteady roll moment control using active flow control on a delta wing. InActive Flow and Combustion Control 2018, pp. 19–32. Springer International Publishing

    work page 2019

  14. [14]

    He, Xiaowei & Williams, David R.2023 Pressure feedback control of aerodynamic loads on a delta wing in transverse gusts.AIAA Journal61(4), 1659–1674

    work page 2023

  15. [15]

    IEEE Journal of Oceanic Engineering25(1), 121–129

    Kato, N.2000 Control performance in the horizontal plane of a fish robot with mechanical pectoral fins. IEEE Journal of Oceanic Engineering25(1), 121–129

    work page 2000

  16. [16]

    Lauder, George V.2015 Fish locomotion: Recent advances and new directions.Annual Review of Marine Science7, 521–545

    work page 2015

  17. [17]

    & Triantafyllou, Michael S.2003 The K ´arm´an gait: novel body kinematics of rainbow trout swimming in a vortex street.Journal of Experimental Biology206(6), 1059–1073

    Liao, James C., Beal, David N., Lauder, George V. & Triantafyllou, Michael S.2003 The K ´arm´an gait: novel body kinematics of rainbow trout swimming in a vortex street.Journal of Experimental Biology206(6), 1059–1073

    work page 2003

  18. [18]

    M¨uller, Ulrike K., Stamhuis, Eize J

    Maertens, A P, Triantafyllou, M S & Yue, D K P2015 Efficiency of fish propulsion.Bioinspiration & Biomimetics10(4), 046013. M¨uller, Ulrike K., Stamhuis, Eize J. & Videler, John J.2002 Riding the waves: the role of the body wave in undulatory fish swimming1.Integrative and Comparative Biology42(5), 981–987. 17

    work page 2002

  19. [19]

    & Piva, R.2021 On the role of added mass and vorticity release for self-propelled aquatic locomotion.Journal of Fluid Mechanics918, A45

    Paniccia, D., Graziani, G., Lugni, C. & Piva, R.2021 On the role of added mass and vorticity release for self-propelled aquatic locomotion.Journal of Fluid Mechanics918, A45

    work page 2021

  20. [20]

    E., Domel, A

    Saadat, M., Fish, F. E., Domel, A. G., Di Santo, V., Lauder, G. V. & Haj-Hariri, H.2017 On the rules for aquatic locomotion.Physical Review Fluids2, 083102

    work page 2017

  21. [21]

    & Grosenbaugh, M.A.1993 Optimal thrust development in oscillating foils with application to fish propulsion.Journal of Fluids and Structures7(2), 205–224

    Triantafyllou, G.S., Triantafyllou, M.S. & Grosenbaugh, M.A.1993 Optimal thrust development in oscillating foils with application to fish propulsion.Journal of Fluids and Structures7(2), 205–224

    work page 1993

  22. [22]

    S., Triantafyllou, G

    Triantafyllou, M. S., Triantafyllou, G. S. & Yue, D. K. P.2000 Hydrodynamics of Fishlike Swimming. Annual Review of Fluid Mechanics32(1), 33–53. Van Buren, Tyler, Floryan, Daniel & Smits, Alexander J.2019 Scaling and performance of simultaneously heaving and pitching foils.AIAA Journal57(9), 3666–3677

    work page 2000

  23. [23]

    J., M ¨uller, U

    Videler, J. J., M ¨uller, U. K. & Stamhuis, E. J.1999 Aquatic vertebrate locomotion: wakes from body waves.Journal of Experimental Biology202(23), 3423–3430

    work page 1999

  24. [24]

    Walker, Jeffrey A. & Westneat, Mark W.1997 Labriform propulsion in fishes: Kinematics of flapping aquatic flight in the bird wrasse gomphosus varius (labridae).Journal of Experimental Biology 200(11), 1549–1569

    work page 1997