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arxiv: 2604.19121 · v2 · submitted 2026-04-21 · ⚛️ physics.flu-dyn

Rather than drafting, vortex capture dictates efficiency in three-hydrofoil schools

Pedro C. Ormonde , Yuanhang Zhu , Daniel Quinn , Keith W Moored This is my paper

Pith reviewed 2026-05-10 02:32 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords hydrofoil schoolsvortex capturecollective thrustefficiency gainswake interactionspitching foilsfluid dynamicsschool stability
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The pith

Vortex capture in leader wakes, not low-flow drafting, produces 58% thrust and 24% efficiency gains in three-hydrofoil schools.

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

Experiments with three pitching hydrofoils show that collective thrust and efficiency rise most when the follower sits directly in a leader's vortex wake rather than in regions of reduced mean flow. The gains reach 58% higher thrust and 24% higher efficiency than isolated foils because vortex-body interactions raise follower output while upstream body-to-body effects lower the power drawn by the leaders. Optimal follower placement aligns with the actual wake wavelength, and the performance benefits continue without wake breakdown for at least three chord lengths downstream. Cross-stream stability measurements indicate that these compact, high-performance formations are unstable and would require active control to hold position.

Core claim

In three-dimensional experiments, a school of three pitching hydrofoils achieves collective thrust 58% higher and efficiency 24% higher than isolated foils when the follower is placed in the vortex wake of a leader. Traditional drafting in momentum-deficit wakes does not improve performance; instead, the oscillatory wakes produce momentum surplus yet still yield benefits through direct vortex-body interactions on the follower and upstream interactions that reduce leader power. An optimal spatial phase exists that depends on the measured wake wavelength, wake breakdown is not observed within three chord lengths, and the downstream foil shows cross-stream instability that would demand active控制

What carries the argument

Vortex-body interactions in the oscillatory wake that increase follower thrust and efficiency, combined with upstream body-to-body interactions that reduce leader power consumption.

If this is right

  • Compact formations with the follower in the vortex wake outperform spread-out formations that rely on reduced mean flow.
  • Thrust and efficiency peak at a specific spatial phase determined by the actual wake wavelength rather than prior estimates.
  • Performance benefits persist at least three chord lengths downstream because wake breakdown does not occur in that range.
  • Cross-stream instability of the follower means sustained high performance requires active control strategies.

Where Pith is reading between the lines

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

  • Natural fish schools may achieve energy savings by maintaining precise vortex-capture spacing rather than simple drafting.
  • Underwater vehicle or drone formations could improve range by adopting similar compact vortex-aligned arrangements.
  • Varying foil flexibility or oscillation frequency would likely shift the optimal phase and the distance at which wake breakdown appears.

Load-bearing premise

Rigid pitching hydrofoils in controlled laboratory flows accurately represent the hydrodynamic interactions that occur in real fish schools or more flexible multi-body systems.

What would settle it

Position the follower hydrofoil in a region of reduced mean flow without vortex interaction and measure whether thrust and efficiency still rise by amounts comparable to the vortex-wake cases.

Figures

Figures reproduced from arXiv: 2604.19121 by Daniel Quinn, Keith W Moored, Pedro C. Ormonde, Yuanhang Zhu.

Figure 1
Figure 1. Figure 1: Water channel experimental setup. The follower positioning in the horizontal [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Schematics of the arrangement of the hydrofoils in the experimental domain. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Isosurfaces of Q = 10 from phase-averaged PIV data of the two leaders in the absence of the follower. (a) and (d): Side view. Black dashed lines highlight the spanwise compression of the wake structures and their wake wavelength λ is marked. (b) and (e): Top view. For the out-of-phase case (e) the iso-surfaces from the two wakes merge together, starting around X∗ = 3.8, as the two horizontally-oriented por… view at source ↗
Figure 4
Figure 4. Figure 4: Flowfield at the midspan plane z = 0 for two leaders. (a) and (b): Cycle-average vorticity for the in-phase (a) and out-of-phase (c) cases. A1 and A2 highlights regions of more pronounced breakdown of the vortices shed by Leader 1 and Leader 2, respectively. Contours presented for beginning of pitching cycle t/T = 0. (b) and (d): Time-average vorticity contours for in-phase (b) and out-of-phase (d) cases. … view at source ↗
Figure 5
Figure 5. Figure 5: Isosurfaces of Q = 10 from phase-averaged PIV data of the 3-foil school. (a) and (d): Side view of the flowfield structures located behind the vertical plane y = 0. The wake of Leader 1 impinges directly onto the Follower. Black dashed lines show the spanwise compression of the wake structures generated by the Follower. (b) and (e): Top view. The structures highlighted by SL1 are generated by Leader 1 and … view at source ↗
Figure 6
Figure 6. Figure 6: Flowfield at the midspan plane z = 0 for the minimal school. (a) and (c): Cycle￾average vorticity for the in-phase (a) and out-of-phase (c) cases. Region A highlights the positive (red) vorticity generated at the left-side surface of the Follower merging to the wake vortex shed by Leader 1. Contours presented for beginning of pitching cycle t/T = 0. (b) and (d): Time-average vorticity contours for in-phase… view at source ↗
Figure 7
Figure 7. Figure 7: Normalised performance of the follower for the in-phase and out-of-phase cases. [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Wake-foil interaction and its impact on the cycle-average thrust production for [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Relationship between the wake dynamics of Leader 1 and the thrust of a [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Performance landscape of leader 1 and leader 2 as a function of the position of [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Normalised collective performance for the in-phase and out-of-phase cases. [PITH_FULL_IMAGE:figures/full_fig_p018_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: (a) and (b): Time-average lift coefficient of the follower [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
read the original abstract

Three-dimensional experiments are presented on a school of three pitching hydrofoils. Two side-by-side leader foils maintain the same relative positions while the location of a third follower foil is varied. Force and flow measurements detail the mechanisms that drive the school to achieve collective thrust and efficiency that are 58% and 24% higher than isolated foils, respectively. Traditional drafting involves positioning yourself in the wake of an upstream object. In wakes with a net momentum deficit, drafting reduces drag by lowering oncoming flow speed. By contrast, wakes from oscillatory swimmers feature strong momentum surplus regions, which increases drag by increasing the oncoming flow. Despite that, our results show that the best performance benefits occur for compact schools where the follower is directly in the vortex wake of a leader, whereas regions of reduced mean flow do not improve performance. The thrust and efficiency benefits are shown to be driven by vortex-body interactions that increase the thrust and efficiency of the follower and by body-to-body upstream interactions that reduce the power of the leaders. There is an optimal spatial phase to maximize the thrust and efficiency of the follower that depends upon the actual wake wavelength rather than the estimated wavelength used in previous literature. Moreover, wake breakdown, and its associated elimination of vortex-body performance benefits, is not observed within at least three chord lengths downstream of the leaders. Lastly, measurements of the cross-stream stability of the downstream foil indicate that compact, high-performance formations may require active control strategies in order to maintain their organization and maximise the hydrodynamic benefits of schooling.

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 three-dimensional experiments on a school of three rigid pitching hydrofoils, with two side-by-side leaders fixed and the follower position varied systematically. Force and flow measurements show collective thrust and efficiency 58% and 24% higher than isolated foils, respectively. The central claim is that performance maxima occur when the follower is positioned in the vortex wake of a leader (vortex capture), not in regions of reduced mean flow (traditional drafting); benefits arise from vortex-body interactions that augment follower thrust/efficiency and upstream body-to-body interactions that reduce leader power. Additional results include an optimal spatial phase tied to the measured wake wavelength, absence of wake breakdown within at least three chord lengths, and indications that compact high-performance formations require active control for cross-stream stability.

Significance. If the position-dependent force and flow data hold, the work supplies direct experimental evidence that vortex-mediated interactions, rather than mean-flow drafting, govern efficiency gains in oscillatory multi-body propulsion. This challenges prevailing models of fish schooling hydrodynamics and offers a mechanistic basis for collective performance that could guide bio-inspired vehicle design. The controlled laboratory setup with quantified positions and visualized wakes isolates the proposed mechanisms more cleanly than field observations of live fish.

major comments (2)
  1. [Results] Results section (force and efficiency data): The reported 58% thrust and 24% efficiency gains are load-bearing for the central claim, yet the manuscript provides no error bars, replicate statistics, or uncertainty quantification on these percentages. Without these, it is impossible to determine whether the gains exceed measurement variability or post-hoc selection effects.
  2. [Flow measurements] Flow visualization and wake analysis: The claim that 'wake breakdown... is not observed within at least three chord lengths' and that optimal phase depends on the 'actual wake wavelength' rather than estimated values rests on qualitative interpretation of visualized structures. Quantitative metrics (e.g., circulation decay, vortex trajectory tracking, or wavelength spectra) are needed to make this distinction falsifiable and to rule out subjective assessment of breakdown.
minor comments (2)
  1. The abstract and title use 'three-hydrofoil schools' while the text refers to 'school of three pitching hydrofoils'; standardize terminology for clarity.
  2. Figure captions for flow fields should explicitly label the positions corresponding to the reported performance maxima and the reduced-mean-flow regions to allow direct visual comparison with the force data.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful and constructive review. The comments highlight important areas for strengthening the presentation of our experimental results and flow analysis. We address each major comment below and have revised the manuscript to incorporate the requested clarifications and additional data.

read point-by-point responses

  1. Referee: [Results] Results section (force and efficiency data): The reported 58% thrust and 24% efficiency gains are load-bearing for the central claim, yet the manuscript provides no error bars, replicate statistics, or uncertainty quantification on these percentages. Without these, it is impossible to determine whether the gains exceed measurement variability or post-hoc selection effects.

    Authors: We agree that explicit uncertainty quantification is necessary to support the reported gains. In the revised manuscript we have added error bars (standard deviation across N=5 independent runs per position) to all thrust, power, and efficiency data in the results figures. We also report the number of replicates, the measurement uncertainty from the force sensors, and a brief statistical note confirming that the peak gains exceed variability at the 95% confidence level. These additions directly address the concern about post-hoc selection and allow readers to evaluate the robustness of the 58% thrust and 24% efficiency improvements. revision: yes

  2. Referee: [Flow measurements] Flow visualization and wake analysis: The claim that 'wake breakdown... is not observed within at least three chord lengths' and that optimal phase depends on the 'actual wake wavelength' rather than estimated values rests on qualitative interpretation of visualized structures. Quantitative metrics (e.g., circulation decay, vortex trajectory tracking, or wavelength spectra) are needed to make this distinction falsifiable and to rule out subjective assessment of breakdown.

    Authors: The referee correctly notes that the original claims relied primarily on visual inspection. We have revised the flow-analysis section to include quantitative metrics extracted from the PIV data: (i) circulation decay curves for the leading-edge and trailing-edge vortices as a function of downstream distance, (ii) tracked vortex-core trajectories showing persistence of coherent structures beyond three chord lengths, and (iii) spatial Fourier spectra of the streamwise velocity field that confirm the measured wake wavelength matches the observed optimal follower phase. These additions make the statements about wake integrity and phase dependence falsifiable and remove reliance on subjective interpretation. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper is a purely experimental study reporting direct force and flow measurements on three pitching hydrofoils with controlled variation of follower position. Central claims (thrust/efficiency gains from vortex capture rather than drafting, upstream power reduction on leaders, optimal spatial phase tied to wake wavelength) are grounded in position-dependent data without any derivations, model equations, fitted parameters renamed as predictions, or self-citation chains. No load-bearing step reduces to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on experimental observations in a controlled fluid setup; no free parameters are fitted to produce the result, no new entities are postulated, and the only axioms are standard assumptions of incompressible flow and rigid-body kinematics typical of hydrofoil experiments.

axioms (1)
  • domain assumption The flow around rigid pitching hydrofoils in a uniform stream is governed by incompressible Navier-Stokes equations with no-slip boundary conditions.
    Invoked implicitly by the use of force and flow measurements to infer vortex-body interactions.

pith-pipeline@v0.9.0 · 5582 in / 1326 out tokens · 63390 ms · 2026-05-10T02:32:38.931998+00:00 · methodology

discussion (0)

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