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arxiv: 2410.04626 · v2 · pith:QHQWR4XMnew · submitted 2024-10-06 · ⚛️ physics.flu-dyn

On the stability of an in-line formation of hydrodynamically interacting flapping plates

Monika Nitsche , Anand U. Oza , Michael Siegel This is my paper

Pith reviewed 2026-05-23 19:55 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords flapping platesschooling modesvortex sheet modelhydrodynamic interactionsstabilityin-line formationflapping propulsion
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The pith

Flapping plates in an in-line formation reach stable schooling modes with separations quantized to the flapping wavelength.

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

The paper simulates two to four plates placed in a line in an inviscid incompressible fluid and actuated with prescribed vertical heaving. The resulting self-induced thrust and fluid drag cause horizontal acceleration, and in certain parameter ranges the plates settle into equilibria of steady forward speed and fixed separation. These separations are discrete multiples of the wavelength set by the flapping motion. Larger numbers of plates or smaller amplitudes trigger instability in the form of oscillations that begin at the lead plate and travel downstream, producing collisions. A control rule that modulates each plate's amplitude according to its speed relative to the plate ahead restores the equilibria and produces a more regular wake pattern.

Core claim

In certain parameter regimes, the plates adopt equilibrium schooling modes wherein they translate at a steady horizontal velocity while maintaining a constant separation distance between them. The separation distances are found to be quantized on the flapping wavelength. As either the number of plates increases or the oscillation amplitude decreases, the schooling modes destabilize via oscillations that propagate downstream from the leader and cause collisions. A simple control mechanism is implemented wherein each plate accelerates or decelerates according to its velocity relative to the plate directly ahead by modulating its own flapping amplitude, and this mechanism stabilizes the modes.

What carries the argument

The vortex sheet model that represents each plate and its wake, from which the hydrodynamic forces are computed to determine the horizontal motion and the resulting equilibria.

If this is right

  • Equilibrium separations occur only at discrete multiples of the flapping wavelength.
  • Instability takes the form of oscillations that originate at the lead plate and travel rearward, producing collisions.
  • The relative-velocity control on amplitude restores steady schooling and produces a more regular vortex pattern in the wake.
  • The same equilibria and instabilities appear in a reduced model based on linear thin-airfoil theory.

Where Pith is reading between the lines

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

  • The wavelength quantization may indicate a resonance condition between plate motion and the induced wake that could be checked in other self-propelled oscillating bodies.
  • The simple relative-velocity control might be implemented with minimal sensing in physical devices to maintain formation without centralized coordination.
  • In real viscous flows the critical amplitudes or plate counts for stability may shift, requiring adjusted control gains.

Load-bearing premise

The vortex sheet model in an inviscid incompressible fluid together with the prescribed heaving actuation is sufficient to produce the reported equilibria and instabilities without three-dimensional effects or viscosity altering the outcomes.

What would settle it

A laboratory measurement, at matching Reynolds number and heaving parameters, of whether the observed separations remain constant at exact integer multiples of the flapping wavelength or whether the predicted downstream instability appears at the same critical number of plates.

Figures

Figures reproduced from arXiv: 2410.04626 by Anand U. Oza, Michael Siegel, Monika Nitsche.

Figure 1
Figure 1. Figure 1: Schematic of the vortex sheet model. The plates (green), each of length [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Time evolution of the translation velocity [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Plate and vortex sheet position at 𝑡 = 25, for the flapping amplitude 𝐴 = 0.2 and 𝑛 = 1, 2, 3 and 4 plates (from top to bottom). All plates are initially equispaced with a distance near 𝑑0 = 4.2. shed from the trailing edge, which leads to a sequence of counter-rotating vortex pairs shed during each oscillation period. By taking the curl of (2.6), we obtain the regularized vorticity corresponding to the sh… view at source ↗
Figure 4
Figure 4. Figure 4: Same as figure 3, but the regularized vorticity is plotted instead of the vortex sheet. The plates are indicated in black. Absolute vorticity values larger than 2 are represented by the darkest blue and red colors. significantly farther than in the 𝑛 = 2 case shown in figure 4(b). A simulation with 𝑛 = 3 plates is shown in supplementary movie 2. Figures 3(d) and 4(d) show the vortex sheets and correspondin… view at source ↗
Figure 5
Figure 5. Figure 5: Snapshot at 𝑡 = 25 of three of the schooling modes obtained for a pair of plates (𝑛 = 2) with flapping amplitude 𝐴 = 0.2. The plates are initially located near the first, second and third equilibria, which, from top to bottom, correspond to the distances 𝑑 ∞ 1 = 4.18, 8.01 and 11.82, respectively. always be independent of 𝑈0, from here on the initial horizontal velocity of each plate is taken to be the ste… view at source ↗
Figure 6
Figure 6. Figure 6: Time evolution of the distances 𝑑1 (𝑡) (left column), 𝑑2 (𝑡) (middle column) and 𝑑3 (𝑡) (right column) for in-line formations of 𝑛 = 2, 3 and 4 plates, respectively. The plots in the top, middle and bottom rows correspond to the heaving amplitudes 𝐴 = 0.4, 0.3 and 0.1, respectively. The different curves in each plot are obtained by varying the initial distances 𝑑 𝑗(0), and are color-coded according to the … view at source ↗
Figure 7
Figure 7. Figure 7: Same as figure 6, but with the stabilization rule (3.3) and 𝑛 = 4 throughout. The stabilization factor 𝛽 = 0.15, 0.45, and 5.0 for the flapping amplitudes 𝐴 = 0.4, 0.3 and 0.1, respectively. Here the curves are color-coded simply by the equilibrium state they reach. original equilibrium. The basins of attraction of these equilibrium schooling modes appear to have shrunk. The results for 𝐴 = 0.1 are the mos… view at source ↗
Figure 8
Figure 8. Figure 8: Wake at 𝑡 = 25 behind a pair of plates (𝑛 = 2) with heaving amplitude 𝐴0 = 0.2 and initial separation distance 𝑑0 = 4.2. The plots are (a) without stabilization, and (b) stabilized according to (3.3) with 𝛽 = 2 [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Vorticity at 𝑡 = 50, with the same parameters as in figure 8 [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Vorticity at 𝑡 = 50 behind three plates (𝑛 = 3) with heaving amplitude 𝐴0 = 0.2 and initial separation distance 𝑑0 = 4.2. The plots are (a) without stabilization, and (b) stabilized according to (3.3) with 𝛽 = 2. Rokhlin 1987; Wang et al. 2020) can be performed in the future to study this phenomenon further [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Potential flow past a plate of unit length translating vertically with a velocity [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Impulsively started plate that translates vertically with constant velocity [PITH_FULL_IMAGE:figures/full_fig_p025_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Streamlines of the flow in figure 12, shown in a reference frame moving with the plate. the shed vortex sheet is enforced to stay constant at zero. The plate circulation is thus the negative of the circulation in the shed vortex sheet. The figure shows that both the shed sheet strength and the plate circulation approach the steady state values as 𝑡 → ∞, with e𝛾 → −2𝑉 and Γ → −𝜋𝑉. These numerical results i… view at source ↗
Figure 14
Figure 14. Figure 14: (a) Desingularized leading edge sheet strength [PITH_FULL_IMAGE:figures/full_fig_p027_14.png] view at source ↗
read the original abstract

The motion of several plates in an inviscid and incompressible fluid is studied numerically using a vortex sheet model. Two to four plates are initially placed in-line, separated by a specified distance, and actuated in the vertical direction with a prescribed oscillatory heaving motion. The vertical motion induces the plates' horizontal acceleration due to their self-induced thrust and fluid drag forces. In certain parameter regimes, the plates adopt equilibrium "schooling modes," wherein they translate at a steady horizontal velocity while maintaining a constant separation distance between them. The separation distances are found to be quantized on the flapping wavelength. As either the number of plates increases or the oscillation amplitude decreases, the schooling modes destabilize via oscillations that propagate downstream from the leader and cause collisions between the plates, an instability that is similar to that observed in recent experiments on flapping wings in a water tank (Newbolt et al., 2024). A simple control mechanism is implemented, wherein each plate accelerates or decelerates according to its velocity relative to the plate directly ahead by modulating its own flapping amplitude. This mechanism is shown to successfully stabilize the schooling modes, with remarkable impact on the regularity of the vortex pattern in the wake. Several phenomena observed in the simulations are obtained by a reduced model based on linear thin-airfoil theory.

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 paper numerically studies 2–4 in-line heaving plates in an inviscid incompressible fluid via a vortex-sheet model. It reports the emergence of equilibrium schooling modes in which the plates translate at constant horizontal speed while maintaining fixed separations that are quantized in units of the flapping wavelength. These modes destabilize for larger plate counts or smaller amplitudes via downstream-propagating oscillations that lead to collisions, an instability qualitatively similar to recent water-tank experiments. A simple feedback control that modulates each plate’s heaving amplitude based on its speed relative to the plate ahead is shown to stabilize the formations and regularize the wake. Several of the observed phenomena are recovered by a reduced linear thin-airfoil model.

Significance. If the reported equilibria and control law remain robust under the model’s idealizations, the work supplies a concrete hydrodynamic mechanism for wavelength-quantized schooling and a minimal, implementable stabilization strategy. The combination of direct simulation, linear theory, and qualitative experimental agreement is a strength; the control result in particular has clear relevance to bio-inspired multi-agent propulsion.

major comments (2)
  1. [Numerical results and reduced-model sections] The central claim that the schooling modes are genuine attractors rests on the inviscid 2-D vortex-sheet formulation. The manuscript should supply a concrete test (e.g., a short viscous regularization study or a comparison of mean thrust/drag balance with and without small viscosity) to quantify how sensitive the quantized separations and stability thresholds are to the idealizations; without such a test the physical robustness of the headline result remains an open correctness risk.
  2. [Control mechanism description] The control law is reported to stabilize the modes and improve wake regularity, yet the manuscript does not state the precise functional form of the amplitude modulation or the gain values used. Because this mechanism is presented as a practical stabilization tool, the explicit control law and its parameter sensitivity must be documented so that the result can be reproduced or extended.
minor comments (2)
  1. [Figures] Figure captions should explicitly state the non-dimensional parameters (heaving amplitude, initial separation, Reynolds number if any) for each panel so that the quantized separations can be read off without cross-referencing the text.
  2. [Reduced-model section] The linear thin-airfoil reduction is invoked to explain several phenomena; a short appendix or subsection deriving the key dispersion relation or thrust expression would make the agreement with the full simulation transparent.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive review and positive assessment of the work. We address each major comment below, indicating the revisions planned for the manuscript.

read point-by-point responses

  1. Referee: [Numerical results and reduced-model sections] The central claim that the schooling modes are genuine attractors rests on the inviscid 2-D vortex-sheet formulation. The manuscript should supply a concrete test (e.g., a short viscous regularization study or a comparison of mean thrust/drag balance with and without small viscosity) to quantify how sensitive the quantized separations and stability thresholds are to the idealizations; without such a test the physical robustness of the headline result remains an open correctness risk.

    Authors: We acknowledge the referee's concern regarding sensitivity to the inviscid idealization. The vortex-sheet model is chosen precisely to isolate the role of inviscid hydrodynamic interactions, which are expected to dominate at the high Reynolds numbers relevant to the cited experiments. A viscous regularization study would require an entirely different numerical framework and is outside the scope of the present study. That said, the quantized equilibria and downstream instability mechanism are recovered in the viscous water-tank experiments of Newbolt et al. (2024), providing indirect support for robustness. In the revised manuscript we will add a dedicated paragraph in the Discussion section that explicitly addresses the limitations of the inviscid formulation, references the experimental agreement, and notes that the reduced linear thin-airfoil model (which is also inviscid) reproduces the same qualitative thresholds. This constitutes a partial revision. revision: partial

  2. Referee: [Control mechanism description] The control law is reported to stabilize the modes and improve wake regularity, yet the manuscript does not state the precise functional form of the amplitude modulation or the gain values used. Because this mechanism is presented as a practical stabilization tool, the explicit control law and its parameter sensitivity must be documented so that the result can be reproduced or extended.

    Authors: The referee correctly identifies that the explicit form of the control law and the gain values were omitted from the original text. This was an oversight. The revised manuscript will state the precise functional form: each plate's heaving amplitude is modulated linearly with the relative horizontal velocity to the plate immediately ahead, A_i = A_0 + K (U_{i-1} - U_i), together with the specific gain K employed in the simulations. We will also include a brief discussion of the range of K over which stabilization is observed, drawn from our existing parameter sweeps. revision: yes

Circularity Check

0 steps flagged

No circularity: results from direct simulation and external comparison

full rationale

The paper obtains schooling modes, quantized separations, and instabilities by solving the vortex-sheet equations numerically for prescribed heaving actuation, then reproduces selected features with an independent linear thin-airfoil reduction. These outputs are compared against external experiments (Newbolt et al. 2024) rather than against quantities fitted from the same data. No load-bearing step equates a prediction to a fitted input by construction, and no uniqueness claim or ansatz is imported solely via self-citation. The derivation chain therefore remains self-contained against the stated model assumptions.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the vortex sheet discretization being faithful to the inviscid hydrodynamics and on the numerical scheme converging to the reported equilibria without additional modeling assumptions.

free parameters (2)
  • heaving amplitude
    Prescribed oscillatory parameter varied across regimes to locate stable modes
  • initial separation
    Specified distance used to initialize simulations and observe resulting equilibria
axioms (2)
  • domain assumption The fluid is inviscid and incompressible
    Explicitly stated as the setting for the vortex sheet model
  • domain assumption Vortex sheet representation captures the essential hydrodynamics of heaving plates
    Core modeling choice enabling the numerical study

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