Flow interactions and forward flight dynamics of tandem flapping wings

Christiana Mavroyiakoumou; Fang Fang; Leif Ristroph; Michael J. Shelley

arxiv: 2505.13149 · v1 · submitted 2025-05-19 · ⚛️ physics.flu-dyn · physics.bio-ph

Flow interactions and forward flight dynamics of tandem flapping wings

Fang Fang , Christiana Mavroyiakoumou , Leif Ristroph , Michael J. Shelley This is my paper

Pith reviewed 2026-05-22 14:16 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn physics.bio-ph

keywords tandem flapping wingsschooling statesvortex sheet methodwake interactionsforward flightcollective dynamicsfluid dynamics

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The pith

Tandem flapping wings spontaneously settle into stable schooling states with separations at integer multiples of their flapping wavelength.

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

The paper studies the forward flight of two wings flapping in tandem, where each is driven to heave and plunge vertically but can propel horizontally through fluid forces alone. Using simulations with an improved vortex sheet method, the authors find that the wings can lock into schooling states of nearly constant separation. These separations emerge as approximately integer multiples of the wavelength of the wavy trajectory each wing follows. Stability comes from an effective potential experienced by the follower in the leader's wake, where positions with in-phase or out-of-phase motions adjust thrust to restore the separation. The same mechanism allows a weaker-flapping follower to keep up by riding in a beneficial wake position and achieve higher speed than when flying alone.

Core claim

What carries the argument

Effective potential for the follower's position in the leader's wake, where thrust variations due to in-phase or antagonistic motions stabilize equilibria at specific separations.

If this is right

Multiple terminal configurations can be reached depending on the initial conditions of the pair.
The equilibria are corroborated by a linearized theory for the motion of the leader, its wake, and the effect on the follower.
A weakly-flapping follower can still school with the leader by passively assuming a favorable wake position, traveling significantly faster than in isolation.
Smaller separations lead to constructive reinforcement but decreased thrust on the follower, while larger separations increase thrust and weaken the collective wake.

Where Pith is reading between the lines

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

The effective potential approach could apply to analyzing coordinated motion in other systems with oscillatory propulsion.
Laboratory experiments with physical models could test whether the predicted wavelength-multiple separations appear in real fluids.

Load-bearing premise

The model assumes that vertical heaving-and-plunging motions are prescribed while horizontal propulsion and positions emerge solely from fluid interactions, and that the two-dimensional vortex sheet method sufficiently captures the wake dynamics.

What would settle it

Experimental observation of whether tandem flapping wings in a fluid maintain stable separations that are integer multiples of the wavelength of their individual paths.

read the original abstract

We examine theoretically the flow interactions and forward flight dynamics of tandem or in-line flapping wings. Two wings are driven vertically with prescribed heaving-and-plunging motions, and the horizontal propulsion speeds and positions are dynamically selected through aero- or hydro-dynamic interactions. Our simulations employ an improved vortex sheet method to solve for the locomotion of the pair within the collective flow field, and we identify 'schooling states' in which the wings travel together with nearly constant separation. Multiple terminal configurations are achieved by varying the initial conditions, and the emergent separations are approximately integer multiples of the wavelength traced out by each wing. We explain the stability of these states by perturbing the follower and mapping out an effective potential for its position in the leader's wake. Each equilibrium position is stabilized since smaller separations are associated with in-phase follower-wake motions that constructively reinforce the flow but lead to decreased thrust on the follower; larger separations are associated with antagonistic follower-wake motions, increased thrust, and a weakened collective wake. The equilibria and their stability are also corroborated by a linearized theory for the motion of the leader, the wake it produces, and its effect on the follower. We also consider a weakly-flapping follower driven with lower heaving amplitude than the leader. We identify 'keep-up' conditions for which the wings may still 'school' together despite their dissimilar kinematics, with the 'freeloading' follower passively assuming a favorable position within the wake that permits it to travel significantly faster than it would in isolation.

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.

Desk Editor's Note private letter to a colleague

The paper maps stable tandem schooling states at wavelength multiples via 2D vortex-sheet simulations and an effective-potential stability analysis, but the 2D inviscid assumption is the main limit on how far the mechanism travels.

read the letter

The main thing to know is that two flapping wings in line can lock into several steady separations that are roughly integer multiples of the wavelength each traces, and the authors trace the stability to an effective potential felt by the follower in the leader's wake. Smaller gaps produce in-phase motions that cut the follower's thrust while larger gaps produce opposing motions that increase it, so the system settles at the minima. They get this from direct simulations with an improved vortex-sheet method where vertical heaving is prescribed and horizontal propulsion emerges, plus a linearized theory that couples the leader motion, its wake, and the follower perturbation. They also show a weaker follower can still school by riding a favorable spot in the wake and going faster than it would alone. That combination of numerics and perturbation analysis is the clearest new piece; it gives a concrete dynamical explanation rather than just observing the configurations. The simulations look internally consistent for the model they chose, and the theory lines up with the observed terminal states without obvious circularity. The soft spot is the 2D inviscid setup. The in-phase versus antagonistic phasing that sets the potential minima depends on wake structures staying coherent; once tip vortices, spanwise flow, or viscous diffusion enter at realistic Reynolds numbers, those structures can smear or break, and the thrust modulation that stabilizes the states may weaken or shift. The vertical kinematics are also prescribed, so the work does not address fully free flight where both degrees of freedom interact. Without the full methods, grid studies, or experimental comparisons it is hard to judge how sensitive the integer-multiple states are to those modeling choices. Readers working on collective locomotion, bio-inspired propulsion, or wake stability will find the effective-potential framing useful and worth checking. The paper is coherent on its own terms and has enough analytical support to merit referee time rather than a desk reject, though any review should press on the 2D-to-3D and inviscid-to-viscous gaps.

Referee Report

2 major / 2 minor

Summary. The manuscript examines the flow interactions and forward flight dynamics of tandem flapping wings using an improved 2D vortex sheet method. Vertical heaving-and-plunging motions are prescribed while horizontal propulsion and positions emerge from hydrodynamic interactions. The authors identify 'schooling states' in which the wings maintain nearly constant separations that are approximately integer multiples of the wavelength traced by each wing. Stability is explained by perturbing the follower to map an effective potential in the leader's wake, where smaller separations produce in-phase motions that reduce follower thrust and larger separations produce antagonistic motions that increase thrust and weaken the collective wake. These equilibria are corroborated by a linearized theory coupling leader motion, wake, and follower. The work also identifies 'keep-up' conditions for a weakly flapping follower that can school despite dissimilar kinematics by assuming a favorable wake position.

Significance. If the central claims hold, the work offers a mechanistic explanation for stable tandem configurations in collective flapping locomotion, with potential relevance to bio-inspired propulsion and natural schooling behaviors. A notable strength is the combination of direct numerical simulations of the vortex-sheet model with a supporting linearized perturbation analysis that derives the effective potential without fitted parameters or self-referential definitions. The emergence of multiple terminal states from different initial conditions and the extension to dissimilar kinematics further strengthen the contribution.

major comments (2)

[§4] §4 (effective potential construction): the stability mechanism rests on mapping the effective potential by perturbing the follower while holding the leader wake fixed; however, the linearized theory (detailed in the subsequent section) couples leader motion, wake evolution, and follower response. It is unclear whether the fixed-wake approximation remains accurate at the reported separations where wake-follower interactions are strong, which is load-bearing for the claimed thrust modulation and equilibrium stability.
[Methods and §3] Methods and §3 (simulation results): the reported in-phase versus antagonistic phasing that produces the effective-potential minima is obtained in an inviscid 2D vortex-sheet wake. The manuscript does not test or discuss how 3D tip vortices, spanwise flow, or finite-Reynolds-number viscous diffusion would alter wake coherence and the associated thrust modulation; this directly affects whether the identified schooling states and their stability mechanism survive beyond the model assumptions.

minor comments (2)

[Figure captions and §3] Figure captions and §3: the wavelength used to normalize separations should be defined explicitly (e.g., as the streamwise distance per flapping cycle) at first use to avoid ambiguity when comparing emergent separations across cases.
[Methods] Notation in the vortex-sheet formulation: the improved method is referenced but the precise regularization or desingularization parameters are not restated; a brief recap near the governing equations would improve reproducibility.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the constructive and detailed review of our manuscript on tandem flapping wing interactions. The comments raise important points about the consistency of our stability analysis and the scope of the model assumptions. We address each major comment below and indicate where revisions will be made.

read point-by-point responses

Referee: [§4] §4 (effective potential construction): the stability mechanism rests on mapping the effective potential by perturbing the follower while holding the leader wake fixed; however, the linearized theory (detailed in the subsequent section) couples leader motion, wake evolution, and follower response. It is unclear whether the fixed-wake approximation remains accurate at the reported separations where wake-follower interactions are strong, which is load-bearing for the claimed thrust modulation and equilibrium stability.

Authors: The effective potential is constructed as a diagnostic to illustrate how follower thrust varies with position in a fixed leader wake, thereby providing intuition for the restoring mechanism. The actual stability of the equilibria is established by the subsequent linearized theory, which fully couples leader motion, wake evolution, and follower dynamics. We will revise the text in §4 to make this distinction explicit, to note the conditions under which the fixed-wake approximation is used, and to discuss its accuracy at the separations examined. revision: partial
Referee: [Methods and §3] Methods and §3 (simulation results): the reported in-phase versus antagonistic phasing that produces the effective-potential minima is obtained in an inviscid 2D vortex-sheet wake. The manuscript does not test or discuss how 3D tip vortices, spanwise flow, or finite-Reynolds-number viscous diffusion would alter wake coherence and the associated thrust modulation; this directly affects whether the identified schooling states and their stability mechanism survive beyond the model assumptions.

Authors: Our results are obtained with an inviscid two-dimensional vortex-sheet model, which excludes three-dimensional tip vortices, spanwise flows, and viscous diffusion. These effects could modify wake coherence and the strength of the thrust modulation. We will add a dedicated paragraph in the discussion section that explicitly addresses these model limitations and their possible influence on the reported schooling states and stability mechanism, while noting that quantitative assessment requires future three-dimensional viscous computations. revision: yes

standing simulated objections not resolved

Quantitative assessment of how three-dimensional tip vortices, spanwise flow, and viscous diffusion alter wake coherence and the stability mechanism, which lies outside the present two-dimensional inviscid framework.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives schooling states and their stability directly from numerical simulations of the vortex-sheet model with prescribed vertical kinematics and emergent horizontal propulsion, followed by perturbation analysis to construct an effective potential and a linearized theory coupling leader, wake, and follower. These steps are self-contained within the simulation framework and do not reduce any claimed prediction or equilibrium to a fitted input, self-definition, or load-bearing self-citation by construction. The method and results are externally falsifiable via the reported terminal configurations and thrust modulations, with no evidence that central claims collapse to prior author work or ansatz smuggling.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the validity of the vortex sheet approximation for modeling wake interactions and the assumption that prescribed vertical motions allow horizontal dynamics to emerge purely from fluid forces.

axioms (1)

domain assumption The flow around the wings can be modeled as two-dimensional and inviscid using an improved vortex sheet method.
This is the core numerical approach stated in the abstract for solving the collective flow field.

pith-pipeline@v0.9.0 · 5811 in / 1317 out tokens · 58796 ms · 2026-05-22T14:16:09.069878+00:00 · methodology

Flow interactions and forward flight dynamics of tandem flapping wings

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)