Quasi-steady aerodynamics predicts the dynamics of flapping locomotion

Leif Ristroph; Olivia Pomerenk

arxiv: 2508.19899 · v2 · submitted 2025-08-27 · ⚛️ physics.flu-dyn

Quasi-steady aerodynamics predicts the dynamics of flapping locomotion

Olivia Pomerenk , Leif Ristroph This is my paper

Pith reviewed 2026-05-18 21:26 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn

keywords flapping locomotionquasi-steady aerodynamicsheaving flightStrouhal numberReynolds numberpropulsionlift and dragforward dynamics

0 comments

The pith

A quasi-steady model using lift and drag predicts the transition to propulsion in flapping wings along with a conserved Strouhal number.

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

The paper shows that a nonlinear quasi-steady aerodynamic model reproduces key features of heaving flight propulsion without explicitly solving the fluid flows. Simulations capture the transition from a stationary state to a propulsive state as Reynolds number increases. The propulsive state features a Strouhal number that stays conserved over wide parameter ranges. Analyses of parameters, sensitivity, and stability interpret these outcomes and stress the role of flow regimes set by Reynolds number and angle of attack. This broadens the conditions where quasi-steady models apply to unsteady locomotion.

Core claim

A nonlinear model involving lift and drag forces that vary with instantaneous speed and attack angle drives the forward dynamics of a heaving thin plate. Simulations of this model reproduce the transition for increasing Reynolds number from a stationary state to a propulsive state characterized by a conserved Strouhal number across broad parameter ranges.

What carries the argument

The quasi-steady model that represents lift and drag forces as functions of instantaneous speed and attack angle to compute stroke-averaged propulsion and drive the forward dynamics of the heaving plate.

If this is right

The model reproduces the transition from stationary to propulsive state as Reynolds number increases.
The propulsive state is marked by a Strouhal number conserved across broad parameter ranges.
Flow regimes set by Reynolds number and angle of attack must be accounted for in the predictions.
Quasi-steady modeling can explain more phenomena of unsteady locomotion.

Where Pith is reading between the lines

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

Detailed vortex simulations may often be unnecessary for predicting average propulsion performance.
The model could guide efficient design of flapping-wing devices by focusing on speed and angle inputs.
Extensions to three-dimensional or flexible wings would test how far the quasi-steady approximation holds.

Load-bearing premise

Lift and drag forces can be expressed as functions of instantaneous speed and attack angle in a quasi-steady way that accurately gives stroke-averaged propulsion without resolving unsteady vortex formation or shedding.

What would settle it

A full unsteady flow simulation or physical experiment that measures propulsion force and Strouhal number at high Reynolds number and finds large deviations from the quasi-steady predictions.

read the original abstract

The propulsion of a flapping wing or foil is emblematic of bird flight and fish swimming. Previous studies have identified hallmarks of the propulsive dynamics that have been attributed to unsteady effects such as the formation and shedding of edge vortices and wing-vortex interactions. Here we show that several key features of heaving flight are captured by a quasi-steady aerodynamic model that aims to predict stroke-averaged forces from wing motions without explicitly solving for the flows. We address the forward dynamics induced by up-and-down heaving motions of a thin plate with a nonlinear model which involves lift and drag forces that vary with speed and attack angle. Simulations reproduce the well-known transition for increasing Reynolds number from a stationary state to a propulsive state, where the latter is characterized by a Strouhal number that is conserved across broad ranges of parameters. Parametric, sensitivity, and stability analyses provide physical interpretations for these results and show the importance of accounting for the flow regimes which are demarcated by Reynolds number and angle of attack. These findings extend the phenomena of unsteady locomotion that can be explained by quasi-steady modeling, and they broaden the conditions and parameter ranges over which such models are applicable.

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

Quasi-steady lift-drag model reproduces the Re-driven transition and conserved Strouhal in heaving propulsion via forward ODE integration.

read the letter

The main thing to know is that a nonlinear quasi-steady model for lift and drag, evaluated at instantaneous speed and angle, drives the forward dynamics of a heaving plate and produces both the shift from stationary to propulsive motion as Reynolds number rises and a Strouhal number that stays roughly constant over wide parameter ranges. The simulations recover these behaviors without resolving vortex formation or shedding explicitly. Parametric sweeps and stability analysis are included to interpret the outcomes and to mark out the flow regimes set by Re and angle of attack. That is a legitimate step beyond earlier quasi-steady work that focused mainly on time-averaged forces. It shows the model can be closed and integrated in a way that lets the conserved Strouhal emerge from the dynamics rather than being imposed directly. For readers who build reduced-order models for bio-locomotion or flapping vehicles, this gives a concrete example of how far the approximation can stretch before unsteady effects must be added back in. The approach is straightforward and the numerics appear reproducible from the description. The soft spots are the missing quantitative checks. The abstract reports reproduction of known transitions but supplies no error metrics, direct experimental comparisons, or side-by-side results against full CFD. The lift and drag coefficient functions are the load-bearing inputs, and without seeing exactly how they were obtained or how sensitive the Strouhal value is to small changes in their functional form, it is difficult to judge robustness. The stress-test concern about implicit tuning is reasonable to keep in mind until the full methods section is examined. This paper is aimed at people already working on flapping propulsion who want simpler alternatives to unsteady simulation. A reader focused on reduced modeling or design exploration would find it useful. It deserves peer review because the core claim is testable, the setup is clear, and referees can ask for the missing validation numbers and sensitivity plots without starting from scratch.

Referee Report

2 major / 1 minor

Summary. The manuscript presents a quasi-steady aerodynamic model for the forward dynamics of heaving flapping locomotion. A nonlinear ODE system is driven by lift and drag forces that depend instantaneously on speed and angle of attack. Forward simulations reproduce the transition from a stationary to a propulsive state as Reynolds number increases, with the propulsive regime characterized by a Strouhal number that remains conserved across broad parameter ranges. Parametric, sensitivity, and stability analyses interpret the results and highlight the role of flow regimes demarcated by Reynolds number and angle of attack.

Significance. If the central claim is substantiated, the work would demonstrate that several hallmarks of flapping propulsion previously linked to unsteady vortex dynamics can emerge from a simpler quasi-steady closure. The emergence of a conserved Strouhal number as an outcome of the forward integration, rather than an imposed input, is a clear strength. This could extend the range of conditions under which quasi-steady models suffice for predicting stroke-averaged locomotion without explicit resolution of leading-edge vortices or wake capture.

major comments (2)

[Abstract] Abstract: the claim that simulations 'reproduce the well-known transition' and a 'conserved Strouhal number' is not supported by any quantitative error metrics, predicted critical Reynolds number, or direct comparison to experimental or prior numerical values of the conserved St; without these, the degree of agreement remains unquantified.
[Model] Force model description: the explicit functional forms of the lift and drag coefficients C_L and C_D (including their dependence on instantaneous speed, attack angle, and Reynolds number) are not provided; because the transition and conserved-St outcome are direct consequences of integrating these closures, the robustness of the results to reasonable variations in the coefficient functions cannot be assessed.

minor comments (1)

[Abstract] The geometric parameters of the 'thin plate' (chord, span, or aspect ratio) should be stated explicitly to enable independent reproduction of the reported simulations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed and constructive review of our manuscript. Their comments have helped us identify areas where the presentation can be improved. We address each major comment below and indicate the revisions we plan to make.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that simulations 'reproduce the well-known transition' and a 'conserved Strouhal number' is not supported by any quantitative error metrics, predicted critical Reynolds number, or direct comparison to experimental or prior numerical values of the conserved St; without these, the degree of agreement remains unquantified.

Authors: We agree that the abstract would benefit from more quantitative statements to support the claims of reproduction. In the full manuscript, the transition is demonstrated through forward simulations showing the onset of propulsion above a critical Reynolds number, and the Strouhal number is shown to converge to a narrow range independent of parameters. However, to address this, we will revise the abstract to include the approximate critical Reynolds number from our simulations and the typical conserved Strouhal number value, along with references to prior experimental and numerical studies for comparison. This will quantify the degree of agreement. revision: yes
Referee: [Model] Force model description: the explicit functional forms of the lift and drag coefficients C_L and C_D (including their dependence on instantaneous speed, attack angle, and Reynolds number) are not provided; because the transition and conserved-St outcome are direct consequences of integrating these closures, the robustness of the results to reasonable variations in the coefficient functions cannot be assessed.

Authors: We appreciate this observation. The manuscript describes the dependence of lift and drag on speed and angle of attack, but the explicit mathematical expressions for C_L and C_D were omitted for brevity, assuming familiarity with standard quasi-steady models. To allow assessment of robustness, we will include the explicit functional forms in the revised model section, including how they incorporate Reynolds number dependence. This will enable readers to evaluate the sensitivity to these choices. revision: yes

Circularity Check

0 steps flagged

No significant circularity; conserved Strouhal emerges from forward ODE integration

full rationale

The derivation consists of defining a nonlinear ODE system for the forward dynamics of a heaving thin plate, closed by quasi-steady lift and drag coefficients that are functions of instantaneous speed and angle of attack. The transition from stationary to propulsive state and the emergence of a conserved Strouhal number across parameter ranges are reported as simulation outcomes of integrating these equations. No quoted step shows the target Strouhal or bifurcation being used to fit the force laws, nor does any central claim reduce by construction to a self-citation or ansatz that encodes the result. The model is presented as predictive from general quasi-steady closures, with results validated by parametric and stability analyses rather than by tautological fitting. This is the expected non-circular case for a forward-simulation study whose key quantities are not inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The claim rests on the quasi-steady approximation for stroke-averaged forces and on empirical or modeled lift-drag relations that vary with speed and angle; these are standard domain assumptions in aerodynamics but are applied here to close the dynamic model without unsteady terms.

free parameters (1)

lift and drag coefficient functions
The model requires explicit functional forms for lift and drag that depend on speed and attack angle; these are not derived from first principles within the paper and must be supplied or fitted from external data.

axioms (2)

domain assumption Quasi-steady approximation suffices to predict stroke-averaged forces from wing motions without solving the flow field
Invoked in the abstract when stating the model 'aims to predict stroke-averaged forces from wing motions without explicitly solving for the flows'.
domain assumption The nonlinear force laws remain valid across the demarcated Reynolds-number and angle-of-attack regimes
Used to interpret the parametric and stability analyses that separate flow regimes.

pith-pipeline@v0.9.0 · 5731 in / 1558 out tokens · 45058 ms · 2026-05-18T21:26:34.881537+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

C_L,D(α,Re) = f(α,Re) C^A + (1−f) C^S with f = 1 − tanh[(α−α_S)/δ_S]/2 and the attached/separated trigonometric forms given in Eq. (3)
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean J_uniquely_calibrated_via_higher_derivative unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Equilibrium condition C_L(α,Re)/C_D(α,Re) = cot α leading to St* ≈ 0.2

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.