The inviscid Euler limit as a critical boundary for moment-based aerodynamic system identification

Sarasija Sudharsan

arxiv: 2604.16633 · v1 · submitted 2026-04-17 · ⚛️ physics.flu-dyn

The inviscid Euler limit as a critical boundary for moment-based aerodynamic system identification

Sarasija Sudharsan This is my paper

Pith reviewed 2026-05-10 06:53 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn

keywords inviscid Euler equationsaerodynamic system identificationimpulse responsetemporal momentsmemory time scalepower-law decayvorticity persistence

0 comments

The pith

The two-dimensional inviscid Euler equations have no window-independent memory time because their impulse response makes the second temporal moment diverge logarithmically.

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

The paper demonstrates that the impulse response of the two-dimensional inviscid Euler equations decays as t to the power of negative three-halves. This power-law tail causes the second temporal moment to grow as the logarithm of the observation window T. As a direct result the apparent memory time extracted from moment ratios scales as the square root of ln T, so that any finite-dimensional model fitted to the data is effectively parameterizing the length of the record rather than an intrinsic property of the flow. The authors introduce a windowed moment ratio diagnostic to quantify the growth and confirm the predicted scaling in compressible Euler simulations until numerical dissipation intervenes.

Core claim

In the two-dimensional inviscid limit the second temporal moment of the impulse response kernel diverges logarithmically with the observation window T. This divergence implies that the characteristic memory time scales as sqrt(ln T) and that no fixed, window-independent time scale exists for moment-based system identification.

What carries the argument

The temporal-moment diagnostic ν_t(T), the ratio of the second to the zeroth windowed moment of the impulse response kernel, which directly exposes the logarithmic divergence produced by the t^{-3/2} tail.

If this is right

Exponential models exhibit stable memory-time plateaus because their decay is fast enough for moments to converge.
Compressible Euler simulations reproduce the sqrt(ln T) growth of memory time at intermediate times.
Numerical dissipation present in any discretization acts as an artificial regularizer that forces moment convergence at late times.
Finite-dimensional state-space models fitted to inviscid data parameterize the observation horizon rather than intrinsic flow physics.

Where Pith is reading between the lines

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

Flows with even small viscosity will cross over to exponential decay after a Reynolds-number-dependent cutoff time, restoring a fixed memory scale.
System identification routines applied to nearly inviscid data must treat the data-window length as an explicit parameter rather than an arbitrary choice.
The same moment-divergence mechanism may appear in other two-dimensional vortical systems whose far-field response decays slower than t to the minus two.

Load-bearing premise

The known t to the minus three-halves asymptotic decay of the two-dimensional Euler impulse response remains the dominant contribution over the finite but growing observation windows used in system identification.

What would settle it

Extract the second moment of the impulse response from two-dimensional inviscid Euler simulations over successively longer time windows and test whether the moment increases in proportion to the logarithm of the window length.

Figures

Figures reproduced from arXiv: 2604.16633 by Sarasija Sudharsan.

**Figure 1.** Figure 1: Vertical displacement h(t) and corresponding angle of attack α(t) profiles in the CFD simulation, shown for the early part of the manuever (t ≤ 1). To compute this derivative, a Savitzky–Golay filter is applied to the lift signal. This approach ensures sufficient smoothness for differentiation while preserving the physically relevant high-frequency transients present in the earlytime response. 5.2 Models … view at source ↗

**Figure 2.** Figure 2: Impulse response kernel decay on log-log scale, comparing the analytical Euler model, CFD Euler simula [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Figure 3: Temporal spread scale νt(T) comparing CFD-derived results with the stable plateau of the Wagner approximation and the logarithmic divergence of the analytical Euler limit. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: Effective scaling exponent βeff characterizing the growth rate of the memory time. To map the transition where the CFD data separates from analytical theory, the effective scaling exponent is defined, βeff = d ln νt d ln T , which characterizes the localized growth rate of the characteristic memory time. βeff theoretically scales as 1/(2 ln T), and βeff → 0 does not imply the convergence of νt; it only ind… view at source ↗

**Figure 5.** Figure 5: Classical Spectral Rice Frequency νs(T) showing rapid convergence for all models. In CFD simulations, numerical dissipation (arising from spatial discretization, artificial viscosity, and timestepping schemes) acts as an effective regularizer that enforces exponential decay at long times. In physical systems, molecular viscosity fulfills this role, dissipating the kinetic energy of shed vorticity and ensu… view at source ↗

read the original abstract

Finite-dimensional state-space representations of unsteady aerodynamics implicitly assume a system with fading memory. However, the impulse response of the two-dimensional inviscid (Euler) equations is characterized by an asymptotic $t^{-3/2}$ power-law decay due to the persistence of shed vorticity. The present work demonstrates that this decay rate constitutes a critical boundary for moment convergence: the second temporal moment diverges logarithmically, causing the characteristic memory time to grow as $\sqrt{\ln T}$ with the observation window $T$. As a result, no window-independent characteristic time scale exists, and finite-dimensional models fitted to inviscid data effectively parameterize the observation horizon rather than intrinsic flow physics. To quantify this behavior, a temporal-moment diagnostic, $\nu_t(T)$, is introduced based on the ratio of the second and zeroth windowed moments of the impulse response kernel. Exponential models exhibit stable memory time plateaus, as their sufficiently fast decay ensures convergence of the moment diagnostic. Compressible Euler simulation results confirm the predicted $\sqrt{\ln T}$ scaling at intermediate times, while numerical dissipation inherent to the discretization acts as an artificial regularizer that enforces convergence at late times. These results establish the two-dimensional inviscid limit as a critical boundary for moment-based system identification, where the absence of a dissipative mechanism prevents the definition of a window-independent characteristic memory time.

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 flags a real issue for moment-based ROMs in inviscid aerodynamics but miscalculates the second-moment scaling.

read the letter

The main thing to know is that the work connects the known t^{-3/2} tail of the 2D Euler impulse response to problems in finite-dimensional aerodynamic models, but the claimed logarithmic divergence does not follow from the stated decay rate. The paper introduces a windowed-moment diagnostic nu_t(T) and runs compressible Euler simulations to illustrate how effective memory time grows with observation window T. That diagnostic is straightforward and parameter-free, and the contrast with exponentially decaying kernels is useful for thinking about when moment convergence actually occurs. The simulations are a reasonable check on intermediate-time behavior before dissipation dominates. These pieces are the parts that hold up and could be worth discussing. The central claim does not. For h(t) ~ t^{-3/2}, the second moment scales as T^{3/2} and any characteristic time extracted from sqrt(m2/m0) scales as T^{3/4}, not sqrt(ln T). Logarithmic growth would require a steeper tail near t^{-3}. The abstract states the t^{-3/2} decay yet asserts the log result, which is an internal inconsistency in the derivation. The simulations are said to confirm the sqrt(ln T) scaling, but that only raises the question of whether the numerics have reached the true asymptotic regime or are still influenced by other effects. The citation pattern for the decay rate itself is standard and not the problem. This paper is for people who build reduced-order models from CFD data and care about the limits of fading-memory assumptions. A reader could take the diagnostic idea and apply it correctly even if the specific scaling needs correction. I would bring it to a reading group to check the moment integrals step by step. I would not cite the current version. It does not deserve peer review until the scaling is fixed, because the main result rests on a calculation that does not match the given decay.

Referee Report

2 major / 1 minor

Summary. The manuscript claims that the asymptotic t^{-3/2} decay of the impulse response for the 2D inviscid Euler equations marks a critical boundary for moment-based system identification. It asserts that this decay causes the second temporal moment to diverge logarithmically (rather than as a power law), so that a characteristic memory time defined from the windowed moments grows as sqrt(ln T) with observation window T. Consequently, no window-independent time scale exists, and finite-dimensional models fitted to such data effectively encode the horizon T instead of intrinsic physics. A diagnostic nu_t(T) based on the ratio of second to zeroth windowed moments is introduced; exponential kernels are shown to produce stable plateaus while Euler simulations are said to confirm the sqrt(ln T) scaling at intermediate times before numerical dissipation enforces convergence.

Significance. If the central scaling result held, the work would usefully delineate a theoretical limit for the applicability of moment-based reduced-order models in unsteady aerodynamics, showing that the 2D inviscid case lacks a dissipative mechanism sufficient to produce a finite, window-independent memory time. The introduction of the nu_t(T) diagnostic and the explicit contrast with exponentially decaying kernels would provide a practical tool for assessing when system-identification assumptions break down.

major comments (2)

[Abstract] Abstract (and the central derivation): the stated t^{-3/2} decay of h(t) does not produce logarithmic divergence of the second moment. With m_2(T) = ∫_0^T t^2 h(t) dt and h(t) ∼ t^{-3/2}, the integrand scales as t^{1/2}, so m_2(T) ∼ T^{3/2} (power-law divergence). Logarithmic divergence of m_2 instead requires h(t) ∼ t^{-3}. The claimed sqrt(ln T) growth of the memory time (presumably sqrt(m_2/m_0)) is therefore incompatible with the given decay rate under standard moment definitions. This directly affects the central claim that no window-independent scale exists and the interpretation of the simulation results.
[Abstract] Abstract and simulation discussion: the manuscript states that compressible Euler simulations confirm the predicted sqrt(ln T) scaling at intermediate times. However, no quantitative details (grid resolution, time-stepping scheme, domain size, or how the impulse response is extracted) are provided to allow verification that the observed growth is not an artifact of the numerical dissipation that the text itself identifies as an artificial regularizer at late times.

minor comments (1)

The precise definition of the temporal-moment diagnostic nu_t(T) (ratio of which moments, normalization, and how the characteristic time is extracted from it) should be stated explicitly in the main text rather than only alluded to in the abstract.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed and insightful report. The two major comments identify a mathematical inconsistency in the moment scaling and a lack of numerical details. We agree with both points and will revise the manuscript accordingly. The core qualitative conclusion—that the 2D inviscid Euler equations lack a window-independent memory time for moment-based system identification—remains valid under the corrected scaling, as the second moment still diverges with T.

read point-by-point responses

Referee: [Abstract] Abstract (and the central derivation): the stated t^{-3/2} decay of h(t) does not produce logarithmic divergence of the second moment. With m_2(T) = ∫_0^T t^2 h(t) dt and h(t) ∼ t^{-3/2}, the integrand scales as t^{1/2}, so m_2(T) ∼ T^{3/2} (power-law divergence). Logarithmic divergence of m_2 instead requires h(t) ∼ t^{-3}. The claimed sqrt(ln T) growth of the memory time (presumably sqrt(m_2/m_0)) is therefore incompatible with the given decay rate under standard moment definitions. This directly affects the central claim that no window-independent scale exists and the interpretation of the simulation results.

Authors: We thank the referee for catching this error in our derivation. Re-examination confirms that for an asymptotic decay h(t) ∼ t^{-3/2}, the integrand t^2 h(t) ∼ t^{1/2} yields m_2(T) ∼ T^{3/2} (power-law divergence), not logarithmic. The sqrt(ln T) scaling for the memory time is therefore incorrect. We will revise the abstract, introduction, and derivation sections to state the proper T^{3/2} divergence of m_2 and the resulting T^{3/4} growth of a memory time defined via sqrt(m_2/m_0). The central claim is unaffected: because m_2 diverges with the observation window T, no finite, window-independent characteristic time exists, and finite-dimensional models fitted to such data still parameterize the horizon rather than intrinsic physics. We will also re-analyze the simulation data against the corrected scaling. revision: yes
Referee: [Abstract] Abstract and simulation discussion: the manuscript states that compressible Euler simulations confirm the predicted sqrt(ln T) scaling at intermediate times. However, no quantitative details (grid resolution, time-stepping scheme, domain size, or how the impulse response is extracted) are provided to allow verification that the observed growth is not an artifact of the numerical dissipation that the text itself identifies as an artificial regularizer at late times.

Authors: We agree that the current manuscript lacks sufficient numerical details for independent verification. In the revised version we will add a dedicated methods subsection (or appendix) specifying: grid resolution and refinement strategy, time-stepping scheme and CFL condition, computational domain size with far-field boundary treatment, and the precise procedure for extracting the impulse response (including the form of the impulsive forcing). We will also include a brief resolution study or comparison of dissipation effects to demonstrate that the intermediate-time growth is not an artifact of numerical viscosity. These additions will allow readers to assess the results directly. revision: yes

Circularity Check

0 steps flagged

Derivation starts from externally known t^{-3/2} decay; moment diagnostic defined directly without fitting or self-definition

full rationale

The paper takes the t^{-3/2} asymptotic decay of the 2D Euler impulse response as an externally known input (not derived or fitted within the work). It then defines the diagnostic ν_t(T) explicitly as the ratio of the second and zeroth windowed moments of the impulse response kernel and applies the known decay to obtain the claimed logarithmic divergence and √(ln T) scaling. No parameter is fitted to the target result, no self-citation is invoked as a load-bearing uniqueness theorem, and no ansatz is smuggled in. The central claim therefore remains independent of its own output, qualifying as no significant circularity (minor score assigned only because the scaling derivation is presented as a first-principles consequence of the input decay).

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The paper rests on the established asymptotic decay of the 2D Euler wake and on the definition of temporal moments; no new free parameters or invented entities are introduced beyond the diagnostic itself.

axioms (1)

domain assumption The impulse response of the 2D inviscid Euler equations decays asymptotically as t^{-3/2} due to persistent shed vorticity.
Invoked in the first paragraph of the abstract as the starting point for the moment analysis.

invented entities (1)

temporal-moment diagnostic nu_t(T) no independent evidence
purpose: Ratio of second to zeroth windowed moments of the impulse response kernel to detect the growth of memory time.
Defined directly from the kernel; no independent evidence supplied beyond the scaling argument.

pith-pipeline@v0.9.0 · 5543 in / 1534 out tokens · 27360 ms · 2026-05-10T06:53:11.609601+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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work page doi:10.2514/3.20031 1985

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Jer-Nan Juang, Minh Phan, Lucas G. Horta, and Richard W. Longman. Identification of observer/kalman filter markov parameters - theory and experiments. Journal of Guidance, Control, and Dynamics, 16 0 (2): 0 320--329, 1993. doi:10.2514/3.21006. URL https://doi.org/10.2514/3.21006

work page doi:10.2514/3.21006 1993

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Joseph Katz and Allen Plotkin. Low-speed aerodynamics. Cambridge university press, 2001

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Heaslet, Franklyn B

Harvard Lomax, Max A. Heaslet, Franklyn B. Fuller, and Loma Sluder. Derivation and application of a general formula for directly computing the aerodynamic forces on wings in generalized motion. NACA Report, 1066, 1952

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L2roe: a low dissipation version of roe's approximate riemann solver for low mach numbers

Kai O wald, Alexander Siegmund, Philipp Birken, Volker Hannemann, and Andreas Meister. L2roe: a low dissipation version of roe's approximate riemann solver for low mach numbers. International Journal for Numerical Methods in Fluids, 81 0 (2): 0 71--86, 2016

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Stephen O. Rice. Mathematical analysis of random noise. The Bell System Technical Journal, 23 0 (3): 0 282--332, 1944. doi:10.1002/j.1538-7305.1944.tb00874.x

work page doi:10.1002/j.1538-7305.1944.tb00874.x 1944

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work page doi:10.1002/j.1538-7305.1945.tb00453.x 1945

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Rumsey, Brian R

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General theory of aerodynamic instability and the mechanism of flutter

Theodore Theodorsen. General theory of aerodynamic instability and the mechanism of flutter. Report 496, National Advisory Committee for Aeronautics, Washington, D.C., 1935

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Towards the ultimate conservative difference scheme

Bram van Leer. Towards the ultimate conservative difference scheme. v. a second-order sequel to godunov's method. Journal of Computational Physics, 32 0 (1): 0 101--136, 1979

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uber die E ntstehung des dynamischen A uftriebes von T ragfl\

Herbert Wagner. \"uber die E ntstehung des dynamischen A uftriebes von T ragfl\"ugeln. Zeitschrift f\"ur Angewandte Mathematik und Mechanik, 5 0 (1): 0 17--35, 1925. doi:10.1002/zamm.19250050103

work page doi:10.1002/zamm.19250050103 1925