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arxiv: 2605.25539 · v3 · pith:MRXBEWCJnew · submitted 2026-05-25 · ⚛️ physics.flu-dyn

Finite-Time Relaxation of Inertial Particle Clustering in Non-Equilibrium Turbulence

Taketo Tominaga , Ryo Onishi This is my paper

Pith reviewed 2026-06-29 20:47 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords inertial particlesparticle clusteringnon-equilibrium turbulencerelaxation modeldirect numerical simulationhomogeneous isotropic turbulenceStokes number
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The pith

Clustering intensity of inertial particles shows hysteresis with turbulence dissipation rate when forcing varies slowly.

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

The paper examines inertial particle clustering under periodic unsteady forcing in homogeneous isotropic turbulence. It shows that the link between instantaneous energy dissipation and clustering strength displays clear hysteresis once the forcing period grows longer than several large-eddy turnover times, exceeding the scatter seen in stationary cases. For the highest-inertia particles the clustering intensity reaches 0.80 and 1.56 times the stationary reference value at identical dissipation rates. A linear relaxation model is then fitted, in which clustering intensity relaxes toward the instantaneous-equilibrium value over a time that scales as one large-eddy turnover time multiplied by the Stokes number to the power 0.4; this cuts the maximum relative error from 49 percent to 10 percent.

Core claim

The instantaneous-equilibrium approximation fails for inertial-particle clustering in non-equilibrium turbulence; instead clustering intensity relaxes toward the equilibrium value with a finite time au_g = 1.0 T_e(t) St(t)^{0.40}.

What carries the argument

Linear relaxation model in which clustering intensity approaches the instantaneous-equilibrium value over a characteristic relaxation time.

If this is right

  • Hysteresis exceeds stationary fluctuations once the forcing period exceeds several large-eddy turnover times.
  • For the largest inertia particles clustering intensity reaches 0.80 and 1.56 times the reference value at the same instantaneous dissipation rate.
  • The relaxation model reduces maximum relative error from 49% to 10% for the largest inertia particles and from 76% to 22% in an independent validation case.

Where Pith is reading between the lines

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

  • Models of particle transport in any flow whose large-eddy time scale changes over several turnover times may need similar relaxation corrections.
  • The reported scaling could be checked by repeating the periodic-forcing experiment at different Reynolds numbers or with different particle size distributions.
  • Collision-rate predictions in unsteady turbulence would inherit the same relaxation lag once the clustering model is updated.

Load-bearing premise

The linear form of the relaxation model and the 0.40 exponent in the scaling law remain valid outside the specific forcing periods and Stokes numbers tested in the simulations.

What would settle it

A simulation or measurement that finds no measurable hysteresis between clustering intensity and instantaneous dissipation rate for forcing periods several times the large-eddy turnover time would falsify the finite-relaxation claim.

Figures

Figures reproduced from arXiv: 2605.25539 by Ryo Onishi, Taketo Tominaga.

Figure 1
Figure 1. Figure 1: shows Cε as a function of Reλ for periodic unsteady forcing. Here, Cε ≡ εL/u ′3 . The simulation condition corresponds to the P1 case in Table I. In P1, phase averaging was performed using time series over 100 forcing cycles. The maximum relative standard errors of the phase￾averaged Cε and Reλ were less than 2% for all conditions in the P1 case in Table I. For all forcing periods, the non-equilibrium diss… view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Raw and phase-averaged trajectories of [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Phase-averaged [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Cross-correlation function [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Time evolution of [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
read the original abstract

Inertial particles in turbulence form clusters, which strongly affect particle collisions and transport properties. Clustering models based on statistically stationary turbulence implicitly assume the instantaneous-equilibrium approximation when applied to time-varying non-equilibrium turbulence. However, the validity of this approximation remains unclear. In this study, the temporal response of inertial particle clustering in non-equilibrium turbulence was investigated using direct numerical simulation of homogeneous isotropic turbulence with unsteady forcing. Periodic responses of the flow and clustering intensity were evaluated by varying the forcing period. The flow showed non-equilibrium scaling for all forcing periods. The relationship between instantaneous energy dissipation rate and clustering intensity showed hysteresis exceeding statistically stationary fluctuations when the forcing period exceeded several large-eddy turnover times. For the particles with the largest inertia, clustering intensity took values of 0.80 and 1.56 times the reference value at the same instantaneous energy dissipation rate. This shows that the instantaneous-equilibrium approximation is not appropriate under such conditions. A linear relaxation model was constructed from transient responses, in which clustering intensity approaches the instantaneous-equilibrium value with a finite relaxation time. The relaxation time scaling was identified as $\tau_g = 1.0 T_\mathrm{e}(t)\,\mathrm{St}(t)^{0.40}$, where $T_\mathrm{e}(t)$ and $\mathrm{St}(t)$ are the instantaneous large-eddy turnover time and Stokes number. The model reduced the maximum relative error from 49% to 10% for the particles with the largest inertia and from 76% to 22% in an independent validation case. These results demonstrate that finite-time relaxation improves prediction accuracy for clustering intensity in non-equilibrium turbulence.

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 / 1 minor

Summary. The manuscript investigates the temporal response of inertial particle clustering in non-equilibrium homogeneous isotropic turbulence via DNS with periodic unsteady forcing. It reports hysteresis in clustering intensity versus instantaneous energy dissipation rate (exceeding stationary fluctuations) for forcing periods longer than several large-eddy turnover times, with values of 0.80 and 1.56 times the reference for the highest-inertia particles. A first-order linear relaxation model is introduced in which clustering intensity approaches the instantaneous-equilibrium value with relaxation time scaling τ_g = 1.0 T_e(t) St(t)^{0.40}; this reduces maximum relative error from 49% to 10% on the primary cases and from 76% to 22% on an independent validation case.

Significance. If the results hold, the work supplies concrete DNS evidence that the instantaneous-equilibrium approximation fails for clustering in time-varying turbulence and that a simple finite-time relaxation correction improves predictions. The quantified hysteresis factors and the error reductions (including the independent validation case) are clear strengths that would be relevant to modeling of particle-laden unsteady flows.

major comments (2)
  1. [Abstract and model construction] Abstract and the paragraph introducing the relaxation model: the scaling τ_g = 1.0 T_e(t) St(t)^{0.40} (prefactor and exponent 0.40) is identified directly from the transient responses of the same periodic-forcing DNS runs that demonstrate the hysteresis; the manuscript supplies neither the fitting procedure (data segments, metric, uncertainty on the exponent) nor a sensitivity test when forcing period or Stokes-number range is altered, which is load-bearing for the claim that the model generalizes and improves accuracy.
  2. [Hysteresis results] Hysteresis results: the statement that observed hysteresis (0.80 and 1.56 factors) exceeds statistically stationary fluctuations is central, yet the manuscript does not specify how the stationary fluctuation baseline was computed or the number of independent realizations used to establish the comparison threshold.
minor comments (1)
  1. [Model equation] Clarify whether T_e(t) and St(t) are evaluated instantaneously at every time step or averaged over some window when inserted into the relaxation-time formula.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify key aspects of the model derivation and statistical analysis. We address each major comment below and will revise the manuscript to incorporate the requested details.

read point-by-point responses

  1. Referee: [Abstract and model construction] Abstract and the paragraph introducing the relaxation model: the scaling τ_g = 1.0 T_e(t) St(t)^{0.40} (prefactor and exponent 0.40) is identified directly from the transient responses of the same periodic-forcing DNS runs that demonstrate the hysteresis; the manuscript supplies neither the fitting procedure (data segments, metric, uncertainty on the exponent) nor a sensitivity test when forcing period or Stokes-number range is altered, which is load-bearing for the claim that the model generalizes and improves accuracy.

    Authors: We agree that the fitting procedure and sensitivity information should be provided to support the generalization claim. The scaling was obtained by extracting relaxation times from the transient segments of the periodic-forcing DNS cases and performing a fit across the available Stokes numbers. In the revised manuscript we will add a dedicated subsection describing the data segments selected, the fitting metric, uncertainty estimates on the exponent, and a sensitivity test by restricting the forcing-period and Stokes-number ranges. revision: yes

  2. Referee: [Hysteresis results] Hysteresis results: the statement that observed hysteresis (0.80 and 1.56 factors) exceeds statistically stationary fluctuations is central, yet the manuscript does not specify how the stationary fluctuation baseline was computed or the number of independent realizations used to establish the comparison threshold.

    Authors: We agree that the baseline computation must be specified. In the revised manuscript we will detail the procedure used to compute the stationary fluctuation baseline from the equilibrium DNS runs, including the number of independent realizations and the statistical threshold applied. revision: yes

Circularity Check

1 steps flagged

Relaxation-time scaling τ_g = 1.0 T_e(t) St(t)^{0.40} identified by fit to the same DNS transients used to show hysteresis

specific steps
  1. fitted input called prediction [Abstract]
    "A linear relaxation model was constructed from transient responses, in which clustering intensity approaches the instantaneous-equilibrium value with a finite relaxation time. The relaxation time scaling was identified as τ_g = 1.0 T_e(t) St(t)^{0.40}... The model reduced the maximum relative error from 49% to 10% for the particles with the largest inertia and from 76% to 22% in an independent validation case."

    The scaling parameters and functional form are obtained by fitting to the transient responses in the periodic-forcing DNS; the same class of data is then used to demonstrate that the fitted model reduces error relative to the instantaneous-equilibrium approximation. The 'prediction' improvement is therefore a direct consequence of the fit rather than an independent test of the scaling.

full rationale

The paper constructs a linear relaxation model directly from the transient responses observed in its periodic-forcing DNS runs. The specific scaling (prefactor 1.0 and exponent 0.40) is stated as 'identified' from those responses, after which the model is applied to the same data class to report error reduction (49%→10% and 76%→22% in the validation case). This constitutes a fitted-input-called-prediction pattern: the functional form and parameters are extracted from the very transients whose non-equilibrium behavior the model is then invoked to 'improve.' The hysteresis observation itself remains independent, but the claimed predictive improvement reduces to the fit. No a-priori derivation or external benchmark for the exponent is supplied.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on DNS data showing hysteresis and on an empirically fitted linear relaxation model whose parameters (prefactor and exponent) are determined from the same transient responses. No new physical entities are postulated.

free parameters (2)
  • prefactor 1.0 = 1.0
    Chosen to match the observed relaxation rate in the periodic-forcing DNS cases.
  • exponent 0.40 = 0.40
    Identified by fitting the dependence of relaxation time on Stokes number across the simulated particle inertias.
axioms (2)
  • domain assumption Homogeneous isotropic turbulence with unsteady forcing produces representative non-equilibrium statistics
    Invoked to justify the simulation setup as a model for general non-equilibrium turbulence.
  • ad hoc to paper Clustering intensity dynamics can be captured by a first-order linear relaxation equation
    The model form is constructed from the transient responses rather than derived from the Navier-Stokes equations.

pith-pipeline@v0.9.1-grok · 5830 in / 1631 out tokens · 41792 ms · 2026-06-29T20:47:59.271088+00:00 · methodology

discussion (0)

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Reference graph

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