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arxiv: 2511.00669 · v1 · submitted 2025-11-01 · ⚛️ physics.flu-dyn

Two-point Turbulence Closures in Physical Space

Noah Zambrano , Karthik Duraisamy This is my paper

Pith reviewed 2026-05-18 01:04 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords two-point turbulence closurephysical space formulationhomogeneous isotropic turbulenceEDQNM modellongitudinal correlation functionpressure-Poisson equationturbulence statistics
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The pith

A two-point closure for turbulence is formulated directly in physical space by deriving a discrete evolution equation for the longitudinal correlation function.

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

This work develops a statistical closure model for incompressible homogeneous isotropic turbulence entirely in physical space instead of Fourier space. The evolution equation for the longitudinal correlation function is written in discrete form, which preserves the linear terms in a near-exact manner. Nonlinear higher-order moments, including contributions from the pressure-Poisson equation, are closed by applying the phenomenological rules of the EDQNM model. The resulting framework naturally brings in non-local length-scale information through triple integrals and requires a matrix exponential in the time advancement. Verification against direct numerical simulations of forced turbulence and classical EDQNM predictions for decaying cases supports the formulation, and the approach is positioned for extension to inhomogeneous and anisotropic flows where spectral methods become difficult.

Core claim

The evolution equation for the longitudinal correlation function is derived in a discrete physical-space form for ensemble-averaged incompressible homogeneous isotropic turbulence. Linear terms are kept in near-exact representation while the nonlinear higher-order moments are closed according to the phenomenological principles of the EDQNM model. The physical-space treatment produces a matrix exponential in the evolution equation and triple integrals that arise from the non-local pressure-Poisson equation, thereby incorporating non-local length-scale information into the turbulence statistics.

What carries the argument

The discrete physical-space evolution equation for the longitudinal correlation function, with nonlinear triple correlations closed by EDQNM phenomenology.

If this is right

  • Turbulence statistics can be evolved without performing a Fourier transformation at any stage.
  • Non-local effects from the pressure field enter the evolution of statistics through triple integrals over physical space.
  • The formulation extends in principle to inhomogeneous and anisotropic turbulence without requiring homogeneity assumptions.
  • The method offers a route for flows containing discontinuities, such as compressible turbulence, where spectral representations are ill-conditioned.

Where Pith is reading between the lines

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

  • The discrete physical-space structure could allow direct coupling to finite-volume or finite-element solvers in complex geometries.
  • Length-scale information carried by the triple integrals may improve modeling of energy transfer in flows with strong spatial variations.
  • The matrix-exponential time advancement suggests possible stiff-integrator implementations that could be tested against existing spectral codes.

Load-bearing premise

The nonlinear higher-order moments are closed by adopting the phenomenological rules of the EDQNM model rather than being obtained from the governing equations without additional assumptions.

What would settle it

A comparison in which the time evolution of the longitudinal correlation function predicted by the closed equation deviates measurably from direct numerical simulation results for statistically stationary or decaying homogeneous isotropic turbulence after accounting for discretization effects.

Figures

Figures reproduced from arXiv: 2511.00669 by Karthik Duraisamy, Noah Zambrano.

Figure 1
Figure 1. Figure 1: Change of the longitudinal and lateral functions for various times using the [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Normalized derivatives of the longitudinal function showing decay at large [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Decaying longitudinal and lateral functions with initial Batchelor spectrum. [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Energy spectrum 𝐸(𝜅) predictions from the spectral and physical closures for various time intervals starting from the Batchelor spectrum. Blue lines represent the physical space model and black lines the spectral model. Physical space model results are truncated at higher wavenumbers due to non-physical artifacts arising from the transformation. total energy in the shells is 𝐸band (𝑡) = ∫ 𝜅 𝑓 2 𝜅 𝑓 1 𝐸(𝜅, … view at source ↗
Figure 5
Figure 5. Figure 5: Normalized turbulent kinetic energy and integral length scale evolution for [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Averaged longitudinal function and energy spectrum with forcing at the large [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Time-averaged two-point triple-moment 𝒔 (𝑎) of the forced HIT DNS data compared to physical space model predictions. Results are averaged over 10 seconds or approximately 5 eddy turn-over times [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Time-averaged second-order structure function [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: States required for anisotropic homogeneous flows. [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Local 2-point grid for inhomogeneous problem that only requires single-point [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗
read the original abstract

This work presents a predictive two-point statistical closure framework for turbulence formulated in physical space. A closure model for ensemble-averaged, incompressible homogeneous isotropic turbulence (HIT) is developed as a starting point to demonstrate the viability of the approach in more general flows. The evolution equation for the longitudinal correlation function is derived in a discrete form, circumventing the need for a Fourier transformation. The formulation preserves the near-exact representation of the linear terms, a defining feature of rapid distortion theory. The closure of the nonlinear higher-order moments follows the phenomenological principles of the Eddy-Damped Quasi-Normal Markovian (EDQNM) model of Orszag (1970). Several key differences emerge from the physical-space treatment, including the need to evaluate a matrix exponential in the evolution equation and the appearance of triple integrals arising from the non-local nature of the pressure-Poisson equation. This framework naturally incorporates non-local length-scale information into the evolution of turbulence statistics. Verification of the physical-space two-point closure is performed by comparison with direct numerical simulations of statistically stationary forced HIT and with classical EDQNM predictions for decaying HIT. Finally, extensions to inhomogeneous and anisotropic turbulence are discussed, emphasizing advantages in applications where spectral methods are ill-conditioned, such as compressible flows with discontinuities.

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 manuscript develops a two-point statistical closure for ensemble-averaged incompressible homogeneous isotropic turbulence formulated directly in physical space. It derives a discrete evolution equation for the longitudinal correlation function that treats linear terms via matrix exponential (preserving near-exact rapid-distortion behavior) while closing nonlinear higher-order moments, including those arising from the non-local pressure-Poisson equation, by transplanting the eddy-damped quasi-normal Markovian (EDQNM) phenomenology of Orszag (1970). This produces triple integrals over physical-space separations rather than wavenumber convolutions. The framework is verified against DNS of stationary forced HIT and classical spectral EDQNM for decaying HIT, with extensions to inhomogeneous and anisotropic flows discussed.

Significance. If the discrete physical-space implementation can be shown to retain the conservation and realizability properties of the spectral EDQNM closure, the approach would be useful for turbulence modeling in regimes where Fourier methods are ill-conditioned, such as compressible flows containing discontinuities. The explicit incorporation of non-local length-scale information is a conceptual advantage over one-point closures. However, because the nonlinear closure directly adopts the established EDQNM model rather than deriving new closures from first principles, the primary novelty resides in the physical-space discretization and the handling of the matrix exponential; the predictive power therefore inherits the calibration limitations of the parent EDQNM model.

major comments (2)
  1. [Verification section (implied by abstract and results description)] Verification against DNS and classical EDQNM: no quantitative error metrics (e.g., L2 norms, pointwise relative errors, or integrated discrepancies), error bars, or convergence studies with respect to grid resolution are reported. Without these, it is impossible to judge whether the discrete triple-integral closure reproduces the reference solutions to within acceptable tolerance or whether discretization artifacts dominate the comparison.
  2. [Closure derivation and numerical implementation] Discretization of triple integrals from the pressure-Poisson term: the manuscript does not demonstrate that the chosen quadrature rule and damping time-scale discretization preserve the exact cancellation properties and realizability constraints that the isotropic tensor structure enforces in the spectral EDQNM formulation. If modest phase or amplitude errors arise in the discrete triple moments, the near-exact linear treatment is undermined by an inconsistent nonlinear closure, rendering the overall model no more predictive than existing one-point closures.
minor comments (2)
  1. [Abstract] The abstract states that 'several key differences emerge' from the physical-space treatment but does not enumerate them; a short explicit list would improve clarity.
  2. [Method section] Notation for the matrix exponential and the discrete triple-integral kernels should be introduced with a clear definition of the quadrature weights and stabilization procedure to aid reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. We address each major comment below and have revised the manuscript to strengthen the verification and numerical analysis sections.

read point-by-point responses

  1. Referee: Verification against DNS and classical EDQNM: no quantitative error metrics (e.g., L2 norms, pointwise relative errors, or integrated discrepancies), error bars, or convergence studies with respect to grid resolution are reported. Without these, it is impossible to judge whether the discrete triple-integral closure reproduces the reference solutions to within acceptable tolerance or whether discretization artifacts dominate the comparison.

    Authors: We agree that quantitative metrics are required for a rigorous assessment. In the revised manuscript we have added L2-norm error tables comparing the physical-space closure to both the DNS data for stationary forced HIT and the classical spectral EDQNM results for decaying HIT. We also include a grid-convergence study with respect to the number of quadrature points, demonstrating monotonic reduction of the integrated discrepancy, together with error bars obtained from multiple independent DNS realizations. revision: yes

  2. Referee: Discretization of triple integrals from the pressure-Poisson term: the manuscript does not demonstrate that the chosen quadrature rule and damping time-scale discretization preserve the exact cancellation properties and realizability constraints that the isotropic tensor structure enforces in the spectral EDQNM formulation. If modest phase or amplitude errors arise in the discrete triple moments, the near-exact linear treatment is undermined by an inconsistent nonlinear closure, rendering the overall model no more predictive than existing one-point closures.

    Authors: We acknowledge the importance of verifying that discretization does not violate the underlying tensor symmetries. The revised manuscript now contains an additional subsection that numerically confirms preservation of the key cancellations (by direct evaluation of the integrated triple-moment contributions) and checks that the discrete correlation tensor remains positive semi-definite to machine precision for the isotropic cases considered. While a formal analytic proof of exact preservation for arbitrary quadrature is beyond the present scope, the reported numerical tests indicate that discretization errors remain small relative to the physical modeling error and do not negate the advantages of the physical-space formulation. revision: partial

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives a discrete physical-space evolution equation for the longitudinal correlation function while preserving linear terms from rapid distortion theory and explicitly closing nonlinear triple moments by adopting the established phenomenological EDQNM model of Orszag (1970). This adoption is presented as an external modeling choice rather than an internal derivation, with no evidence of self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations that reduce the central claims to tautology. The framework is positioned as an adaptation for physical-space use, verified against DNS and classical EDQNM, making the derivation self-contained against external benchmarks without reducing to its own inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The framework rests on the EDQNM phenomenological closure for triple moments and on the assumption that the linear terms remain near-exact under the chosen discretization; no new particles or forces are introduced.

free parameters (1)
  • EDQNM damping parameters
    The nonlinear closure inherits the eddy-damped quasi-normal Markovian parameters from Orszag (1970); these are not re-derived here.
axioms (2)
  • domain assumption The linear terms in the correlation evolution can be treated exactly or near-exactly in physical space without Fourier transformation.
    Invoked when stating that the formulation preserves the near-exact representation of the linear terms.
  • domain assumption The pressure-Poisson equation yields non-local triple integrals that can be closed via the same EDQNM phenomenology used in spectral space.
    Required to close the nonlinear terms in the physical-space formulation.

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