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arxiv: 2509.02570 · v2 · submitted 2025-08-19 · ⚛️ physics.flu-dyn · math.PR

Random field reconstruction of inhomogeneous turbulence. Part II: Numerical approximation and simulation

Markus Antoni , Quinten K\"urpick , Felix Lindner , Nicole Marheineke , Raimund Wegener This is my paper

Pith reviewed 2026-05-18 21:50 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn math.PR
keywords inhomogeneous turbulencerandom field modelstochastic Fourier integralsrandomized quadraturenumerical discretizationKolmogorov scalingturbulence Reynolds numberlocal linearization
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The pith

A discretization scheme using randomized quadrature and local linearization converges to the continuous random field model for inhomogeneous turbulence.

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

The paper presents a numerical method to simulate turbulent velocity fluctuations in flows whose mean properties change with position. It starts from a continuous model that represents these fluctuations through stochastic Fourier-type integrals and then discretizes that model for practical computation. The discretization pairs randomized quadrature for the integrals with a local linearization that approximates how the mean flow carries turbulent structures. Simulations using the scheme reproduce the spatial variation of fluctuation statistics, including the way Kolmogorov scaling depends on the local turbulence Reynolds number. A reader would care because the method offers an efficient way to generate realistic inhomogeneous turbulence data for engineering applications without solving the full Navier-Stokes equations at every point.

Core claim

The authors construct a discretization scheme that combines randomized quadrature for the stochastic integrals with local linearization of the non-uniform advection of turbulent structures by the mean flow. They establish analytically that the scheme converges to the underlying continuous random field model. In simulations the discrete fields reproduce the influence of inhomogeneous model parameters on the generated fluctuations, preserve spatio-temporal ergodicity properties, and satisfy Kolmogorov's two-thirds law with dependence on the local turbulence Reynolds number. The implementation supports flexible local evaluation of the turbulence field at arbitrary points in space and time.

What carries the argument

Randomized quadrature for stochastic integrals combined with local linearization of non-uniform advection in the random field model.

If this is right

  • The generated fields accurately reflect spatial changes in mean-flow quantities through the model parameters.
  • Kolmogorov's two-thirds law holds locally and varies with the local turbulence Reynolds number.
  • Spatio-temporal ergodicity properties of the continuous model carry over to the discrete approximation.
  • The scheme permits efficient local evaluation of the turbulence field without global recomputation.
  • Inhomogeneous features such as position-dependent intensity and length scales are reproduced directly from the input flow quantities.

Where Pith is reading between the lines

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

  • The same quadrature-plus-linearization pattern could be applied to other stochastic integral representations of random fields in fluid mechanics.
  • The method offers a practical route to synthetic turbulence generation for computational fluid dynamics in engineering geometries where mean-flow inhomogeneity matters.
  • Quantitative error bounds on the linearization step could be derived by comparing against exact transport solutions in simple shear flows.
  • The approach may reduce computational cost in large-eddy simulations that require inflow turbulence with realistic spatial variation.

Load-bearing premise

The local linearization of advection by the inhomogeneous mean flow introduces negligible error relative to the exact transport of turbulent structures.

What would settle it

Direct comparison of statistics from the linearized scheme against a reference simulation that uses the full nonlinear advection in regions of strong mean-flow gradients would falsify the claim if systematic deviations appear in the fluctuation spectra or correlation functions.

Figures

Figures reproduced from arXiv: 2509.02570 by Felix Lindner, Markus Antoni, Nicole Marheineke, Quinten K\"urpick, Raimund Wegener.

Figure 4.1
Figure 4.1. Figure 4.1: Illustration of the indices used in Algorithm 4.1. The sample path of u ′ N is evaluated at time t and one or multiple spatial evaluation points x, specifying one or multiple scaled time integration kernels s 7→ η(t − s; x, t) with possibly different supports. The range indicated by the horizontal solid blue line comprises those intervals Ij that contribute to the evaluations at time t. See the text for … view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: Streamline plot showing an approximate realization of a two￾dimensional analogue of Model 2.1 at a fixed time point, with spatial scaling func￾tion σx = k 3/2/ε increasing along the x1-axis by a factor of four. The scaling functions and underlying flow quantities are depicted in semi-log plots. See the text for details. with the behavior of the spatial scaling factor σx , it can be seen that the turbulen… view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: Streamline plot showing an approximate realization of a two￾dimensional analogue of Model 2.1 at a fixed time point, with viscosity scaling function σz = εν/k2 increasing along the x1-axis by a factor of ten. The scaling functions and underlying flow quantities are depicted in semi-log plots. See the text for details. wavenumbers than the comparably narrow spectra corresponding to the structures on the r… view at source ↗
Figure 5.3
Figure 5.3. Figure 5.3: Streamline plot showing an approximate realization of a two￾dimensional analogue of Model 2.1 at a fixed time point, with velocity scaling function σu = k 1/2 increasing along the x1-axis by a factor of four. The scaling functions and underlying flow quantities are depicted in semi-log plots. See the text for details. vectors depicted in the vector plots is scaled by a factor of 0.01 in order to facilita… view at source ↗
Figure 5.4
Figure 5.4. Figure 5.4: Approximate realizations of the turbulence field in Model 2.1 at a fixed time point. The heat maps show the first velocity component, and the plotted velocity vectors are scaled by a factor of 0.01. Top: Spatial scaling function σx = k 3/2/ε increasing along the x1-axis by a factor of four (scenario from [PITH_FULL_IMAGE:figures/full_fig_p015_5_4.png] view at source ↗
Figure 5.5
Figure 5.5. Figure 5.5: Approximate sample paths of the components u ′ i (x, t) of the turbu￾lence field in Model 2.1 on the line segment {x = (x1, 0, 0): 0 ≤ x1 ≤ 1}, at a fixed time point t. Top: Increasing spatial scaling factor σx = k 3/2/ε (scenario from [PITH_FULL_IMAGE:figures/full_fig_p016_5_5.png] view at source ↗
Figure 5.6
Figure 5.6. Figure 5.6: Approximate sample paths w.r.t. time of the components u ′ i (x, t) of the turbulence field in Model 2.1 at a fixed spatial point x, with temporal scaling function σt = k/ε increasing along the x1-axis by a factor of four. The scaling functions and underlying flow quantities are depicted in semi-log plots. See the text for details. As all simulation results discussed up to this point concern snapshots of… view at source ↗
Figure 5.7
Figure 5.7. Figure 5.7: Approximate realizations of the first component u ′ 1 of the turbulence field in Model 2.1 at a fixed time point. The fluctuations are subject to advection by a stationary non-uniform mean flow (streamlines in blue color) in two different szenarios for the underlying flow data k, ε, and ν. See the text for details. A further essential feature of our model is its ability to consistently capture the advect… view at source ↗
Figure 5.8
Figure 5.8. Figure 5.8: Spatial averaging at a fixed time point t. Left: Approximate sample path kω of the instantaneous turbulent kinetic energy ∥u ′∥ 2/2 along the x1-axis, together with associated moving averages over line segments and three-dimensional balls, estimating the underlying turbulent kinetic energy k. Right: Approximate sample path εω of the instantaneous dissipation rate δ 2 z ν ∥∇xu ′ + (∇xu ′ ) ⊤∥ 2/2, togethe… view at source ↗
Figure 5.9
Figure 5.9. Figure 5.9: Spatio-temporal averaging. Left: Approximate sample path kω of the instantaneous turbulent kinetic energy ∥u ′∥ 2/2 along a mean flow pathline xt, together with associated moving averages over pathline segments and environments thereof generated by three-dimensional balls, estimating the underlying turbulent kinetic energy k. Right: Approximate sample path εω of the instantaneous dissi￾pation rate δ 2 z … view at source ↗
read the original abstract

A novel random field model or the reconstruction of turbulent velocity fluctuations from inhomogeneous characteristic flow quantities in terms of stochastic Fourier-type integrals has recently been introduced and analyzed by the authors. This article concerns the numerical discretization and implementation of the model and discusses its key features by means of numerical simulations. We present a suitable discretization scheme that combines a randomized quadrature method for stochastic integrals with a local linearization of the non-uniform advection of the turbulent structures by the mean flow. The convergence of the scheme towards the continuous model is verified analytically. Moreover, we describe an efficient algorithmic implementation that allows for flexible local evaluations of the simulated turbulence field. The main features of the model are illustrated by a variety of simulation results, each highlighting specific aspects such as the influence of the inhomogeneous model parameters on the generated fluctuations, spatio-temporal ergodicity properties under inhomogeneous flow conditions, and the validity of Kolmogorov's two-thirds law in dependence on the local turbulence Reynolds number.

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

1 major / 2 minor

Summary. The manuscript presents a numerical discretization for the random field model of inhomogeneous turbulence from Part I. The scheme combines randomized quadrature for the stochastic Fourier integrals with local linearization of advection by the inhomogeneous mean flow. Analytical convergence of the discretization to the continuous model is claimed, an efficient implementation for local field evaluations is described, and simulations illustrate the effects of inhomogeneous parameters, spatio-temporal ergodicity, and the validity of Kolmogorov's two-thirds law as a function of local turbulence Reynolds number.

Significance. If the claimed analytical convergence holds with controlled linearization error, the work supplies a practical, locally evaluable method for generating synthetic inhomogeneous turbulence that respects spatially varying mean-flow statistics. This is useful for CFD validation and engineering flows with strong inhomogeneity. The combination of an analytical convergence result with targeted numerical demonstrations of ergodicity and Reynolds-number dependence is a clear strength.

major comments (1)
  1. [discretization scheme and convergence analysis] The central convergence claim for the combined discretization (randomized quadrature plus local linearization) is load-bearing. The manuscript must supply the explicit error bound or theorem showing that the local-linearization error for the advection operator vanishes in the discretization limit; otherwise the scheme converges at best to a modified transport model whose statistics differ from the exact inhomogeneous advection in the Part I continuous model. This is especially relevant when mean-flow gradients are not small.
minor comments (2)
  1. [algorithmic implementation] Clarify the precise quadrature rule and the number of random samples used in the stochastic integrals; the current description is too terse for reproducibility.
  2. [simulation results] The simulation figures would benefit from explicit error bars or ensemble sizes to support the reported ergodicity and two-thirds-law observations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address the major concern on the convergence analysis below and will strengthen the presentation accordingly.

read point-by-point responses

  1. Referee: The central convergence claim for the combined discretization (randomized quadrature plus local linearization) is load-bearing. The manuscript must supply the explicit error bound or theorem showing that the local-linearization error for the advection operator vanishes in the discretization limit; otherwise the scheme converges at best to a modified transport model whose statistics differ from the exact inhomogeneous advection in the Part I continuous model. This is especially relevant when mean-flow gradients are not small.

    Authors: We appreciate the referee highlighting the need for an explicit combined error bound. Section 3 of the manuscript already verifies convergence analytically by bounding the randomized quadrature error via standard Monte Carlo estimates and controlling the local linearization error through the Lipschitz constant of the mean flow, showing that this contribution vanishes as the local stencil size tends to zero. The analysis establishes that the scheme converges to the continuous model of Part I rather than a modified transport equation. To make the result fully explicit and address cases with non-small gradients, we will add a dedicated theorem in the revised manuscript that states the total discretization error (quadrature plus linearization) and its vanishing rate in the joint limit of increasing quadrature points and decreasing linearization scale. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to prior model; numerical convergence and simulations remain independent

full rationale

The paper cites the authors' own prior work (Part I) for the introduction of the underlying random field model via stochastic Fourier-type integrals, but this citation supports the starting point rather than load-bearing the central numerical claims. The discretization scheme, randomized quadrature, local linearization, analytical convergence verification, algorithmic implementation, and simulation results (including Kolmogorov scaling dependence on local Reynolds number) are developed and justified within this manuscript without reducing to self-referential definitions, fitted inputs renamed as predictions, or ansatzes smuggled via self-citation. The derivation chain for the numerical approximation is self-contained against external benchmarks such as standard quadrature convergence and ergodicity checks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the random field model introduced in the authors' previous work and on standard assumptions for convergence of randomized quadrature methods applied to stochastic integrals; no new free parameters or invented entities are introduced in the numerical part itself.

axioms (2)
  • domain assumption The random field model from Part I correctly represents the statistics of inhomogeneous turbulence.
    The entire numerical development and verification is predicated on this prior model being appropriate.
  • standard math Randomized quadrature converges for the stochastic Fourier-type integrals under the stated regularity conditions.
    Invoked to justify the discretization scheme and its convergence proof.

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Reference graph

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