Effects of Reynolds number and spatial resolution on the pressure source terms in turbulent boundary layers

Aditya Agarwal; Rahul Deshpande

REVIEW 4 major objections 5 minor 16 references

Effects of Reynolds number and spatial resolution on the pressure source terms in turbulent boundary layers

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read In a zero-pressure-gradient turbulent boundary layer, the RMS of every pressure source term increases with friction Reynolds number across the layer, and turbulence-turbulence terms grow faster than mean shear.

desk verdict Useful first look at PIV resolution effects on pressure source terms; the Re_tau trend is plausible but statistically thin. read the letter →

arxiv 2412.19474 v1 pith:C7Q2A4ZM submitted 2024-12-27 physics.flu-dyn

Aditya Agarwal , Rahul Deshpande This is my paper · ORCID

classification physics.flu-dyn PACS 47.27.nb

keywords turbulentboundarylayerpressuresourcetermsfrictionReynoldsnumberdirectnumericalsimulationparticleimagevelocimetryspatialresolutionwall-pressurefluctuationsPoissonequationfor

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper asks what feeds wall-pressure fluctuations in a turbulent boundary layer and whether those feeding terms grow with Reynolds number. Using a published DNS database at friction Reynolds numbers near 1000 and 2000, it computes the root-mean-square of every pressure source term in the Poisson equation for pressure—the mean-shear term and the six turbulence-turbulence interaction terms. It reports that the RMS of all source terms increases across the whole boundary-layer thickness as Re_tau rises, with the nonlinear turbulence-turbulence terms growing faster than the mean-shear term. It then box-filters the DNS velocity fields to mimic the spatial resolution of planar and stereo PIV, showing that every recoverable source term is strongly attenuated and that the cross-plane stereo-PIV orientation suffers roughly 10–20% larger errors than the streamwise-wall-normal plane.

What carries the argument

The Poisson equation for the fluctuating pressure, written as $partial^{2}$ p / partial x_i partial x_i = -(T_MS + T^TT), is the central object; it is obtained by taking the divergence of the momentum equation, applying the Reynolds decomposition, and subtracting the mean pressure. T_MS = 2 (partial U / partial y)(partial v / partial x) + 2 (partial V / partial x)(partial u / partial y) is the linear mean-shear source for a boundary layer, and T^TT is the sum of six turbulence-turbulence terms T_ij^TT, such as (partial u / partial x)^2 - $partial^{2}$ $u^{2}$ / partial $x^{2}$ and the cross-gradient products. The analysis computes RMS wall-normal profiles of these terms from the DNS data and compares them across Reynolds numbers; the spatial-resolution study applies a box filter over interrogation volumes matching typical PIV configurations and recomputes the same terms from the filtered fields.

What would settle it

Computing the same wall-normal RMS profiles from a DNS at a higher Reynolds number (say Re_tau approximately 4000) or from sub-blocks at several intermediate Re_tau would falsify the paper if the source-term magnitudes fail to keep increasing across the full layer or if the turbulence-turbulence terms stop growing faster than the mean-shear term; separately, a laboratory PIV measurement at the two nominal resolutions could check the predicted 10–20% error gap between the planar and cross-plane configurations.

Watch

Extended reading notes

Core claim

For a zero-pressure-gradient turbulent boundary layer at Re_tau approximately 1000 and 2000, the RMS of the mean-shear source term T_MS and of each turbulence-turbulence term T_ij^TT, normalized by delta and $U_tau^{2}$, increases with Re_tau at every wall-normal location across the layer, while the wall-normal locations of the peaks stay fixed (y+ approximately 10 for T_MS and y+ approximately 20–30 for the TT terms). The turbulence-turbulence term T_yz^TT grows more rapidly than T_MS near the wall, so that at Re_tau approximately 2000 its maximum exceeds the T_MS maximum, indicating the growing importance of nonlinear pressure sources at higher Reynolds numbers. When the DNS fields are box-filtered to the resolution of a planar PIV measurement (10 x 10 x 22 in viscous units) or a stereo-PIV measurement (22 x 10 x 10), all recoverable source terms are attenuated across the entire layer, with errors largest near the wall (y+ < 10); the stereo-PIV case has errors about 10–20% larger than the planar case, and the locations of the maxima are preserved.

Load-bearing premise

The analysis assumes that the two 10-delta-long sub-blocks taken from each DNS volume produce converged root-mean-square values for the source terms at Re_tau approximately 1000 and 2000, and that these two Reynolds numbers are enough to establish the claimed monotonic growth and the faster growth of the turbulence-turbulence terms.

Editorial extensions

If this is right

If the increase of wall-pressure fluctuations with Re_tau is driven by the growth of these source terms, then models of wall-pressure noise at flight Reynolds numbers should put more weight on turbulence-turbulence interactions rather than mean shear alone.
Because the peak locations of the source terms stay at the same y+ values as Re_tau grows and as resolution degrades, the wall-normal scaling of the dominant source regions is robust, which helps in designing wall-normal sampling strategies for experiments.
Box-filtering mimics what PIV actually measures, so high-Re_tau PIV experiments that estimate pressure from velocity gradients will systematically underpredict source-term magnitudes, more so for cross-plane stereo-PIV than for planar x-y PIV by about 10–20%.
Correction schemes that recover under-resolved turbulence intensities, following the approach cited in the paper, would need to be extended to the source terms, since simple filtering leaves the peak locations intact but not the amplitudes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural extension the paper does not carry out is to test the same trends at Re_tau of order 10^4, where DNS cannot currently reach; if the faster growth of turbulence-turbulence terms persists, pressure fluctuations at practical Reynolds numbers would be dominated by nonlinear interactions rather than by mean-shear production.
The attenuation results imply that pressure-from-PIV reconstructions, which solve the Poisson equation for p from measured velocity fields, will inherit a small-scale bias even if mean velocity statistics are accurate; the roughly 10–20% larger error in the cross-plane case suggests that experimenters choosing between planar and stereo configurations should weight the in-plane orientation error mo
Because box-filtering preserves the y+ locations of the source-term maxima, a resolution-correction curve that depends only on the ratio of interrogation volume to local viscous scale might be transferable across Reynolds numbers, which could be tested by applying the same box filter to higher-Re DNS or experimental fields.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper does two things: it reports the first Re_tau variation of pressure source terms in a ZPG TBL, and it quantifies how box-filtering to mimic planar and stereo PIV attenuates those terms. The resolution part is the more solid half. Because it compares the same flow realization before and after filtering, the attenuation numbers are controlled, and the ~10-20% larger errors for the stereo-PIV case are a genuine, useful result for experiment design.

The Re_tau part is where I'd push back. The central claim—all source terms grow with Re_tau and the turbulence-turbulence terms grow faster—rests on exactly two conditions, Re_tau ~1000 and ~2000, with no measure of sampling variability. The sub-blocks are ~10 delta long; without bootstrap or scatter across blocks, the difference between the two profiles could be noise. The abstract states the trend flatly, while the body hedges by saying confirmation requires higher Re_tau. That mismatch needs to be fixed: either add uncertainty quantification or reframe the claim as a preliminary observation.

I also had trouble following the sub-block extraction. "Two subsets from each of thirteen volumes, such that we obtained thirteen volumes at two Re_tau" isn't clear enough to reproduce. That's a minor fix but worth making.

One more thing: the box-filter model is taken from Lee et al. without validation here, which is fine given the precedent, but a sentence acknowledging its limitations would help.

Overall, the paper is honest, the math is standard, and the resolution study is genuinely informative. The Re_tau trend is plausible and consistent with the known growth of wall-pressure fluctuations, but it needs more statistical support before I'd treat it as more than a hint. I'd send it to review if submitted to a journal, with the expectation that the authors add convergence evidence or soften the abstract. For now, it's a useful conference contribution, not a definitive scaling law.

Referee Report

4 major / 5 minor

Summary. The manuscript analyzes the pressure source terms (mean-shear T_MS and turbulence-turbulence T_TT_ij) in zero-pressure-gradient turbulent boundary layers using the Sillero et al. DNS database for friction Reynolds numbers between about 1000 and 2000. It reports that the RMS of all source terms increases with Re_tau across the entire boundary-layer thickness and that the nonlinear TT terms grow faster than the mean-shear term. The paper also box-filters the DNS velocity fields to mimic the spatial resolution of planar PIV (Case 1) and stereo-PIV (Case 2), finding significant attenuation of all recoverable source terms, with Case 2 errors larger than Case 1 by roughly 10%–20%. The authors argue these results are the first to quantify the Re_tau variation of pressure source terms in a TBL and the first to assess PIV spatial-resolution effects on these terms.

Significance. If the claims are correct, the paper provides a useful contribution to understanding the Reynolds-number scaling of pressure sources and the spatial-resolution limitations of PIV-based pressure estimation. The use of a well-established DNS database and the direct computation of source terms from velocity gradients are strengths, and the PIV-simulation exercise is a controlled comparison that usefully highlights measurement challenges. The spatial-resolution part of the study is more robust because it compares the same flow realization before and after filtering. However, the central Reynolds-number trend rests on only two Reynolds numbers with no uncertainty quantification, and the mathematical definitions of the source terms contain ambiguities that must be resolved. The paper's own statement that the faster-growth hypothesis can only be confirmed at much higher Re_tau (Section III.A) also weakens the abstract's definitive claim.

major comments (4)

[Section III.A, Figure 3, Section II.A] The claim that all source terms increase with Re_tau and that the turbulence-turbulence terms grow faster than the mean-shear term is based on only two Reynolds numbers (Re_tau ≈ 1000 and 2000) with no estimate of sampling uncertainty. The source terms are computed from finite 10-delta streamwise sub-blocks, but the paper does not report standard errors, confidence intervals, or a convergence test over block length or number of independent samples. Without such information, the observed differences between the two profiles could be within sampling noise, especially for the higher-order, noisier terms. The authors themselves state in Section III.A that the faster-growth hypothesis 'can be confirmed only after estimating the pressure source terms at much higher Reτ', which is inconsistent with the definitive wording in the abstract. Please add convergence or uncertainty estimates, or temper the claims accordingly.
[Equations (5)-(7) and Sections III.A-III.B] The simplified TT source terms in Eq. (5) as written, T_TT_ij = ∂u_i/∂x_j ∂u_j/∂x_i − ∂²(u_i u_j)/∂x_i∂x_j, are not equivalent to the definition in Eq. (3), which subtracts the Reynolds-averaged product <<u_i u_j>>. Unless the last term in Eq. (5) is intended to be ∂²(<<u_i u_j>>)/∂x_i∂x_j, the implementation is inconsistent with the stated definition. Equation (7) repeats the same ambiguity for all six components (e.g., T_xx = (∂u/∂x)² − ∂²(u²)/∂x² instead of − ∂²(<<u²>>)/∂x²). Since all TT profiles and the central comparisons depend on this definition, the authors should clarify or correct the expressions and verify which form was actually computed.
[Equation (6)] The mean-shear term for a ZPG TBL is given as T_MS = 2(∂U/∂y)(∂v/∂x) + 2(∂V/∂x)(∂u/∂y). This omits the terms 2(∂U/∂x)(∂u/∂x) and 2(∂V/∂y)(∂v/∂y), which are generally non-zero in a spatially developing boundary layer (with ∂V/∂y = -∂U/∂x by continuity). If these terms are negligible, the paper needs to justify that; otherwise they must be included for T_MS to be correctly evaluated and for the comparison between the two Reynolds numbers to be meaningful.
[Section II.A] The description of the sub-block extraction is unclear: 'We extracted two subsets of the full computational domain along the streamwise direction ... from each of the thirteen 3-D volumes ... such that we obtained thirteen 3-D volumes associated with two different Reτ'. Please clarify how many independent blocks are used for each Reynolds number and how the RMS statistics are computed (e.g., over homogeneous x-z planes, over multiple snapshots, or both). This information is necessary to assess the statistical convergence of the presented profiles.

minor comments (5)

[Abstract and Section III.B] The statement that Case 2 errors are larger than Case 1 by about 10%–20% should be tied to a specific figure panel or to a quantitative range taken from the profiles; the current wording is a qualitative summary that is hard to verify from Figure 4 alone.
[Nomenclature, Equations (2)-(8)] The notation for T_MS is inconsistent: it appears as a scalar in Eq. (6) and Eq. (8), while T_TT_ij carries two indices. Please define whether T_MS is a scalar or a tensorial component, and use consistent notation throughout.
[Equation (7)] The expression for T_TT_xy contains a typo: '∂2uv/∂xy' should read '∂²(uv)/(∂x∂y)' or '∂²(uv)/∂x∂y'. Please check the typesetting for all derivative terms.
[Section II.A and III.A] The paper uses 'RMS values of the pressure source terms' but never explicitly defines the averaging operator. Specify whether the RMS is computed over the homogeneous spanwise direction, the streamwise direction within each block, time (if multiple snapshots are used), or some combination of these.
[General presentation] The manuscript has several minor grammatical and formatting issues (e.g., 'It permits investigation of the Reτ-variation' in Section II.A, and inconsistent use of italics for variables). A careful proofread is recommended.

Circularity Check

0 steps flagged · score 1.0 of 10

No circularity: the source terms are computed directly from DNS gradients with no fitted parameters, and the minor self-citations are background only.

full rationale

The central derivation chain is self-contained against an external benchmark. The pressure source terms T_MS and T_TT are computed directly from the Sillero et al. DNS velocity fields via the standard Poisson-equation decomposition in equations (6)-(8), with no parameter fitted to any subset of the data. The Re_tau comparison at two Reynolds numbers is a direct measurement, not a prediction from an adjusted model. The PIV-mimicking exercise is likewise non-circular: the box-filter resolutions (10x10x22 and 22x10x10) are adopted from published PIV grid sizes following the external procedure of Lee et al. [11], and the resulting attenuation is an observed outcome rather than a constructed one. The only self-citations are [4] and [10], used for background motivation on wall-pressure scaling and superstructure evidence; neither is load-bearing for the source-term results, which depend on external sources [1], [3], and [11]. The abstract's monotonic Re_tau trend is supported by only two profiles without sampling-error estimates, but that is a statistical robustness concern, not circularity: the claim does not reduce to the inputs by construction. Accordingly, no circular step meeting the required evidentiary standard was identified.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central results rest on standard fluid mechanics equations, a public DNS database, and two hand-chosen filter scenarios. There are no fitted constants, no new physical entities, and no new governing laws. The main burden is the assumption that the DNS samples and the box-filter model are representative, which is a data-quality and modeling assumption rather than a circular fitting procedure.

free parameters (3)

Case 1 box filter dimensions (simulated planar PIV) = Delta_x+ x Delta_y+_min x Delta_z+ = 10 x 10 x 22
Chosen by the authors to mimic the interrogation volume of a planar PIV experiment in the x-y plane, based on typical resolutions in prior PIV studies [11]. Not fitted to any target result, but the specific values affect all attenuation numbers.
Case 2 box filter dimensions (simulated stereo-PIV) = Delta_x+ x Delta_y+_min x Delta_z+ = 22 x 10 x 10
Chosen to mimic a stereo-PIV experiment in the y-z plane, with the thicker filter along the streamwise direction. The choice directly influences the reported 10 to 20 percent error difference between cases.
Streamwise sub-block length = 10 delta
The authors limit each DNS sub-block to 10 boundary-layer thicknesses to keep Re_tau approximately constant. The specific length is a methodological choice that affects statistical convergence but is not fitted to data.

assumptions (5)

domain assumption Incompressible Navier-Stokes equations govern the flow (Eq. 1).
The pressure Poisson equation is derived from this starting point; the paper does not justify incompressibility beyond the turbulent boundary layer context.
domain assumption Reynolds decomposition and the divergence of the momentum equation yield the pressure Poisson equation for fluctuations (Eqs. 2-5).
This is a standard derivation, cited to Chang et al. [3]. The paper uses it without derivation.
domain assumption The ZPG TBL is two-dimensional in the mean, so the mean-shear term reduces to Eq. 6.
The authors neglect mean gradients in the spanwise direction, which is appropriate for a canonical ZPG TBL but is not explicitly justified in the text.
domain assumption The Sillero et al. DNS database is statistically converged and spatially resolved enough to compute velocity-gradient statistics at Re_tau 1000 and 2000.
The paper relies on the published validations of Sillero et al. [1], [14] without independent verification of convergence of the source-term RMS.
ad hoc to paper Box filtering the DNS velocity fields within interrogation volumes faithfully represents the spatial attenuation of a PIV experiment.
This is the central modeling assumption for the resolution-error analysis. The paper adopts the strategy of Lee et al. [11] but provides no experimental validation of the attenuation magnitude.

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Cite this review

Pith. "Pith review of Effects of Reynolds number and spatial resolution on the pressure source terms in turbulent boundary layers." pith.science (2026). https://pith.science/paper/C7Q2A4ZM

@misc{pith2026241219474,
  author       = {Pith},
  title        = {Pith review of: Effects of Reynolds number and spatial resolution on the pressure source terms in turbulent boundary layers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C7Q2A4ZM}},
  note         = {Machine review of arXiv:2412.19474}
}

abstract

The increase in wall-pressure fluctuations with increasing friction Reynolds number ($Re_{\tau}$) of a turbulent boundary layer (TBL) is well known in the literature. However, very few studies have investigated the $Re_{\tau}$-variation of the source terms of the pressure fluctuations, which are solely a function of the spatial velocity gradients within the TBL. This study quantifies the pressure source terms in a zero-pressure gradient TBL by utilizing a published direct numerical simulation (DNS; Sillero et al. 2013, Phys. Fluids) database across 1000 $\lesssim$ $Re_{\tau}$ $\lesssim$ 2000. It is found that the magnitude of all source terms increases with $Re_{\tau}$ across the entire TBL thickness, with the turbulence-turbulence (non-linear) interaction terms growing faster than the mean-shear (linear) source terms. Further, we use the simulation database to mimic the scenario of particle image velocimetry (PIV) experiments that are typically spatially under-resolved compared to DNS data. It is used to quantify the effect of spatial resolution on the accuracy of pressure source terms, which are estimated here for two common PIV scenarios: (i) planar PIV in the streamwise-wall-normal plane, and (ii) stereo-PIV in the spanwise-wall-normal plane of a ZPG TBL. This exercise reveals significant attenuation of all pressure source terms compared to those estimated from the original DNS, highlighting the challenges of accurately estimating these source terms in a high $Re_{\tau}$ PIV experiment.

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Reference graph

Works this paper leans on

16 extracted references · 15 canonical work pages

[1]

J. A. Sillero, J. Jim ´enez, and R. D. Moser. One-point statistics for turbulent wall-bounded flows at Reynolds numbers up toδ+ ≈ 2000. Phys. Fluids , 25(10), 2013

work page 2000
[2]

Tennekes and J

H. Tennekes and J. L. Lumley. A First Course in Turbulence . The MIT Press, 1972

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[3]

P. A. Chang, U. Piomelli, and W. K. Blake. Relationship between wall pressure and velocity-field sources. Phys. Fluids, 11:3434–3448, 1999

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[4]

Deshpande, R

R. Deshpande, R. Vinuesa, J. Klewicki, and I. Marusic. Active and inactive contributions to the wall pressure and wall-shear stress in turbulent boundary layers. ArXiv preprint, arXiv:2406.15733, 2024

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[5]

R. L. Panton, M. Lee, and R. D. Moser. Correlation of pressure fluctuations in turbulent wall layers. Phys. Rev. Fluids , 2(9):094604, 2017

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[6]

Tsuji, J.H.M

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R. de Kat and B. W. van Oudheusden. Instantaneous planar pressure determination from PIV in turbulent flow. Exp. Fluids , 52:1089– 1106, 2012

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R. Ma, K. Alam ´e, and K. Mahesh. Direct numerical simulation of turbulent channel flow over random rough surfaces. J. Fluid Mech. , 908:A40, 2021

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[9]

Raffel, C

M. Raffel, C. E. Willert, F. Scarano, C. J. K ¨ahler, S. T. Wereley, and J. Kompenhans. Particle image velocimetry: a practical guide . springer, 2018

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J. A. Sillero, J. Jim ´enez, and R. D. Moser. Two-point statistics for turbulent boundary layers and channels at Reynolds numbers up to δ+ ≈ 2000. Phys. Fluids , 26(10), 2014

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

J. A. Sillero, J. Jim ´enez, and R. D. Moser. One-point statistics for turbulent wall-bounded flows at Reynolds numbers up toδ+ ≈ 2000. Phys. Fluids , 25(10), 2013

work page 2000

[2] [2]

Tennekes and J

H. Tennekes and J. L. Lumley. A First Course in Turbulence . The MIT Press, 1972

work page 1972

[3] [3]

P. A. Chang, U. Piomelli, and W. K. Blake. Relationship between wall pressure and velocity-field sources. Phys. Fluids, 11:3434–3448, 1999

work page 1999

[4] [4]

Deshpande, R

R. Deshpande, R. Vinuesa, J. Klewicki, and I. Marusic. Active and inactive contributions to the wall pressure and wall-shear stress in turbulent boundary layers. ArXiv preprint, arXiv:2406.15733, 2024

arXiv 2024

[5] [5]

R. L. Panton, M. Lee, and R. D. Moser. Correlation of pressure fluctuations in turbulent wall layers. Phys. Rev. Fluids , 2(9):094604, 2017

work page 2017

[6] [6]

Tsuji, J.H.M

Y . Tsuji, J.H.M. Fransson, P.H. Alfredsson, and A.V . Johansson. Pres- sure statistics and their scaling in high-Reynolds-number turbulent boundary layers. J. Fluid Mech. , 585:1–40, 2007

work page 2007

[7] [7]

de Kat and B

R. de Kat and B. W. van Oudheusden. Instantaneous planar pressure determination from PIV in turbulent flow. Exp. Fluids , 52:1089– 1106, 2012

work page 2012

[8] [8]

R. Ma, K. Alam ´e, and K. Mahesh. Direct numerical simulation of turbulent channel flow over random rough surfaces. J. Fluid Mech. , 908:A40, 2021

work page 2021