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REVIEW 3 major objections 3 minor 54 references

Scaling of wall-pressure--velocity correlations in high Reynolds number turbulent pipe flow

T0 review · 3 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Wall-pressure coherence with velocity collapses on $\lambda_x/y$ and stays Reynolds-number invariant from $Re_\tau \approx 4\,800$ to $47\,000$, and quadratic stochastic estimation from wall pressure alone recovers wall-attached…

desk verdict Solid experimental extension of wall-pressure–velocity coherence scaling to Re_tau ~ 47,000 in pipe flow, but the headline QSE correlation is computed in-sample and should be read as an upper bound until cross-validated. read the letter →

arxiv 2501.08018 v1 pith:JLZFUHG4 submitted 2025-01-14 physics.flu-dyn

Giulio Dacome , Lorenzo Lazzarini , Alessandro Talamelli , Gabriele Bellani , Woutijn J. Baars This is my paper · ORCID
classification physics.flu-dyn
keywords wall-pressurefluctuationscoherencespectrahighReynoldsnumberpipeflowquadraticstochasticestimationamplitudemodulationwall-attachedturbulenceTaylor'shypothesislarge-scalemotions
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

Experiments in a long-pipe facility, spanning friction Reynolds numbers $Re_\tau$ from roughly 4,800 to 47,000, test whether wall-pressure fluctuations can serve as a proxy for velocity fluctuations in the logarithmic layer. The paper shows that the coherence between wall pressure and streamwise or wall-normal velocity collapses onto a single curve when the streamwise wavelength is scaled by the distance-from-the-wall, $\lambda_x/y$, and is unchanged across the entire Reynolds-number range. Squaring the wall pressure adds low-frequency content, raising the coherence at large scales and pointing to an amplitude-modulation link between large-scale velocity and wall-pressure intensity. A quadratic stochastic estimator that uses wall pressure and its square as inputs recovers a wall-attached, large-scale-filtered streamwise velocity with correlation $\rho \approx 0.6$ at every Reynolds number, while the linear-only estimator stays below about 0.4. The practical stake is that a sparse wall-pressure sensor plus a fixed nonlinear estimator could provide a Reynolds-number-independent observer of energetic off-wall velocity fluctuations for feedback control at application-level conditions.

What carries the argument

The linear coherence spectrum $\gamma^2_{u p_w}$ between velocity and wall pressure, defined as the magnitude-squared cross-spectrum normalized by the two autospectra, is the primary diagnostic; the paper also uses $\gamma^2_{u p_w^2}$ for the de-meaned squared wall pressure. Frequency is converted to streamwise wavelength with Taylor's hypothesis, $\lambda_x = U_c/f$ at $U_c^+ = 10$, to compare with spatial DNS data. The stochastic estimator is quadratic stochastic estimation (QSE): a time-domain convolution of $p_w$ and $p_w^2$ with kernels built from cross-spectra divided by input autospectra. The reference signal $u_W$ is a large-scale pass-filtered streamwise velocity with a Reynolds-number-invariant cutoff at $\lambda_x/y = 14$, and accuracy is scored by the Pearson correlation between $u_W$ and the QSE estimate.

What would settle it

Recompute the coherence spectra using a scale-dependent convection velocity, such as one extracted from resolvent analysis or from direct spatial measurements, and ask whether the $\lambda_x/y$ collapse survives; if it does not, the fixed $U_c^+ = 10$ conversion was the cause of the apparent collapse. A second check is to train the QSE kernel at one Reynolds number and evaluate it on measurements at another: $\rho \approx 0.6$ should persist across $Re_\tau$ if the estimator is genuinely Reynolds-number invariant.

Watch

Extended reading notes

Core claim

The paper's central claim is that wall-pressure–velocity coherence spectra are a function of $\lambda_x/y$ alone and are Reynolds-number invariant from $Re_\tau = 4\,794$ to $47\,015$. The quadratic term $p_w^2$ is more coherent with large-scale streamwise velocity than the linear term $p_w$: at $\lambda_x/y = 60$ the coherence rises from roughly 0.1 to 0.3, an effect the authors interpret as large-scale velocity modulating the intensity of the smaller-scale wall pressure. This modulation is confirmed by showing that coherence with $p_w^2$ matches coherence with the Hilbert envelope of the wall pressure. Using quadratic stochastic estimation with $p_w$ and $p_w^2$ as the only inputs, the estimated streamwise velocity correlates with the wall-attached filtered velocity $u_W$ at $\rho \approx 0.6$ for all test cases; the linear-only estimate is about 20% lower. The authors conclude that wall-pressure sensing alone gives a meaningful, Reynolds-number-invariant estimate of off-the-wall velocity fluctuations at high Reynolds numbers.

Load-bearing premise

The load-bearing premise is that frequencies can be mapped to streamwise wavelengths by Taylor's hypothesis with the single convection velocity $U_c^+ = 10$ for all scales and Reynolds numbers; the paper itself notes that the wall-pressure convection velocity is scale-dependent, so if the true mapping varies with scale or $Re_\tau$, the collapse in $\lambda_x/y$ could be partly a product of that fixed conversion.

Editorial extensions

If this is right

  • A wall-pressure sensor paired with the quadratic estimator can be used for real-time control of energetic log-layer velocity fluctuations at high Reynolds numbers, because the estimation accuracy does not degrade from $Re_\tau \approx 4\,800$ to $47\,000$.
  • The quadratic wall-pressure term is the source of the usable correlation: removing it lowers the estimator's correlation from about 0.6 to below 0.4.
  • Coherence spectra for pipe flow agree with those from channel-flow DNS, supporting a common $\lambda_x/y$ scaling across wall-bounded flow geometries.
  • The large-scale coherence region, linked to inactive and global modes, means wall-pressure sensors can track very-large-scale motions even when the small-scale coherence is low.
  • The modulation interpretation implies that the wall-pressure intensity envelope, not just the pressure itself, carries the information needed to estimate large-scale velocity fluctuations.

Reading between the lines

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

  • If the $\lambda_x/y$ collapse survives a scale-dependent convection-velocity correction, the result strengthens the case for a universal wall-attached pressure–velocity coupling; applying such a correction would also test whether the single-velocity Taylor's-hypothesis mapping is hiding the Reynolds-number invariance.
  • The QSE estimator was demonstrated for streamwise velocity at two wall-normal positions with a single pressure point; a spanwise array of pressure sensors could extend the approach to wall-normal velocity or to a wider band of scales, and its training at one Reynolds number could be tested for transfer to another.
  • Comparing the estimator against the full unfiltered streamwise velocity, rather than only the wall-attached filtered signal $u_W$, would separate how much of the $\rho \approx 0.6$ ceiling comes from the filter definition versus from genuine coherence limits.
  • A natural next step is to train the QSE kernel at low Reynolds number and apply it at high Reynolds number to see whether the same kernels remain valid, which would make the estimator effectively calibration-free across operating conditions.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The manuscript reports synchronized wall-pressure and hot-wire velocity measurements in the CICLoPE long-pipe facility at seven friction Reynolds numbers between 4,794 and 47,015. It documents wall-pressure statistics, linear coherence spectra between streamwise or wall-normal velocity and linear or squared wall-pressure fluctuations, and stochastic estimation of streamwise velocity from wall-pressure time series. The central claims are that the coherence spectra collapse when plotted against lambda_x/y and are Reynolds-number invariant, that the squared wall-pressure term substantially increases large-scale coherence via an amplitude-modulation mechanism, and that quadratic stochastic estimation from wall-pressure alone yields a normalized correlation of about 0.6 with the wall-attached streamwise velocity for all cases.

Significance. If fully substantiated, the reported Reynolds-number-invariant coherence scaling and the QSE correlation of about 0.6 would be valuable: they extend wall-pressure-velocity correlation scaling to application-level Reynolds numbers and support wall-pressure-based sensing for real-time control of energetic off-the-wall fluctuations. The experimental program is a clear strength: synchronized multi-sensor pressure and hot-wire measurements over a decade of Re_tau, careful treatment of pinhole-cavity resonance, and hPOD-based acoustic-noise removal. The agreement of the coherence spectra for cases 3-7 with the spatial DNS reference data of Baars et al. (2024) is convincing. However, the QSE accuracy is reported in-sample, and the temporal-to-spatial mapping relies on a single fixed convection velocity that the manuscript itself acknowledges to be scale-dependent; both points bear directly on the two main applied claims.

major comments (3)
  1. [§6, Eqs. (6.3)-(6.4) and Fig. 8] The QSE kernels HL and HQ are estimated from the same synchronized pressure and velocity records that are subsequently used to compute the reported Pearson correlation rho. The text does not state that any portion of the 480 s records was held out or that the kernels were cross-validated. Because these are complex frequency-dependent filters with a linear and a quadratic input channel, the reported rho approximately 0.6 is an in-sample accuracy metric and may overstate the predictive skill obtainable in real-time control, where kernels must be fixed before the control input arrives. Please estimate the kernels on a calibration subset and evaluate the correlation on independent validation data, or perform leave-one-block-out cross-validation, and report the resulting out-of-sample rho.
  2. [§3-§5, Eq. (4.1) and Figs. 3, 5, 7] The temporal-to-spatial conversion lambda_x = U_c/f with fixed U_c+ = 10 is applied for all scales, wall-normal positions, and Reynolds numbers. The manuscript itself acknowledges on pages 8-9 that the convection velocity of the wall-pressure field is scale-dependent and that Taylor's hypothesis is not strictly valid for near-wall fluctuations. Because the central collapse in lambda_x/y and the quantitative comparison with spatial DNS data depend on this mapping, a fixed U_c could impart or mask Reynolds-number trends. Please quantify this sensitivity, for example by recomputing the collapse with U_c+ in a plausible range (e.g., 8-12) or by using the two-point pressure-sensor pair to estimate a scale-dependent convection velocity.
  3. [Abstract, §3, §4 and Fig. 8] The abstract and conclusions state the results for 4,794 <= Re_tau <= 47,015 and report rho for each Re_tau, but the two lowest-Reynolds-number cases are explicitly excluded from the QSE correlation and show attenuated coherence due to incomplete facility-noise removal (§2.4 and Fig. 5). The Reynolds-invariance claim is therefore actually supported by cases 3-7, and no uncertainty estimates are provided for either the coherence spectra or the rho values. Please state the supporting cases explicitly in the abstract/conclusions and add error estimates, such as bootstrap confidence intervals over spectral blocks or coherence confidence bounds, so that the claimed collapse can be assessed quantitatively.
minor comments (3)
  1. [§6 and Fig. 8] The text states that "Figure 8b presents values of rho[uW, uhat_QSE]" and later that a lower correlation appears in "Fig. 8b" for LSE, but the caption labels panel (a) as QSE and panel (b) as LSE; please correct the cross-references so the text matches the figure.
  2. [§6] The sentence referencing "points A and F (see Fig. 1b)" appears to refer to Fig. 1(d), which shows the measurement points in the area of interest; please update the reference.
  3. [Appendix A] The hPOD mode-selection criterion (retaining modes 3 and 4) is illustrated for test case 3 only; please state explicitly whether the same mode indices were selected for all seven cases and how the cutoff frequency f_c was chosen for each case.

Circularity Check

1 steps flagged · score 6.0 of 10

Coherence collapse is benchmarked externally, but the QSE rho≈0.6 is reported in-sample: the estimator kernels are fitted to the same records used for the correlation.

  1. fitted input called prediction [Sec. 6, Eqs. (6.3)–(6.4) and Fig. 8; also Sec. 6 opening paragraph and Conclusions]
    "The linear kernel equals the cross-spectrum between u and pw, divided by the auto-spectra of pw (the input quantity during the estimation method), HL(ye,f)=<U~(ye,f)P~w*(f)>/<|P~w(f)|^2>, (6.3) whereas the quadratic kernel includes the wall-pressure–squared term ... HQ(ye,f)=... (6.4) ... To evaluate the accuracy of the estimation with respect to the reference time series, u(y0,t), the Pearson correlation coefficient is employed ... Figure 8b presents values of rho[uW(y,t), uhat_QSE(y,t)]."

    The QSE/LSE kernels in Eqs. (6.3)–(6.4) are constructed from the cross-spectra of the same wall-pressure and velocity records on which Fig. 8 computes rho. The estimator is therefore the optimal linear/quadratic projection of u onto pw for those very records; no held-out split or cross-validation is reported. The resulting rho≈0.6 is an in-sample goodness-of-fit (the fraction of variance explained by the fitted spectral filter), not an independent prediction on unseen data. Section 6 frames this as 'the accuracy of the prediction of u fluctuations', and the Conclusions convert the in-sample fit into the claim that wall-pressure sensing is 'scalable to application-level conditions'.

full rationale

The coherence-scaling portion of the paper is not circular: Eq. (4.1) is a measured coherence, and the collapse in lambda_x/y is tested against spatial DNS data from Baars et al. (2024) and other literature, providing an external benchmark. The references to Baars et al. (2017, 2024) are self-citations but they are not load-bearing in a circular way: they are backed by independent DNS data and are used as comparisons or filters, not as the source of the present collapse. Taylor's hypothesis with Uc+ = 10 is a stated assumption, explicitly acknowledged as imperfect, and is a modeling limitation rather than a definitional equivalence. The genuine circularity is concentrated in Sec. 6: the stochastic-estimation kernels are fitted to the same records used to evaluate the reported rho, so the claimed prediction accuracy is in-sample. This makes the QSE result a partial circularity, while the coherence findings remain independently supported.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central scaling claim depends on Taylor's hypothesis with a fixed Uc+ = 10, on hPOD mode selection for facility-noise removal, and on the assumption that residual acoustic noise is uncorrelated with velocity. The QSE accuracy claim depends on kernels fitted to the same records used for evaluation. No new physical entities are introduced.

free parameters (4)
  • Convection velocity Uc+ = 10 = 10
    Used to convert temporal frequency to streamwise wavelength via lambda_x = Uc/f for all spectral and coherence plots; chosen as a constant, acknowledged scale-dependent, and directly affects the lambda_x/y collapse.
  • hPOD retained modes and cutoff frequency fc = modes 3 and 4, fc = 70 Hz for the reference case
    Facility-noise filtering reconstructs wall pressure from modes 3 and 4 only, and fc is adjusted per test case; this hand-selected mode set determines the noise-corrected pressure input to all coherence and estimation analyses.
  • Resonance correction kernel Hr = second-order model with resonance at 4,350 Hz
    Pinhole-cavity transfer kernel fitted to flow-off acoustic data and used to correct wall-pressure spectra; the paper notes the kernel changes with grazing turbulence, making the correction imperfect.
  • QSE kernels HL and HQ = per-case cross-spectral ratios from Eqs. (6.3) and (6.4)
    Optimal linear and quadratic filters estimated from the same synchronized wall-pressure and velocity records used to compute the reported correlation; no cross-validation is reported.
assumptions (5)
  • domain assumption Taylor's hypothesis maps temporal frequency to streamwise wavelength with a single convection velocity Uc+ = 10.
    Invoked in Sections 3 and 4 for all spectra; acknowledged as scale-dependent and not strictly valid for near-wall pressure.
  • ad hoc to paper Facility acoustic noise is spatially correlated across sensors and is absent in hPOD modes 3 and 4, so reconstruction with those modes yields hydrodynamic wall-pressure.
    Appendix A; the mode-selection rule is specific to this facility and dataset and is not independently verified.
  • domain assumption Additive facility noise is uncorrelated with velocity fluctuations, so its removal can only increase true coherence and correlation.
    Used in Section 2.4 and Section 6 to interpret attenuation in test cases 1 and 2.
  • domain assumption Wall-attached eddy model and the cutoff at lambda_x/y = 14 define the reference signal u_W for evaluating QSE.
    Section 6; u_W is not the full velocity signal but a large-scale pass-filtered version, which changes the meaning of rho about 0.6.
  • standard math Standard spectral estimation via FFT ensembles and cross-spectra is valid.
    Used throughout for coherence and spectra; no proof is required, but convergence is asserted from record length rather than demonstrated with confidence intervals.

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Pith. "Pith review of Scaling of wall-pressure--velocity correlations in high Reynolds number turbulent pipe flow." pith.science (2026). https://pith.science/paper/JLZFUHG4

@misc{pith2026250108018,
  author       = {Pith},
  title        = {Pith review of: Scaling of wall-pressure--velocity correlations in high Reynolds number turbulent pipe flow},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JLZFUHG4}},
  note         = {Machine review of arXiv:2501.08018}
}
abstract

An experimental study was conducted in the CICLoPE long-pipe facility to investigate the correlation between wall-pressure and turbulent velocity fluctuations in the logarithmic region, at high friction Reynolds numbers ($4\,794 \lesssim Re_\tau \lesssim 47\,015$). Hereby we explore the scalability of employing wall-pressure to effectively estimate off-the-wall velocity states (e.g., to be of use in real-time control of wall-turbulence). Coherence spectra for wall-pressure and streamwise (or wall-normal) velocity fluctuations collapse when plotted against $\lambda_x/y$ and thus reveals a Reynolds-number-independent scaling with distance-from-the-wall. When the squared wall-pressure fluctuations are considered instead of the linear wall-pressure term, the coherence spectra for the wall-pressure--squared and velocity are higher in amplitude at wavelengths corresponding to large-scale streamwise velocity fluctuations (e.g., at $\lambda_x/y = 60$ the coherence value increases from roughly 0.1 up to 0.3). This higher coherence typifies a modulation effect, because low-frequency content is introduced when squaring the wall-pressure time series. Finally, quadratic stochastic estimation is employed to estimate turbulent velocity fluctuations from the wall-pressure time series only. For each $Re_\tau$ investigated, the estimated time series and a true temporal measurement of velocity inside the turbulent pipe flow, yield a normalized correlation coefficient of $\rho \approx 0.6$ for all cases. This suggests that wall-pressure sensing can be employed for meaningful estimation of off-the-wall velocity fluctuations, and thus for real-time control of energetic turbulent velocity fluctuations at high $Re_\tau$ applications.

Figures

Figures reproduced from arXiv: 2501.08018 by the authors.

Figure 1
Figure 1. (a,b) Photographs of the CICLoPE laboratory, with in (b) the test section at the downstream end f thlifilit() Shtif thihlt (M1 tM4 td i Figure 1: (a,b) Photographs of the CICLoPE laboratory, with in (b) the test section at the downstream end [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Probability density functions of the wall-pressu Fi2Pbbilitditftif thllfl [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. (a,b) Premultiplied energy spectra of wallpressure fluctuations, for a (an increase in colour intensity of corresponds to an increase in Refollowin Figure 3: (a,b) Pre-multiplied energy spectra of wall-pressure fluctuations, for all [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Wall-pressure intensity inferred from integrating the wall-pressure spectra. Current results are compared to several datasets available from the literatureData are taken from the DNS studies of Panton t l Figure 4: Wall-pressure intensity inferred from integrating the …
Figure 5
Figure 5. Figure 5: (a) Coherence spectra for the fluctuations in strea Fi5() Chfhflii [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: (a) Normalised wall-pressure time series of micro Fi6() Nlid lltiif i [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: (a) Coherence spectra for the fluctuations in wall- Fi7() Chfhfliill [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: (a) Correlation coefficient computed with the QSE-b Fi8() Cltiffiit td ith thQSEb [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 9
Figure 9. Figure 9: (a) Spectra of the eigenvalue obtained from the complexvalued 14 004)(b) Normalised magnitude of the complex modes Φ(n)for n = 1 Figure 9: (a) Spectra of the eigenvalue obtained from the complex-valued [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]

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Pith tools

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