{"id":"21e1d86a-b7a0-466f-844b-2faa188967ee","arxiv_id":"2501.08018","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Wall-pressure-velocity coherence collapses on a distance-from-wall scaling across a decade of Reynolds numbers, and quadratic stochastic estimation from wall pressure alone reaches about 0.6 correlation with a filtered velocity signal.","lead":"In a long pipe experiment at friction Reynolds numbers up to 47,000, researchers measured wall pressure and velocity together. They found that wall pressure can estimate large-scale turbulent velocity fluctuations, with a correlation of about 0.6, supporting wall-pressure-based flow control.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"QSE correlation is reported in-sample; without cross-validation the claimed rho≈0.6 may reflect overfitting, and the Taylor-hypothesis mapping may bias the stated coherence collapse.","rationale":"The reader's verdict was CONDITIONAL, with the weakest assumption being the Taylor-hypothesis conversion. I agree that a single U_c+ = 10 for all scales is a load-bearing assumption for the collapse claim, and the paper itself flags its limitations. However, in my reading the more decision-relevant concern for the practical claim (wall-pressure-based velocity estimation at application Re) is the in-sample nature of the reported QSE correlation. The coherence-collapse claim has independent support: the collapse in lambda_x/y is compared against spatial DNS (Baars et al. 2024) at Re_tau = 5200, and the qualitative trends are consistent with prior work; even if the absolute wavelength mapping carries a systematic error, the relative scaling across the seven CICLoPE cases is less sensitive to the choice of U_c. By contrast, the rho≈0.6 QSE result has no independent support, and the paper's own Fig. 8 shows lower values for the two lowest-Re cases, attributed to noise. Both concerns are addressable: a simple hold-out cross-validation would settle the overfitting question, and a scale-dependent convection-velocity sensitivity test would settle the mapping question. For these reasons I recommend CONDITIONAL, matching the reader, rather than ACCEPT or REJECT: the central claims are plausible and the underlying data are unique, but the estimation claim currently rests on an in-sample metric and the scaling claim on an acknowledged approximation.","tokens_in":26896,"tokens_out":1949,"duration_ms":17828,"concrete_test":"Recompute the QSE correlation rho using leave-one-run-out or temporal hold-out cross-validation: fit kernels (Eq. 6.3–6.4) on the first half of each 480 s record and evaluate rho[u_w, u_hat] on the second half, for all seven Reynolds numbers. If the held-out rho drops materially below 0.6 or becomes Reynolds-number dependent, the headline estimation claim must be weakened. Additionally, repeat the coherence-collapse analysis using a scale-dependent convection velocity (e.g., the resolvent-based U_c(lambda_x) of Luhar et al. 2014 or the measured two-point coherence phase velocity) instead of the fixed U_c+ = 10, and check whether the collapse of Figs. 5 and 7 and the lambda_x/y thresholds remain within the stated scatter.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The paper's central applied claim is that wall-pressure alone yields a Reynolds-number-independent estimator of wall-attached streamwise velocity with rho≈0.6. The QSE kernels (Eq. 6.3–6.4) are constructed from the very same pressure and velocity time series used to compute the reported Pearson correlation in Fig. 8. This is an in-sample accuracy metric: a multi-input spectral filter (linear term plus quadratic term, with complex kernels at every frequency) fitted to the estimation data will, by construction, achieve higher correlation than its true predictive skill on independent data. The paper states that the correlation is between the estimated and true time series, but does not state that the kernels were cross-validated or that a held-out portion of the 480 s records was used. Real-time control relevance requires out-of-sample performance, since in an application the kernels must be fixed before the control input arrives. Separately, the abscissa of the coherence collapse is lambda_x/y, obtained from frequency via lambda_x = U_c/f with fixed U_c+ = 10 (Eq. 4.1, Fig. 3 caption, Fig. 5 caption). The paper acknowledges that wall-pressure convection velocity is scale-dependent and Taylor's hypothesis is not strictly valid, yet a single U_c is applied for all scales, wall-normal positions, and Reynolds numbers. If the true scale-dependent convection velocity differs from U_c+ = 10, the apparent collapse in lambda_x/y, and the agreement with spatial DNS, could be partly an artifact of the mapping. The paper notes test cases 1 and 2 are contaminated by imperfect noise removal, and those points are excluded from the rho collapse claim, weakening the claim of universality across all seven cases.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":27259,"tokens_out":3605,"duration_ms":38016,"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":[{"comment":"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.","section":"§6, Eqs. (6.3)-(6.4) and Fig. 8"},{"comment":"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.","section":"§3-§5, Eq. (4.1) and Figs. 3, 5, 7"},{"comment":"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.","section":"Abstract, §3, §4 and Fig. 8"}],"minor_comments":[{"comment":"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.","section":"§6 and Fig. 8"},{"comment":"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.","section":"§6"},{"comment":"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.","section":"Appendix A"}],"recommendation":"major_revision","confidential_remarks":"This is a strong experimental contribution with a valuable dataset, and the coherence collapse for cases 3-7 is convincing. The main load-bearing issue is the in-sample nature of the QSE correlation, which is fixable by cross-validation; the fixed-Taylor-hypothesis mapping is a second, also addressable concern. I therefore recommend major revision rather than rejection. The paper is well within the scope of JFM."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does something genuinely useful: it takes the wall-pressure–velocity coherence scaling that Baars et al. (2024) established in DNS channel flow up to Re_tau ~ 5,200 and demonstrates it experimentally in a pipe at Re_tau up to 47,015. That is a real reach, and the experiment is careful. The lambda_x/y collapse for cases 3–7 is convincing, the comparison against external DNS is honest and non-circular, and the Hilbert-transform analysis supporting the modulation interpretation of the pressure-squared coherence is a nice touch. I also appreciate the transparency about the two lowest-Re cases being contaminated by facility noise; they do not hide it.\n\nThe soft spots are real but not fatal. The biggest one is the QSE correlation. Equations (6.3) and (6.4) build the estimator kernels from the same pressure and velocity records that are then used to compute the Pearson rho in Fig. 8. That is an in-sample accuracy metric. A multi-frequency linear plus quadratic filter fitted to the estimation data will almost always look better on that same data than it would on independent records. Since the abstract and conclusions sell rho ~ 0.6 as the basis for real-time control, the authors need to either cross-validate (e.g., split the 480 s records, or show that the kernels are stable across subsets) or at least label it as in-sample skill. This is a standard and addressable issue, not a reason to reject.\n\nThe Taylor-hypothesis mapping is a smaller concern. The authors acknowledge that the wall-pressure convection velocity is scale-dependent and that Taylor's hypothesis is not strictly valid, yet they use a single Uc+ = 10 everywhere. The collapse is partly enforced by this choice. Still, the agreement with spatial DNS data suggests the mapping is not wildly wrong for the scales of interest, and a sensitivity study (e.g., varying Uc+ within a plausible range) would settle it. I would not block publication on this, but it should be discussed more explicitly.\n\nMinor: there are no error bars or uncertainty estimates anywhere, and the universal claim rests on five clean cases rather than seven. Neither is disqualifying, but both weaken the quantitative claims.\n\nBottom line: the central scaling result holds up; the estimation result needs a caveat. This deserves a serious referee, and I would send it out, asking specifically for cross-validation of the QSE and sensitivity to the convection velocity. The paper is a solid contribution to the high-Re experimental literature and worth citing once those points are addressed.","headline":"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.","tokens_in":27817,"tokens_out":1881,"would_cite":true,"duration_ms":22127,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["wall-pressure fluctuations","coherence spectra","high Reynolds number pipe flow","quadratic stochastic estimation","amplitude modulation","wall-attached turbulence","Taylor's hypothesis","large-scale motions"],"falsifier":"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.","tokens_in":26722,"feed_emoji":"🌊","tokens_out":9051,"duration_ms":82349,"temperature":0.7,"pith_summary":"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.","feed_headline":"Wall pressure alone estimates off-wall velocity at high Re","feed_subtitle":"Coherence collapses on one distance scaling from Re_tau 4,800 to 47,000; quadratic estimator hits correlation 0.6.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the spatial DNS coherence reference data and the previous Reynolds-number scaling result that the current experiment extends to pipe flow at higher $Re_\\tau$.","marker":"Baars et al. 2024"},{"why":"Defines the self-similar wall-attached eddy framework and the $\\lambda_x/y = 14$ cutoff used to construct the reference velocity signal $u_W$.","marker":"Baars et al. 2017"},{"why":"Introduces the quadratic stochastic estimation approach and the idea that including $p_w^2$ improves velocity estimates from wall pressure.","marker":"Naguib et al. 2001"},{"why":"Provides the 7:1 structure aspect ratio and Reynolds-invariance evidence used to set measurement resolution and interpret coherence scaling.","marker":"Baidya et al. 2019"},{"why":"Supplies the wall-pressure intensity scaling and atmospheric-layer probability density used to validate the experimental wall-pressure statistics.","marker":"Klewicki et al. 2008"},{"why":"Provides boundary-layer wall-pressure spectral peak and pdf spread used to validate the current pressure spectra.","marker":"Tsuji et al. 2007"},{"why":"Earlier measurement of low but significant scale-dependent coherence between wall pressure and velocity, the baseline that the current coherence behavior is compared against.","marker":"Gibeau & Ghaemi 2021"},{"why":"Documents the limitation of Taylor's hypothesis for near-wall fluctuations, the caveat the paper itself flags for its frequency-to-wavelength conversion.","marker":"Dennis & Nickels 2008"}],"fun_headline_variants":["Wall-pressure squared lifts velocity coherence to 0.3","One scaling law ties pressure and velocity at high Re","Quadratic wall pressure estimates velocity: rho=0.6","Reynolds-invariant coherence from wall pressure alone","Pressure-only sensing hits rho 0.6 for velocity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Wall-pressure squared lifts velocity coherence to 0.3","One scaling law ties pressure and velocity at high Re","Quadratic wall pressure estimates velocity: rho=0.6","Reynolds-invariant coherence from wall pressure alone","Pressure-only sensing hits rho 0.6 for velocity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000275,"raw_usage":{"total_tokens":1710,"prompt_tokens":1082,"completion_tokens":628,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":698,"completion_tokens_details":{"reasoning_tokens":546}},"tokens_in":698,"tokens_out":628,"duration_ms":6258,"temperature":1.0,"reasoning_tokens":546,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T20:29:26.934899+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the spatial DNS coherence reference data and the previous Reynolds-number scaling result that the current experiment extends to pipe flow at higher $Re_\\tau$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the self-similar wall-attached eddy framework and the $\\lambda_x/y = 14$ cutoff used to construct the reference velocity signal $u_W$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the quadratic stochastic estimation approach and the idea that including $p_w^2$ improves velocity estimates from wall pressure."},{"cited_title":", Baars, W","cited_arxiv_id":null,"evidence_quote":"Provides the 7:1 structure aspect ratio and Reynolds-invariance evidence used to set measurement resolution and interpret coherence scaling."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the wall-pressure intensity scaling and atmospheric-layer probability density used to validate the experimental wall-pressure statistics."},{"cited_title":", Fransson, J","cited_arxiv_id":null,"evidence_quote":"Provides boundary-layer wall-pressure spectral peak and pdf spread used to validate the current pressure spectra."},{"cited_title":"& Ghaemi, S","cited_arxiv_id":null,"evidence_quote":"Earlier measurement of low but significant scale-dependent coherence between wall pressure and velocity, the baseline that the current coherence behavior is compared against."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Documents the limitation of Taylor's hypothesis for near-wall fluctuations, the caveat the paper itself flags for its frequency-to-wavelength conversion."}],"review_version":1}