{"id":"5e65ec53-dc02-4c3a-b59f-48fef7c03c62","arxiv_id":"2412.09562","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Spanwise-coherent wavepackets in the turbulent boundary layer are shown by LES and SPOD to drive broadband trailing-edge noise, and a two-mode acoustic SPOD reconstruction reproduces the far-field sound.","lead":"This paper uses a large computer simulation of airflow over a NACA 0012 airfoil to identify the turbulent structures that produce broadband trailing-edge noise, finding spanwise-coherent, streamwise-travelling wavepackets as the source. It also shows the far-field sound is low-rank and can be reconstructed with just two data-derived modes, which can simplify noise prediction and control.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The M=0.3 simulation may not certify the wavepacket mechanism at the experimental M=0.088–0.133; the radiating wavenumber set and the low-rank acoustic model depend strongly on the Mach-scaled spanwise domain.","rationale":"The paper is a careful numerical study with credible validation: mean velocity profiles, surface-pressure spectra, coherence lengths, and far-field PSD all match the companion experiment well, and the SPOD convergence analysis in Appendix B supports the statistical reliability of the leading modes. The central claim, however, is that the identified wavepackets are the structures driving the noise at the experimental conditions, and that the acoustic field is low-rank. This claim is load-bearing because the simulation is performed at M=0.3 with a span of 0.4375c, while the experiments are at M=0.088–0.133 with a span of 4c. The M^5 scaling collapses the total sound power but does not certify the wavepacket source structure, and the scattering condition makes the set of radiating spanwise wavenumbers strongly dependent on both Mach number and spanwise domain size. Thus the observed dominance of the k_z=0 wavepacket and the two-mode A-SPOD reconstruction could be a consequence of the narrow, wavenumber-selective numerical domain rather than a universal property of low-Mach airfoil noise. This is a genuine soft spot, but it is a known and acknowledged compromise, and the paper's correlational evidence is strong enough to warrant a conditional acceptance pending a direct check at the experimental Mach number. Changing the verdict to accept would require the proposed low-Mach test; rejecting the paper would ignore the substantial internal consistency and prior literature supporting wavepacket mechanisms. Therefore the reader's CONDITIONAL verdict is appropriate, and the recommended action is to keep it unchanged.","tokens_in":27041,"tokens_out":15654,"duration_ms":151198,"concrete_test":"Run a companion LES at Mach 0.1 with the same Reynolds number and tripping geometry but with a spanwise domain enlarged to include the experimental radiating wavenumber range (e.g., L_z=4c), and repeat the spanwise-Fourier H-SPOD/A-SPOD analysis. If the leading modes no longer show the same streamwise wavepacket envelope and phase speed, or if the far-field acoustic spectrum requires more than two A-SPOD modes per wavenumber to match the M^5-scaled experimental data, then the identified mechanism is a compressibility/domain-size artifact rather than the low-Mach trailing-edge noise source.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that spanwise-coherent, streamwise-travelling wavepackets drive trailing-edge noise is established in a simulation at M=0.3 with L_z=0.4375c, while the experiments are at M=0.088–0.133 with L_z=4c. The far-field comparison uses the M^5 scaling (section 3.3), which validates the total sound power but does not validate the wavepacket source structure itself. Under the scattering condition k_z < k_0 = ω/a, the set of radiating spanwise wavenumbers is strongly Mach- and span-dependent: for the LES, n_z=1 radiates only above He≈47.9 (St≈7.6), whereas in the experiment (L_z=4c, M=0.1) n_z up to about 4 radiate at St=10. Consequently, the paper's observations that the acoustic field is low-rank (section 4.4.3) and dominated by the k_z=0 wavepacket (section 4.3.2) are conditioned on a numerical domain that admits only a few radiating wavenumbers. If at the experimental Mach number the acoustic field is a superposition of a wider set of wavenumbers, the two-mode A-SPOD reduced-order model and the dominance of the k_z=0 wavepacket may not transfer. The paper acknowledges the Mach compromise in section 2.2 ('some compromises are made...') and argues that the good PSD agreement justifies it, but no independent check at the experimental Mach number is provided. Additionally, the causal claim that wavepackets 'drive' the noise is inferred from SPOD correlation; the time-delay analysis that supports causality is performed only for k_z=0 up to St≈10 (section 3.6), leaving the higher-wavenumber and higher-frequency evidence correlational.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper reports a wall-resolved compressible implicit LES of a NACA 0012 airfoil at chord Reynolds number 200,000, Mach 0.3 and 3 degrees angle of attack, with zig-zag trip strips and a spanwise length of 0.4375c, designed to match companion experiments at Mach 0.088-0.133, Re=2-3e5 and span 4c. The simulation is validated against experimental mean-flow profiles, surface-pressure and far-field spectra, coherence lengths, and the scattering condition. Spanwise Fourier decomposition and spectral proper orthogonal decomposition (SPOD), including hydrodynamic- and acoustic-weighted extended SPOD (H-SPOD and A-SPOD), are used to identify coherent structures. The central claims are that (i) the dominant hydrodynamic structures are spanwise-coherent, streamwise-travelling wavepackets concentrated near the trailing edge; (ii) their radiation obeys the scattering condition k_z < k_0; and (iii) A-SPOD yields a low-rank reconstruction of the far-field acoustics, with two modes matching the LES spectra within about 1 dB up to He≈25.","tokens_in":27478,"tokens_out":8111,"duration_ms":72261,"significance":"If correct, the paper provides a strong confirmation of the wavepacket mechanism for broadband trailing-edge noise and a promising basis for reduced-order modelling. The manuscript is careful in its validation: mean flow, surface-pressure and far-field spectra, coherence lengths, and the scattering condition are all compared with experiment, and Appendix B confirms SPOD convergence for the leading modes (σ≥0.98 for He<25). The use of the spanwise Fourier basis is justified via CSD eigenvalue analysis in Appendix A, and the A-SPOD energy ranking is transparent. The main caveat is that the quantitative low-rank result is demonstrated for a specific Mach-number/span combination and may not transfer directly to the experimental configuration; this limits the generality of the title-level claim that wavepackets drive the noise, although the core identification is plausible.","major_comments":[{"comment":"The claim that the acoustic field is low rank and that two A-SPOD modes suffice is conditioned on the LES spanwise domain and Mach number. For the LES parameters (M=0.3, L_z=0.4375c), Eq. (3.2) gives cut-on Helmholtz numbers He_nz = 2π M St n_z = 14.36 n_z, so that at St=10 only n_z=0 and 1 radiate. At the experimental conditions (M=0.1, L_z=4c), the same Strouhal number admits n_z=0,...,4 as radiating wavenumbers. The statement in §3.3 that the larger experimental span has 'a small impact on the spectrum' is based on the total PSD at one far-field location, not on a mode-resolved energy budget. I ask the authors to either compute the cumulative acoustic energy of the radiating wavenumbers from the experimental frequency-wavenumber CSD (Fig. 8b) or provide an error bound for the two-mode A-SPOD model at the experimental span, and to qualify the 'low rank' conclusion as domain-specific if no such evidence can be provided.","section":"§3.4 and §4.4.3"},{"comment":"The causal language 'structures driving trailing-edge noise' is only directly supported for k_z=0. The time-delay analysis in §3.6 (Fig. 10c) is performed exclusively for the spanwise-averaged (k_z=0) signals and is reported to hold up to St≈10. For n_z>0, the relationship between the SPOD-identified wavepackets and the radiated sound is inferred from spatial correlation and the scattering condition, not from a time-resolved causal test. The paper should either extend the time-delay/coherence analysis to n_z>0 (the simulation data are available to do this) or explicitly state that for non-zero spanwise wavenumbers the identification is correlational, thereby tempering the conclusion that wavepackets 'drive' the noise across all wavenumbers.","section":"§3.6 and §4.3"},{"comment":"The Mach-number compromise (LES at M=0.3 vs experiments at M=0.088–0.133) is acknowledged but not quantitatively assessed. The M^5 scaling in §3.3 validates the total radiated sound power level, but the wavepacket source structure, the radiation directivity, and the set of cut-on spanwise wavenumbers are Mach-dependent through Eq. (3.2). The paper does not provide an argument or a calculation showing that the dominant wavepacket mechanism and the two-mode low-rank property persist at the experimental Mach number. A resolvent analysis or a companion low-Mach simulation (even at reduced span) would directly address this; at minimum, the authors should discuss the expected Mach dependence of the scattering and of the wavepacket convection speed (here taken as c_ph=0.6U_inf) and why the identified mechanism is expected to be invariant.","section":"§2.2 and §3.3"}],"minor_comments":[{"comment":"In the paragraph after Fig. 7, 'the acoustic radiation on the suction side is stronger than on the suction side' should presumably read 'stronger on the suction side than on the pressure side'.","section":"§3.3"},{"comment":"The concluding sentence, 'The current work stats the ground for future resolvent analysis...' appears to contain a typo; it should likely read 'sets the ground'.","section":"§5"},{"comment":"Equation (3.2) uses the Helmholtz number He without a definition; please define He = 2π St M at first use, and ensure the notation is consistent with the Strouhal number St used elsewhere in the paper.","section":"§3.4"},{"comment":"The companion paper entry (Demange et al. 2024b) contains the placeholder 'arXiv:Number here' and needs the complete reference.","section":"References"},{"comment":"In Figure 16b, the vertical dashed lines are labeled as the scattering condition of Eq. (3.2), but the text notes that the observed cut-offs are somewhat lower; please clarify in the caption whether the lines are the theoretical values, and state the interpretation of the offset. In §4.4.1, the phrase 'the ratio ... exceeds 80%' should specify that this is the energy share of the leading A-SPOD mode.","section":"Figure 16b and §4.4.1"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern about Mach/span dependence is, in my reading, the key issue. The paper's validation is strong and the wavepacket identification is plausible, but the low-rank acoustic conclusion is broader than the evidence presented. I recommend major revision, asking for a mode-resolved experimental energy budget or an equivalent quantitative qualification of the transferability. The paper fits the journal's scope and is likely to be a valuable contribution once this point is addressed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core result — spanwise-coherent wavepackets at n_z=0,1,2,3 in the turbulent boundary layer correlate with trailing-edge noise and radiate only when k_z<k_0 — is well supported by the LES, and the A-SPOD reduced-order model is a genuinely useful step beyond the earlier n_z=0 studies. What is actually new: the extension to higher spanwise wavenumbers, the explicit design of the spanwise domain to satisfy the scattering condition, and the acoustic-weighted ESPOD that reconstructs the far field with two modes. The validation is thorough: mean flow, surface and farfield spectra, coherence lengths, and the frequency-wavenumber decomposition all line up with the companion experiment, and Appendix B checks SPOD convergence with sigma >= 0.98 for leading modes below He=25. That is real evidence.\n\nThe soft spots are proportionate. The biggest one is the Mach compromise: the LES runs at M=0.3 with L_z=0.4375c, which admits only n_z=0 and n_z=1 as radiating wavenumbers over the He<25 range shown. So the striking low-rankness of the acoustic field in section 4.4.3 is partly a property of the numerical domain, not a proven property of the low-Mach experiment. The paper is honest about the compromise in section 2.2 and argues that the M^5-scaled PSD collapse justifies it, but that validates total sound power, not the wavepacket source structure. The stress-test note is right to flag this as load-bearing. It is addressable with a discussion of how many cut-on wavenumbers the experiment admits at the same He and why those extra wavenumbers do not appear to matter — the paper gestures at this in section 3.3 but does not quantify it.\n\nSecond, the causal language ('wavepackets as the driver', 'structures driving') is stronger than the evidence. The time-delay analysis in section 3.6, which supports convection-scattering causality, is performed only for k_z=0 and up to St~10; for higher wavenumbers and higher frequencies the evidence is SPOD correlation. That is a wording issue, not a flaw in the SPOD itself. Third, the companion paper reference still says 'arXiv:Number here', which needs fixing.\n\nOverall, the central argument holds up: the wavepackets are real, they obey the scattering condition, and the A-SPOD reconstruction is a useful reduced-order model — within the conditions simulated. A serious referee should push for tempering the conclusions and adding a check on the Mach dependence, but this is solid, well-executed work that deserves to go through peer review. I would bring it to a reading group and expect a good discussion about domain effects on low-rank acoustic models.","headline":"Careful, well-validated LES that extends wavepacket identification to nonzero spanwise wavenumbers; the low-rank acoustic claim is real but partly conditioned on the Mach- and span-limited domain, and the causal wording outruns the correlational evidence.","tokens_in":27997,"tokens_out":3449,"would_cite":true,"duration_ms":34411,"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":"This paper shows that spanwise-coherent, streamwise-travelling wavepackets in the turbulent boundary layer drive broadband trailing-edge noise, and that two acoustic-weighted SPOD modes reproduce the far-field spectrum.","keywords":["trailing-edge noise","spectral proper orthogonal decomposition","wavepackets","scattering condition","large eddy simulation","airfoil self-noise","spanwise coherence","low-rank model"],"falsifier":"Perform an identical LES and A-SPOD analysis at the experimental freestream Mach number near 0.1 while keeping the same tripping and spanwise domain; if two A-SPOD modes then fail to reconstruct the far field within about 1 dB, or if modes with $k_z \\ge k_0$ radiate appreciable sound, the wavepacket and scattering claim does not generalize.","tokens_in":26841,"feed_emoji":"🔊","tokens_out":6820,"duration_ms":64947,"temperature":0.7,"pith_summary":"This numerical study claims that broadband trailing-edge noise from an airfoil is generated by a small set of streamwise-travelling, spanwise-coherent wavepackets in the turbulent boundary layer, not by the many small incoherent eddies assumed in classical coherence-length models. Using a wall-resolved compressible large-eddy simulation of a tripped NACA 0012 airfoil at Reynolds number 200,000, the authors validate the simulation against the companion experiment and then decompose the flow by spanwise wavenumber and frequency. They show that only modes with spanwise wavenumber $k_z$ below the acoustic wavenumber $k_0$ radiate sound, and that an acoustic-weighted spectral proper orthogonal decomposition captures the far-field noise with two modes. If correct, this gives a physics-based, low-rank route to trailing-edge noise prediction and control.","feed_headline":"Two wavepacket modes capture the full airfoil noise field","feed_subtitle":"Spanwise-wide turbulent structures, not small eddies, scatter sound at a trailing edge—and just two modes model it.","key_machinery":"The central object is the extended spectral proper orthogonal decomposition (ESPOD), a data-driven modal decomposition in which the compressible-energy inner product is evaluated in one subdomain, here the turbulent boundary layer near the trailing edge (H-SPOD) or the acoustic far field (A-SPOD), while the resulting modes are projected over the whole flow. The analysis is carried out on spanwise Fourier-transformed snapshots, so each mode carries a definite spanwise wavenumber $k_z$ and frequency. The scattering condition $k_z < k_0$, with $k_0 = \\omega/a_0$, is the criterion that separates radiating from evanescent modes; it is used both to design the spanwise domain width and to interpret the mode shapes.","core_discovery":"The paper's central claim is that the structures driving broadband trailing-edge noise are spanwise-coherent, streamwise-travelling wavepackets concentrated near the trailing edge, with acoustic radiation governed by the scattering condition $k_z < k_0$. The authors establish this by applying spectral proper orthogonal decomposition to spanwise Fourier-transformed LES data: for each low spanwise wavenumber the leading mode is a wavepacket that extends from the trip to the wake, and its far-field signature is propagative only when the scattering condition is met. Weighting the decomposition in the acoustic region (A-SPOD) shows that the acoustic field is strongly low-rank: the leading mode carries most of the sound power, and two modes reproduce the LES far-field spectrum within about 1 dB. The same modes show that only a small fraction of the near-wall hydrodynamic energy is actually radiating.","pith_inferences":["If this low-rank picture generalizes to full-scale Reynolds and Mach numbers, trailing-edge noise prediction could shift from measuring surface-pressure coherence lengths to tracking a handful of wavepacket amplitudes, lowering the cost of both experiments and numerical models.","The same A-SPOD basis could be combined with resolvent analysis to identify which frequencies and spanwise wavenumbers most need control, giving a physical target for serrations or other trailing-edge treatments instead of an empirical one.","A direct test of the scattering condition at a higher Mach number, for example near $M=0.5$, would reveal whether the $k_z < k_0$ cutoff remains sharp or shifts with mean-flow convection, a detail the paper already hints at when it notes that the observed cutoffs sit slightly below the theoretical values.","The observation that only a small fraction of hydrodynamic energy radiates suggests why coherence-length-based models are hard to calibrate: what matters is the projection of the turbulent field onto a few radiating wavenumbers, not the total turbulence amplitude."],"forward_implications":["Only spanwise wavenumbers satisfying $k_z < k_0$ radiate: for a fixed spanwise domain each wavenumber turns on at a distinct cutoff frequency, and the integrated far-field spectrum is assembled wavenumber by wavenumber according to that rule.","The far-field acoustic field is low-rank: the leading A-SPOD mode carries up to about 80% of the acoustic energy in the broadband range $3 \\le He \\le 25$, and two modes reconstruct the LES far-field spectrum within about 1 dB.","Hydrodynamic SPOD is inefficient for acoustics, needing roughly 24 modes for 1 dB accuracy, because most turbulent kinetic energy near the trailing edge does not radiate; acoustic-weighted SPOD is the better basis for reduced-order noise models.","Wavepacket structures exist for nonzero spanwise wavenumbers as well, appearing as oblique three-dimensional wavepackets whose radiation obeys the same scattering condition, extending earlier analyses that were limited to $k_z = 0$.","The coherence between span-averaged surface pressure and far-field acoustics reaches about 0.8 in the broadband noise range, showing that the radiating part of the flow is the spanwise-coherent component rather than individually incoherent eddies."],"supporting_citations":[{"why":"Supplies the scattering condition $k_z < k_0$ that the paper validates and uses to choose the spanwise domain width.","marker":"Nogueira et al. (2017)"},{"why":"Companion experimental study that provides the validation data and the spanwise-coherence observations the LES reproduces.","marker":"Demange et al. (2024b)"},{"why":"Prior numerical identification of spanwise-coherent structures scattering at the trailing edge, which this work extends to higher spanwise wavenumbers with experimental validation.","marker":"Sano et al. (2019)"},{"why":"Earlier demonstration that spanwise-coherent hydrodynamic waves around airfoils correlate with radiating acoustics, providing the baseline for the wavenumber-resolved analysis.","marker":"Abreu et al. (2021)"},{"why":"Introduced extended proper orthogonal decomposition, the method behind the ESPOD used to correlate hydrodynamic and acoustic regions.","marker":"Borée (2003)"},{"why":"Provides the classic trailing-edge scattering theory and the $M^5$ acoustic scaling used to compare simulation and experiment.","marker":"Williams & Hall (1970)"},{"why":"Supplies the spectral estimation procedure used throughout the paper for CSD, PSD, and SPOD computations.","marker":"Welch (1967)"},{"why":"Defines the compressible energy norm used as the SPOD weighting operator.","marker":"Chu (1965)"}],"fun_headline_variants":["Two wavepacket modes predict airfoil noise within 1 dB","Spanwise wavepackets, not small eddies, drive trailing-edge noise","Just two modes capture full trailing-edge noise field","Wavepackets scatter sound at airfoil trailing edge","Low-rank acoustics: two modes model trailing-edge noise"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The numerical validation transfers from Mach 0.3 to the experimental Mach 0.088–0.133 through $M^5$ acoustic scaling and by ignoring the open-jet and side-plate installation; if compressibility at Mach 0.3 or the installation changes the noise-generation mechanism, the identified wavepackets are not certified as the true low-Mach sources.","fun_headline_variants_meta":{"raw":{"variants":["Two wavepacket modes predict airfoil noise within 1 dB","Spanwise wavepackets, not small eddies, drive trailing-edge noise","Just two modes capture full trailing-edge noise field","Wavepackets scatter sound at airfoil trailing edge","Low-rank acoustics: two modes model trailing-edge noise"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000208,"raw_usage":{"total_tokens":1445,"prompt_tokens":1030,"completion_tokens":415,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":646,"completion_tokens_details":{"reasoning_tokens":329}},"tokens_in":646,"tokens_out":415,"duration_ms":4346,"temperature":1.0,"reasoning_tokens":329,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T16:55:15.552939+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Perform an identical LES and A-SPOD analysis at the experimental freestream Mach number near 0.1 while keeping the same tripping and spanwise domain; if two A-SPOD modes then fail to reconstruct the far field within about 1 dB, or if modes with $k_z \\ge k_0$ radiate appreciable sound, the wavepacket and scattering claim does not generalize.","supporting_citations":[{"cited_title":", Cavalieri, André V.G","cited_arxiv_id":null,"evidence_quote":"Supplies the scattering condition $k_z < k_0$ that the paper validates and uses to choose the spanwise domain width."},{"cited_title":", Cavalieri, André V.G","cited_arxiv_id":null,"evidence_quote":"Prior numerical identification of spanwise-coherent structures scattering at the trailing edge, which this work extends to higher spanwise wavenumbers with experimental validation."},{"cited_title":", Tanarro, Alvaro , Cavalieri, André V.G","cited_arxiv_id":null,"evidence_quote":"Earlier demonstration that spanwise-coherent hydrodynamic waves around airfoils correlate with radiating acoustics, providing the baseline for the wavenumber-resolved analysis."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the classic trailing-edge scattering theory and the $M^5$ acoustic scaling used to compare simulation and experiment."},{"cited_title":"1967 The use of fast fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms","cited_arxiv_id":null,"evidence_quote":"Supplies the spectral estimation procedure used throughout the paper for CSD, PSD, and SPOD computations."},{"cited_title":"Acta mechanica 1 (3), 215--234","cited_arxiv_id":null,"evidence_quote":"Defines the compressible energy norm used as the SPOD weighting operator."}],"review_version":1}