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Identification of structures driving trailing-edge noise. Part II -- Numerical investigation

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2412.09562 v1 pith:7G2R64R4 submitted 2024-12-12 physics.flu-dyn

Zhenyang Yuan , Simon Demange , Kilian Oberleithner , André V. G. Cavalieri , Ardeshir Hanifi This is my paper · ORCID
classification physics.flu-dyn
keywords trailing-edgenoisespectralproperorthogonaldecompositionwavepacketsscatteringconditionlargeeddysimulationairfoilself-noisespanwisecoherencelow-rankmodel
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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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 / 5 minor

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.

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 (3)
  1. [§3.4 and §4.4.3] 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.
  2. [§3.6 and §4.3] 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.
  3. [§2.2 and §3.3] 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.
minor comments (5)
  1. [§3.3] 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'.
  2. [§5] 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'.
  3. [§3.4] 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.
  4. [References] The companion paper entry (Demange et al. 2024b) contains the placeholder 'arXiv:Number here' and needs the complete reference.
  5. [Figure 16b and §4.4.1] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the wavepacket-source identification is cross-validated by independent SPOD weightings and experimental comparisons.

full rationale

The derivation chain is not circular. H-SPOD modes are computed with a hydrodynamic weighting region near the trailing edge, and their acoustic content is then examined; the finding that the leading modes are spanwise-coherent, streamwise-travelling wavepackets that radiate according to k_z < k_0 is an empirical result of the LES data, not an input to the decomposition. A-SPOD is, by construction, optimal with respect to acoustic energy in the far field, so its efficient reconstruction of that same far field with two modes is partly a consequence of the chosen norm; however, the paper frames this as an optimal data-driven reconstruction rather than an out-of-sample prediction, and the eigenvalue dominance that yields the low-rank result is a data-dependent property, not an identity. Moreover, the A-SPOD mode shapes independently reproduce the hydrodynamic wavepackets found by H-SPOD, providing a cross-check across two different weighting regions. The scattering condition is imported from Nogueira et al. (2017), an externally derived and independently falsifiable theoretical result, and the experimental companion paper is used for validation of PSD, coherence, and mean flow rather than as the sole basis for the central claim. The acknowledged compromises in Mach number and spanwise domain (Sections 2.2 and 3.4) are validity limitations that could affect transferability to the experimental regime, but they do not make any derivation equivalent to its inputs. No fitted parameter is renamed as a prediction, and no load-bearing argument reduces to a self-citation. Therefore the circularity score is 0.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

No new physical entities are introduced; wavepackets are observed flow structures, not ad hoc constructs. The free parameters and axioms listed are the main inputs the central claim depends on that the reader did not pay for upstream.

free parameters (1)
  • Hydrodynamic convection velocity ratio c_ph/U_inf = 0.6
    Used to shift the surface and acoustic time series before computing coherence (section 3.6). The value is determined by correlating signals at different streamwise locations, in agreement with the companion experiment; it enters the phase-delay comparison that supports the convection-scattering interpretation.
assumptions (6)
  • standard math Compressible energy norm (Chu 1965; Mack 1984; Hanifi et al. 1996) defines the SPOD inner product (Eq. 4.3).
    The SPOD modes and eigenvalues depend on this norm; it is standard in linear flow stability and resolvent analysis.
  • domain assumption Spanwise periodicity and local homogeneity justify spanwise Fourier decomposition near the trailing edge (Appendix A).
    The LES uses periodic spanwise boundary conditions; the paper validates that eigenmodes of the experimental CSD resemble Fourier modes, but the assumption is load-bearing for the Fourier-wavenumber SPOD analysis.
  • domain assumption Scattering condition k_z < k_0 (Nogueira et al. 2017) governs which wavenumbers radiate.
    Used throughout to interpret spectra and mode shapes (sections 3.4 and 4.3); it is a theoretical result from a cited prior work, not derived here.
  • ad hoc to paper Mach number M=0.3 with M^5 scaling approximates the low-Mach experiments (M=0.088-0.133), and installation effects are negligible.
    The comparison in section 3.3 relies on this scaling; if compressibility or tunnel installation changes the source mechanism, the validation is compromised.
  • ad hoc to paper Time-shifting with phase velocity c_ph=0.6U_inf restores coherence lost by block averaging (Jaunet et al. 2017; Blanco et al. 2022).
    This processing step is needed to isolate the k_z=0 coherence in section 3.6; it assumes the convection-scattering propagation path.
  • domain assumption The implicit LES with the stated grid and polynomial order resolves acoustic waves up to St about 25-30 and the relevant turbulent scales.
    The paper states the mesh cut-off and excludes data for y/c > 10; the source-identification conclusions are limited to the resolved frequency range.

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Pith. "Pith review of Identification of structures driving trailing-edge noise. Part II -- Numerical investigation." pith.science (2026). https://pith.science/paper/7G2R64R4

@misc{pith2026241209562,
  author       = {Pith},
  title        = {Pith review of: Identification of structures driving trailing-edge noise. Part II -- Numerical investigation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7G2R64R4}},
  note         = {Machine review of arXiv:2412.09562}
}
read the original abstract

The aim of the present work is to investigate the mechanisms of broadband trailing-edge noise generation to improve prediction tools and control strategies. We focus on a NACA 0012 airfoil at 3 degrees angle of attack and chord Reynolds number Re = 200,000. A high-fidelity wall-resolved compressible implicit large eddy simulation (LES) is performed to collect data for our analysis. The simulation is designed in close alignment with the experiment described in detail in the companion paper (Demange et al. 2024b). Zig-zag geometrical tripping elements, added to generate a turbulent boundary layer, are meshed to closely follow the experimental setup. A large spanwise domain is used in the simulation to include propagative acoustic waves with low wavenumbers. An in-depth comparison with experiments is conducted showing good agreement in terms of mean flow statistics, acoustic and hydrodynamic spectra, and coherence lengths. Furthermore, a strong correlation is found between the radiated acoustics and spanwise-coherent structures. To investigate the correlation for higher wavenumbers, spectral proper orthogonal decomposition (SPOD) is applied to the spanwise Fourier-transformed LES dataset. The analysis of all SPOD modes for the leading spanwise wavenumbers reveals streamwise-travelling wavepackets as the source of the radiated acoustics. This finding, confirming observations from experiments in the companion paper, leads to a new understanding of the turbulent structures driving the trailing-edge noise. By performing extended SPOD based on the acoustic region, we confirm the low rank nature of the acoustics, and a reduced-order model based on acoustic extended SPOD is proposed for the far-field acoustic reconstruction.

Figures

Figures reproduced from arXiv: 2412.09562 by the authors.

Figure 1
Figure 1. High order computational mesh elements near the airfoil, top: the wall resolved [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Instantaneous iso-surfaces of the Q-criterion colored by streamwise [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Skin friction coefficient (𝐶𝑓 ) map of time averaged field. Colour map is saturated such that blue and red show negative and positive 𝐶𝑓 , respectively. 00 0 0 0 0 0  000 00 00 00 00 00 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (20 more)
Figure 4
Figure 4. Figure 4: Mean velocity profiles obtained at the streamwise/spanwise locations marked as [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: Mean velocity profile at the different streamwise stations: [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: Mean and RMS velocity profiles measure at [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: Power spectra density (PSD) calculated at: (a) the acoustic line array at [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
Figure 8
Figure 8. Figure 8: Frequency-wavenumber spectrum based on spanwise Fourier transform of CSD [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: Coherence analysis for the surface pressure fluctuation at [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]
Figure 10
Figure 10. Figure 10: Theoretical phase delays sketches for two scenarios: (a) a hydrodynamic wave [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]
Figure 11
Figure 11. Figure 11: Coherence between surface pressure fluctuation at [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]
Figure 12
Figure 12. Figure 12: The choice of the subdomain for SPOD weighting matrix [PITH_FULL_IMAGE:figures/full_fig_p020_12.png]
Figure 13
Figure 13. Figure 13: SPOD eigenvalues corresponding to compressible energy for the first five [PITH_FULL_IMAGE:figures/full_fig_p021_13.png]
Figure 14
Figure 14. Figure 14: The leading H-SPOD mode shape ˜𝑝 for the first three spanwise wavenumbers (𝑛𝑧 = 0, 1, 2) at the frequency 𝐻𝑒 = 9.82 (a, c, e) and 𝐻𝑒 = 16.69 (b, d, f), respectively. The ˜𝑝 is premultiplied by the square root of its eigenvalue to give the real amplitude. 25.52, the se…
Figure 15
Figure 15. Figure 15: Pressure iso-surface of three dimensional SPOD mode shapes ˜𝑝 [PITH_FULL_IMAGE:figures/full_fig_p024_15.png]
Figure 16
Figure 16. Figure 16: (a) Region of the acoustic field (dashed circle arc) used to integrate the power [PITH_FULL_IMAGE:figures/full_fig_p024_16.png]
Figure 17
Figure 17. Figure 17: Spectrum of the first four spanwise wavenumbers from the A-SPOD analyses. [PITH_FULL_IMAGE:figures/full_fig_p025_17.png]
Figure 18
Figure 18. Figure 18: The leading A-SPOD mode shapes ˜𝑝 of wavenumbers 𝑛𝑧 = 0, 1 for frequencies 𝐻𝑒 = 9.82 and 16.69. 4.4.2. A-SPOD mode shapes The pressure component of A-SPOD modes are presented in figure 18. The frequencies and wavenumbers presented here are the same as those shown in f…
Figure 19
Figure 19. Figure 19: (a) Ratio of acoustic energy between accumulated SPOD modes ( [PITH_FULL_IMAGE:figures/full_fig_p028_19.png]
Figure 20
Figure 20. Figure 20: Eigenvalue analysis on the CSD matrices constructed from the acoustic line [PITH_FULL_IMAGE:figures/full_fig_p031_20.png]
Figure 21
Figure 21. Figure 21: Correlation coefficient 𝜎1,𝑘 for the leading 10 SPOD modes with 𝐻𝑒 ⩽ 30. Here shows the dataset 𝑖 = 1, 2 with 75% overlapping with the original LES dataset. within the frequency range 𝐻𝑒 < 30 for both hydrodynamic SPOD and acoustic SPOD. Accordingly, the leading SPODs…
Figure 22
Figure 22. Figure 22: The leading SPOD mode shape for the first three leading spanwise wavenumbers (𝑛𝑧 = 0, 1, 2, 3) at the frequency 𝐻𝑒 = 9.82. (a, c, e, g) and (b, d, f, h) show mode shapes corresponding to ˜𝑢, and ˜𝑣, respectively [PITH_FULL_IMAGE:figures/full_fig_p033_22.png]
Figure 23
Figure 23. Figure 23: The leading SPOD mode shape for the first two leading spanwise wavenumbers (𝑛𝑧 = 0, 1) at the frequency 𝐻𝑒 = 11.78. Both acoustic SPOD and hydrodynamic SPOD are presented, indicated by ASPOD and HSPOD, respectively. Chu, Boa-Teh 1965 On the energy transfer to small di…

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Pith tools

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