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REVIEW 3 major objections 4 minor 92 references

Supersonic jet impingement on concave surfaces

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A concave wall facing a supersonic jet can amplify the primary screech tone by up to 23 dB compared with a flat wall, because the curved surface focuses the returning feedback wave back toward the nozzle lip.

desk verdict The helical-mode locking to the H1 guided-jet-mode lower limit is the solid, citable result; the 23 dB curvature-amplification claim is overstated because standoff is not controlled — Flat[L2.1] already gains 20.4 dB purely from distance. read the letter →

arxiv 2608.01542 v1 pith:E5BTHYON submitted 2026-08-02 physics.flu-dyn physics.app-phphysics.comp-ph

Hemanth Chandravamsi , Dhanush Vittal Shenoy , Steven H. Frankel This is my paper · ORCID
classification physics.flu-dynphysics.app-phphysics.comp-ph PACS 47.40.Ki43.28.Ra
keywords aeroacousticsscreechtonessupersonicimpingingjetsguidedjetmodeswallcurvatureeffectslarge-eddysimulationsurfaceloading
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper argues that a concave wall facing an underexpanded supersonic jet is not a passive receiver: the indentation acts as an acoustic mirror and resonator that focuses the feedback wave back toward the nozzle, amplifying the primary screech tone by up to 23 dB relative to a flat wall at comparable standoff. It further claims helical and axisymmetric tones are chosen by different physics — helical tones lock to the lower-frequency edge of the H1 guided jet mode set by the jet's own shear-layer profile, while axisymmetric tones follow Powell's loop-length criterion. If both claims hold, wall curvature becomes a practical control for tone amplitude and, through a mode-selection rule, for the direction of the unsteady surface load: axial pounding for axisymmetric screech, a rotating in-plane bending moment for helical screech. The stakes are concrete because surface shape appears in vertical take-off and landing over contoured ground, jet-blast deflectors, and cold-spray deposition, where tonal loading drives fatigue.

What carries the argument

Four coupled pieces. (1) The Gaussian indentation $z/D=2.08+0.5\exp(-(r/D)^2/2\sigma^2)$ reduces wall shape to one parameter $\sigma$ interpolating between the two flat-wall limits. (2) The Powell-Tam source-transfer budget $\Delta\mathrm{SPL}_{\mathrm{lip}}=\Delta\mathrm{SPL}^{\mathrm{PT}}_{\mathrm{src}}+\Delta T_u$ (Eq. 3.2) partitions curvature-induced amplification: the source term scales with rms Mach-disk displacement times tone Strouhal (Eq. 3.3), and the transfer term $\Delta T_u$ is the residual that measures the efficiency of the upstream return to the lip. (3) The vortex-sheet guided-jet-mode dispersion relation (Tam & Hu 1989; Tam & Ahuja 1990), whose H1 lower limit — where the m

What would settle it

Keep the jet at Mj=1.56 and sweep the flat-wall standoff L/D between about 2.0 and 3.0 in the same computational setup: the helical-tone claim predicts the tone stays pinned near St≈0.325 (the H1 lower limit is set by the jet profile alone), while the axisymmetric tone steps along the Powell ladder; if the helical tone instead moves with L/D, the dispersive-selection claim fails. Separately, computing q_d directly from the shear-layer wavepacket growth for each of the six geometries would settle the source-transfer split: a variation of more than about 1 dB across cases would misattribute the

Watch

Extended reading notes

Core claim

A concave impingement wall actively shapes the screech resonance. Narrowing the indentation spread $\sigma$ from 4.0 to 0.4 raises the primary-tone level by about 23 dB above the Flat[L2.6] reference, to $\approx$176 dB/St. A Powell-Tam budget splits this gain between a stronger Mach-disk tonal source (62% at $\sigma$0.4) and a more efficient upstream return of the feedback wave to the nozzle (38%); direct measurement shows the returning wave strengthens by up to 27 dB, carried mainly inside the jet column. All four helical cases pick tones within St=0.336–0.357, just 3–9% above the H1 guided-jet-mode lower limit, with matching J1 eigenfunctions — evidence a guided jet mode closes the helica

Load-bearing premise

The load-bearing premise is that the downstream shear-layer gain q_d is the same for all six geometries (Section 3.5), so that every curvature effect can be attributed to the Mach-disk source and the upstream return — a premise the paper's own Limitations paragraph hedges, noting that the transfer term is a residual rather than a measured ratio and that the Flat[L2.6] reference mixes standoff with shape.

Editorial extensions

If this is right

  • If the wall-curvature result is correct, a narrow concave indentation is an effective amplifier rather than a suppressor of impingement tones: up to 23 dB of extra tonal level at L/D≈2.6, strongest when the indentation stays within about two jet diameters of the axis (r99/D ≲ 2.4).
  • Helical screech frequency is governed by the jet's shear-layer profile, not the wall: all four helical cases sit within St=0.336–0.357, within 3–9% of the H1 lower limit, so wall geometry can change helical tone amplitude but not its frequency.
  • The screech mode is the fatigue-load switch: axisymmetric screech concentrates tonal energy in the axial force (up to 43% of the mean net load in rms at σ0.4), while helical screech hides the tone from the axial force and puts it in a precessing in-plane bending moment, largest at intermediate indentation spread.
  • The concave wall focuses the returning wave through both the jet column and the ambient, and the channel that benefits depends on how far the indentation extends past the column — 78% of the σ0.4 indentation area sits within r/D=0.7, and the column channel gains 15.8 dB more than the ambient there.

Reading between the lines

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

  • My inference: because the H1 lower limit depends only on the jet's shear-layer profile and operating condition, the paper's claim implies the helical tone frequency should stay nearly fixed as the nozzle-to-wall distance is swept at constant Mj — a test the six-case matrix does not isolate, since standoff and mode type co-vary there.
  • My inference: the Hartmann-whistle analogy the authors invoke suggests indentation depth (fixed at 0.5D here) is the natural second control; varying depth at fixed σ should move the tone through cavity-mode staging, possibly recovering the screech suppression reported for convex cylindrical walls in earlier experiments.
  • My inference: the axisymmetric tone's failure to match any A_n branch may reflect the underexpanded shock structure rather than a true non-GJM mechanism; a shock-free (ideally expanded) jet at the same loop length should land on a guided branch if the guided-mode picture extends.
  • My inference: for applications (VTOL over contoured terrain, jet-blast deflectors, cold-spray deposition), the practical reading is that terrain shape can be treated as a control input: a narrow pocket both amplifies the tone and decides whether fatigue loading pulses along the axis or rotates in-plane.
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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 / 4 minor

Summary. The paper uses compressible large-eddy simulations of an underexpanded round supersonic jet (Mj = 1.56, Re = 6e4) impinging on flat and Gaussian-concave walls, together with a vortex-sheet guided-jet-mode model and Powell's feedback-loop analysis. Six geometries are studied: two flat plates at L/D = 2.08 and 2.58, and four Gaussian indentations of fixed depth with spread sigma = 0.4, 0.8, 1.6, 4.0. The central claims are that (i) narrowing the indentation amplifies the primary screech tone by up to 23 dB relative to the flat wall at L/D = 2.6, with the amplification attributed to a Powell-Tam source-transfer budget; (ii) helical screech tones lock to the lower-limit frequency of the H1 guided jet mode, independent of wall geometry, while axisymmetric tones follow Powell's loop-length criterion; and (iii) the azimuthal symmetry of the screech mode selects whether the unsteady wall loading appears in the axial force or in the transverse force and bending moment.

Significance. If the claims hold, the paper provides a useful demonstration that wall curvature can control both screech amplitude and the direction of unsteady surface loading, and it offers a clean separation between helical and axisymmetric frequency-selection mechanisms. The helical result is especially strong: the H1 lower-limit prediction is computed from an external vortex-sheet benchmark with no fitted constants, and it is supported by both the measured frequencies and the radial eigenfunctions across all four helical cases. The loading-selection rule in Eq. (4.1) is elegant and directly verified by the spectral content of Fz and the bending moment. However, the headline 23 dB curvature-amplification claim is quantitatively confounded by standoff, and the axisymmetric Powell claim is based on only two cases, one of which is poorly predicted. The paper deserves publication only after these load-bearing issues are addressed.

major comments (3)
  1. [§3.2, Table 2, Eq. (2.1), Conclusions (i)] The headline claim of 'up to 23 dB' amplification from concave indentation is not a clean measure of curvature because the concave family also changes the nozzle-to-wall standoff. Concave[σ0.4] has ℓlip/D = 2.31 and Lavg/D = 2.43, versus L/D = 2.58 for Flat[L2.6]. The paper's own Flat[L2.1] case shows the size of the standoff effect: at L/D = 2.08 its SPL1 is 173.1 dB/St, only 2.6 dB below Concave[σ0.4] (175.7 dB/St), while Flat[L2.6] is 152.7 dB/St. The §3.5 limitations admit that the Flat[L2.6] reference 'differs in mean standoff as well as in shape', and the budget's own source term assigns +10.4 dB to Flat[L2.1] purely from standoff. No simulation holds standoff fixed while varying σ, so the 23 dB value in the abstract and conclusions overstates the curvature-specific effect. This should be either reworded to acknowledge the standoff contribution explicitly, or supported by matched-s
  2. [§3.5, Eqs. (3.2)–(3.3), Fig. 8(c)] The source-transfer budget is not a quantitative attribution for most of the configurations. The transfer term ΔTu is a residual by construction, and the source term ΔSPL^PT_src uses the total rms Mach-disk displacement. For the four helical cases, the axial Mach-disk displacement retains only 5–10% of its rms in a band around the primary tone, and the axial motion is not the tonal source for a helical mode, as the paper itself states in the Limitations. Since four of the six cases are helical, the claimed 62%/38% and 40%/60% splits in Fig. 8(c) rest on a source estimate that is not the relevant tonal source for half the configurations. The direct upstream-wave measurements of §5.2 support a real increase in the returning wave, but they do not quantify the source/transfer split. The conclusions should be reworded to present this as a first-order, two-channel observation rather than a mea
  3. [§5.4, Table 3, Fig. 18(b)] The claim that 'axisymmetric frequencies follow Powell's loop-length criterion' is weakly supported. For Flat[L2.1] the Powell prediction is within 2%, but for Concave[σ0.4] it overpredicts the tone by 17.9% using ℓlip and by 12.2% using Lavg. Matching the measured tone would require uc ≈ 0.46–0.50uj, well below the measured 0.599uj, and the paper notes that this case's convection velocity is the least reliable. With only two axisymmetric cases, one degraded by more than 12%, the contrast with the helical H1-lower-limit agreement (3–9% across four cases) is not strong enough to support a distinct 'loop-length selection' mechanism as a firm conclusion. The axisymmetric result should be presented as tentative, or additional axisymmetric cases at other standoffs should be added.
minor comments (4)
  1. [Figure 7 caption] Typos: 'Distructive interference' should be 'Destructive interference'.
  2. [§3.4, Table 2] The mode labels 'A' and 'C' are used without an explicit definition of 'C' at first appearance; state that 'C' denotes the helical (m = ±1) family, for example after Eq. (3.1) or in the Table 2 caption.
  3. [References] The Wagner (1971) entry in the reference list appears to have a mismatched title; please verify that it corresponds to the cited impinging-jet study.
  4. [§3.2, Fig. 3(a)] The vertical offset of the spectra by +30 dB makes the comparison across curves possible, but the y-axis label is unclear; specify which curve is offset and by how much in the caption.

Circularity Check

1 steps flagged · score 3.0 of 10

Helical GJM prediction is an independent external benchmark, but the axisymmetric Powell comparison sets the free integer N per case without an independent measurement; the 23 dB curvature claim is confounded by standoff, though that is a correctness issue rather than a circular reduction.

  1. fitted input called prediction [Section 5.4, discussion after Eq. (5.3); Table 3]
    "The mode number N counts the cells in the standing-wave pattern formed by the downstream and upstream waves over the loop length. It is N=3 for Flat[L2.1] and N=2 for the other cases, with the measured uc entering directly."

    In Eq. (5.3), N is the only free integer. The paper assigns N=3 or N=2 per case without reporting an independent node count, standing-wave cell count, or staging measurement anywhere in §3.4 or §5.4. With N set separately for each configuration, the Powell loop-length formula can place a resonance near the observed tone by construction, so the quoted agreement (1.7% for Flat[L2.1], 12-18% overprediction for Concave[σ0.4]) is not a fully independent test of the axisymmetric loop-length criterion. This is a mild circularity confined to the axisymmetric sub-claim; the helical GJM prediction uses the parameter-free H1 lower limit and is not affected.

full rationale

The paper's central helical-mode result is genuinely self-contained: the H1 lower-limit Strouhal number comes from the Tam-Ahuja vortex-sheet dispersion relation evaluated at Mj=1.56 with no fitted constants, and it is compared against measured helical tones across four configurations, including a flat wall. The radial eigenfunction agreement for helical cases is also an external shape comparison, not a fit. The Powell-Tam source-transfer budget is explicitly framed as a residual-based split, and the paper directly measures the upstream wave to corroborate the transfer term rather than presenting the residual as proof. The self-citations in the methodology concern the numerical solver and validation, not the physical conclusions, and are not load-bearing. The main non-circular concern is the 23 dB curvature-amplification claim: the concave family changes both shape and effective standoff relative to Flat[L2.6], and the paper's own Flat[L2.1] case shows a 20.4 dB increase from standoff alone. However, that is a confounding/variable-isolation problem, not a circular reduction, and the paper acknowledges in its Limitations that the Flat[L2.6] reference 'differs in mean standoff as well as in shape.' The only true circular step is the axisymmetric Powell test, where the integer N is assigned per case without independent determination. Since the helical GJM prediction and the loading-mode analysis retain independent content, the overall circularity is mild rather than structural.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central physical claims rest on the vortex-sheet model and Powell feedback criterion from prior literature, plus the LES modeling assumptions. No new entities are introduced. The main free parameter beyond the simulation setup is the integer mode number N in the Powell prediction.

free parameters (2)
  • mode number N in Powell prediction = 3 (Flat[L2.1]); 2 (other five cases)
    Integer loop-mode number in Eq. (5.3); not measured independently. Changing N by 1 shifts the predicted St substantially, so it acts as a discrete free parameter in the comparison.
  • boundary-layer thickness at nozzle exit = 0.05D
    Chosen following Gojon & Bogey (2017); affects initial shear-layer growth and hence screech, but is a standard simulation-setup parameter, not fitted to the target result.
assumptions (5)
  • domain assumption Vortex-sheet model assumes an inviscid, top-hat jet separated by an infinitesimally thin vortex sheet from a quiescent ambient, and seeks neutral modes (Appendix A).
    Used to compute GJM branches and H1 lower limit in Section 5.3; this idealization neglects finite shear-layer thickness and shock cells.
  • domain assumption Powell feedback-loop criterion: resonance requires the phase accumulated over one loop to be an integer multiple of 2 pi (Eq. 5.2); the classical form assumes the upstream leg is non-dispersive at ambient sound speed.
    Basis of the loop-length predictions in Table 3 and Figure 18; its validity for the axisymmetric cases is the point under test.
  • ad hoc to paper The downstream gain q_d is approximately common to all six configurations (Section 3.5).
    Justifies assigning all curvature-induced amplification to the Mach-disk source and upstream transfer terms; the paper revisits it in the Limitations paragraph but does not test it directly.
  • domain assumption The LES numerical dissipation acts as an implicit subgrid-scale model, and the low-dispersion scheme resolves waves up to St=5.3.
    Standard implicit LES practice (Section 2.2); the resolved frequency range covers the tones of interest.
  • domain assumption The upstream-propagating disturbance is identified by the negative-wavenumber half-plane k_z < 0 and by radial bands that separate the jet column from the ambient (Section 5.2).
    This decomposition underlies the column/ambient amplitude comparison and the dispersion analysis; finite-window effects are not quantified.

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Cite this review

Pith. "Pith review of Supersonic jet impingement on concave surfaces." pith.science (2026). https://pith.science/paper/E5BTHYON

@misc{pith2026260801542,
  author       = {Pith},
  title        = {Pith review of: Supersonic jet impingement on concave surfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E5BTHYON}},
  note         = {Machine review of arXiv:2608.01542}
}
abstract

The aeroacoustic resonance of round supersonic jets impinging on concave surfaces is investigated using compressible large-eddy simulations, vortex-sheet modelling, and Powell's feedback-loop analysis. The choked jets operate at an ideally expanded Mach number of $1.56$ and a Reynolds number of $6\times10^4$. Six geometries are considered: two flat plates at $L/D=2.08$ and $2.58$, where $L$ is the nozzle-to-wall distance and $D$ the nozzle exit diameter, and four Gaussian concave surfaces of fixed depth and indentation spread $\sigma\in\{0.4,0.8,1.6,4.0\}$. As the indentation narrows, the primary-tone amplitude increases by up to $23\,\mathrm{dB}$ relative to the flat-wall reference at $L/D=2.6$, together with larger wall-pressure fluctuations and moments. A Powell-Tam source-transfer budget attributes this amplification to increased Mach-disk source amplitude and more efficient return of the upstream feedback wave to the nozzle. The stronger upstream-propagating waves are consistent with acoustic focusing by the concave wall. For the helical cases, the measured frequencies and radial eigenfunctions agree closely with the guided jet mode predicted by the vortex-sheet model, supporting its role in closing the upstream feedback path. The same selection is recovered for concave and flat walls alike, so this tone is governed by the shear-layer profile of the equivalent ideally expanded jet rather than by the wall geometry. The axisymmetric frequencies, by contrast, coincide with no guided-mode branch and appear instead to follow Powell's classical loop-length criterion. The results identify distinct frequency-selection mechanisms for helical and axisymmetric screech and demonstrate that wall curvature provides effective control of screech amplitude and surface loading.

Figures

Figures reproduced from arXiv: 2608.01542 by the authors.

Figure 1
Figure 1. (a) Geometric profiles of the flat and concave impinging surfaces defined by equation (2.1). (b) Isometric view of the Concave[𝜎0.4] surface illustrating the localised indentation. (c) Instantaneous flow field of the Concave[𝜎0.4] jet, showing the axial velocity 𝑢/𝑎∞ in the meridional 𝑟-𝑧 plane and iso-surfaces of the Q-criterion coloured by the velocity magnitude |u|/𝑎∞. using the isentropic relations: 𝑝 exit 𝜌∞𝑎 2… view at source ↗
Figure 2
Figure 2. Instantaneous fields, statistics and modal content for the six impinging jet configurations. (a) Instantaneous density field. (b) Mean velocity magnitude field (LES); the white box in the Flat[L2.1] panel insets the experimental PIV of Henderson et al. (2005) over −0.95 ⩽ 𝑟/𝐷 ⩽ 0 for side-by-side comparison, all other fields being LES. (c) Standard deviation of the density field. (d) First POD mode based on the dens… view at source ↗
Figure 3
Figure 3. (a) Near-field pressure spectra at 𝑟/𝐷 = 1 for the flat and concave configurations. The cyan curve is the Flat[L2.1] case from the LES of Gojon & Bogey (2017). (b) Spectrograms of the same signals. The rows follow the same top-to-bottom order as the offset spectra in panel (a), from Flat[L2.6] at the top to Flat[L2.1] at the bottom. Arrows indicate intermittent breaks or decreased amplitude in tonal events. band, in… view at source ↗
Figures from the paper (15 more)
Figure 4
Figure 4. Figure 4: Polar distributions of sound pressure level on the nozzle-exit plane at 𝑟/𝐷 = 1: band-pass-filtered power at (a) the primary screech tone 𝑆𝑡1 and (b) the secondary tone 𝑆𝑡2, and (c) the overall level (OASPL). The acoustic loading increases monotonically across the conc…
Figure 5
Figure 5. Figure 5: Mach-disk unsteadiness and associated spectral content. (a) Temporal evolution of the axial Mach-disk location along 𝑟 = 0. (b) Probability density function (PDF) of the Mach-disk location. (c) Power spectral density (PSD) of the Mach-disk motion for Flat[L2.1] and Con…
Figure 6
Figure 6. Figure 6: Phase fields of the density fluctuation field at each configuration’s primary Strouhal number 𝑆𝑡1 (table 2): (a) Flat[L2.1], (b) Concave[𝜎0.4], (c) Concave[𝜎0.8], (d) Concave[𝜎1.6], (e) Concave[𝜎4.0] and (f) Flat[L2.6]. For each configuration the upper sub-panel is the…
Figure 7
Figure 7. Figure 7: Amplitude fields filtered at the dominant screech-tone frequency, shown as twelve equal acoustic-level bands from 𝑙1 to 𝑙12 in dB/St; the green region, for example, denotes values within [𝑙3, 𝑙4]. The 𝑙1 to 𝑙12 range is set separately for each case to bring out the nea…
Figure 8
Figure 8. Figure 8: Decomposition of the spread-dependent primary-tone amplification. (a) Powell-Tam source-strength amplification ΔSPLPT src equation (3.3) versus the lip-line feedback length ℓlip/𝐷. (b) Upstream-wave amplification in the jet column, 𝑟/𝐷 ∈ [0, 0.5] (filled triangles), an…
Figure 9
Figure 9. Figure 9: Wall-flow topology and skin friction on the impinging surface. Top: mean velocity magnitude |u|/𝑎∞ and projected streamlines in the meridional 𝑟-𝑧 plane within 0 ⩽ 𝑟/𝐷 ⩽ 4 for the Flat[L2.1] configuration. Middle: radial distribution of the skin friction coefficient 𝐶𝑓…
Figure 10
Figure 10. Figure 10: Axial and transverse forces on the impinging surface. (a) Time history of the axial force 𝐹𝑧/(𝜌∞𝑎 2 ∞𝐷 2 ). (b,c) Power spectral density (PSD) of 𝐹𝑧 for (b) Flat[L2.1] (black) and Concave[𝜎0.4] (red), and (c) the remaining four configurations. Vertical dashed lines in…
Figure 11
Figure 11. Figure 11: Moments about the centre of the impinging surface. (a) Time history of one in-plane bending-moment component, 𝑀⊥,1/(𝜌∞𝑎 2 ∞𝐷 3 ), for all six configurations. (b) Time-averaged moment magnitude |M|/(𝜌∞𝑎 2 ∞𝐷 3 ) for each configuration; error bars indicate the root-mean…
Figure 12
Figure 12. Figure 12: Mean convection velocity of the downstream wavepacket along the maximum-turbulent-ki￾netic-energy shear-layer trajectories for the six configurations. (a) Local 𝑢𝑐 (𝑧)/𝑢 𝑗 versus axial station 𝑧/𝐷, averaged over the upper and lower trajectories. (b) Trajectory-mean ⟨𝑢…
Figure 13
Figure 13. Figure 13: Wavenumber spectrum |𝜌ˆ(𝑘 𝑧𝐷 𝑗 , 𝑟/𝐷)| at each case’s primary tone 𝑆𝑡1 on the meridional 𝑟-𝑧 plane, restricted to the upstream-propagating half 𝑘 𝑧𝐷 𝑗 < 0 and normalised in each panel by its own maximum. The yellow dashed lines mark the lip line at |𝑟|/𝐷 = 0.5, and th…
Figure 14
Figure 14. Figure 14: Upstream-propagating density fluctuations travelling at the ambient sound speed (𝑐𝜙 = −𝑎∞) for the six configurations, each band-limited to its primary tone 𝑆𝑡1 (table 2): (a) Flat[L2.1], (b) Concave[𝜎0.4], (c) Concave[𝜎0.8], (d) Concave[𝜎1.6], (e) Concave[𝜎4.0] and (…
Figure 15
Figure 15. Figure 15: Column-region envelope of the upstream-propagating pressure field, max|𝑟 |⩽0.5𝐷 |𝑝ˆ − |, along the axial coordinate for the six configurations, all normalised by the overall maximum so that amplitudes are directly comparable. The field ˆ𝑝 − is obtained from the upstre…
Figure 16
Figure 16. Figure 16: Frequency-wavenumber spectrum of the upstream density field on the 𝑟 = 0.35 𝐷 cylinder, projected onto each case’s dominant azimuthal mode, with the vortex-sheet dispersion branches overlaid. Colour: 10 log10 |𝑆(𝑘 𝑧𝐷 𝑗 , 𝑆𝑡)|2 in dB. Solid black: axisymmetric 𝐴𝑛 branc…
Figure 17
Figure 17. Figure 17: Radial eigenfunctions of the upstream 𝑐𝜙 = −𝑎∞ density component over the jet core, 𝑟/𝐷 ∈ [−0.5, 0.5], each at its primary tone 𝑆𝑡1 (table 2): (a) Flat[L2.1], (b) Concave[𝜎0.4], (c) Concave[𝜎0.8], (d) Concave[𝜎1.6], (e) Concave[𝜎4.0] and (f) Flat[L2.6]. Red: phase-mat…
Figure 18
Figure 18. Figure 18: Frequency selection of the dominant screech tone across the six configurations. (a) Screech-tone Strouhal numbers against the nozzle-to-wall distance 𝐿/𝐷; for the two flat-wall configurations both the primary and the secondary tone are plotted. The gray lines are the …

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.