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Stable and Efficient One-Way Modelling of Convective Disturbances in Laminar Boundary Layers: OWNS-Summation

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Evaluating the one-way Navier–Stokes projector as an additive partial-fraction sum instead of a recursive product removes the large-order numerical breakdown, turning the approximation order into an ordinary convergence parameter.

desk verdict A well-controlled experiment shows the additive OWNS-S evaluation removes the recursive-product instability that caps OWNS-R order; the mechanism is convincing, though the 'pure convergence parameter' claim outruns the conditioning analysis. read the letter →

arxiv 2608.01155 v1 pith:7TGUGY3X submitted 2026-08-02 physics.flu-dyn math-phmath.MPmath.OC

Elliot J. Badcock , Shahid Mughal This is my paper · ORCID
classification physics.flu-dynmath-phmath.MPmath.OC MSC 65F1565F6076E0576M20
keywords Boundary-LayerInstabilityParabolisedStabilityEquationsOne-WayNavier-StokesOWNS-SummationOWNS-Recursionspectralprojectionfloating-pointerrortransonicboundarylayer
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 paper argues that the large-$N$ breakdown of the OWNS-R one-way marching method is not caused by its pole placement but by the way it evaluates the rational approximation: as a recursive product of resolvent factors, in which each stage's rounding error is multiplied through all later stages. The paper proposes OWNS-Summation (OWNS-S), which evaluates the same rational approximation as an additive partial-fraction sum, with every resolvent solve acting on the same input state. In exact arithmetic the two are the same operator; in floating point the sum stays bounded while the product grows to about $10^{40}$ in an accumulated Frobenius diagnostic at $N=100$. If correct, the result makes $N$ a genuine convergence parameter, lets the $N$ solves run in parallel, and removes the need to keep the order small, so practitioners can certify convergence by raising $N$ as in ordinary discretisation studies. The paper supports this with projection-quality metrics and spatial marches through incompressible, hypersonic, and transonic boundary layers, including a continuous march through a subsonic-to-supersonic spectral reorganisation.

What carries the argument

The rational approximation of the spectral projector, $P_N^+ = w_0 I + \sum_{k=1}^N w_k (B - i\beta_k^* A)^{-1} A$, evaluated additively rather than as the product $\frac{1}{c+1}\prod_{k=1}^N (B-i\beta_k^* A)^{-1}(B-i\beta_k^- A)$. The auxiliary poles $\beta_k^*$ come from the polynomial identity of Eq. (14), and the weights from residues or a least-squares fit; the additive evaluation is what decouples the solves and stops multiplicative rounding-error amplification.

What would settle it

The paper's own synthetic experiment is the sharpest check: with identical poles, the accumulated Frobenius diagnostic $\rho^R_k$ for the recursive product should grow to about $10^{40}$ at $N=100$ while $\rho^S_k$ for the summation stays bounded; reproducing this on any pencil would confirm the mechanism. To test the $C$-term assumption, compute the retained/leaked eigenvalue sets for the full quadratic pencil $\mathbf{B} - i\alpha\mathbf{A} - \alpha^2\mathbf{C}$ on a strong non-parallel or low-$R$ case and compare with the first-order-pencil projector; any mismatch in the directional classif

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Extended reading notes

Core claim

At fixed auxiliary poles, OWNS-S is shown to reproduce OWNS-R's projected operator in exact arithmetic but to have fundamentally different floating-point error propagation. In OWNS-R, the error committed at stage $k$ is carried through the $N-k$ remaining stage operators, so the accumulated diagnostic $\rho^R_k$ grows roughly to $10^{40}$ at $N=100$; in OWNS-S each solve's error enters only one weighted term, so $\rho^S_k$ stays bounded. The paper therefore identifies the recursive evaluation---not the poles---as the dominant source of the instability that previous work attributed to pole-solving rounding errors and recommended extended precision for. With the same poles, OWNS-S remains accu

Load-bearing premise

The projector is built from the first-order pencil $(B,A)$ only, while the marched equation keeps the second-order $C\,\partial^2_x$ term; the paper assumes this does not affect the upstream–downstream separation, and if that assumption fails in some flow the projection could retain or remove the wrong modes.

Editorial extensions

If this is right

  • Approximation order $N$ becomes a convergence parameter in the ordinary sense: it can be raised until the solution stops changing, so converged results can be certified without an unknown ceiling.
  • The $N$ resolvent solves per station are independent, reducing the per-station critical path to $1+\lceil N/n_t\rceil$ on $n_t$ threads and bringing OWNS-S's critical path to PSE-like levels.
  • The same rational projector used by OWNS-R is evaluated accurately at $N$ where OWNS-R diverges; in the hypersonic case, the two distinct failure modes (parameter-set deficit at low $N$, recursive instability at high $N$) are separated, with OWNS-S surviving both.
  • OWNS-S can march continuously through a transonic spectral reorganisation from subsonic to supersonic topology at streamwise resolutions below the PSE step floor.
  • The paired greedy parameter selection removes the need for eigendecomposition during the march, with the analytic candidate pool matching eigenvalue-based selection in the benchmark case.

Reading between the lines

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

  • If the multiplicative product is the mechanism, then every OWNS-R implementation---regardless of extended-precision poles or factor ordering---should hit the same size ceiling; a direct test would be to run OWNS-R with the least-squares weights that repair retention, and check whether the ceiling survives.
  • The additive-vs-recursive distinction is not specific to OWNS: other contour-integral projectors and one-way methods that currently evaluate products of resolvents could inherit the same cure, and the paper leaves the quadratic-pencil projector (including $C$) as an obvious next step to test the first-order-pencil assumption.
  • Since the $\beta_k^*$ contour is only an estimate of the spectral separation, the method's reliability on strongly non-parallel or low-Reynolds-number flows likely hinges on whether the neglected $C$ term changes the upstream–downstream partition; computing a quadratic-pencil projector would settle that directly.
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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

2 major / 4 minor

Summary. The paper introduces OWNS-Summation (OWNS-S), an additive partial-fraction evaluation of the rational spectral projector used in one-way Navier–Stokes marching, in place of the recursive product evaluation of OWNS-R. With identical poles and weights the two evaluations are equivalent in exact arithmetic, but differ in floating-point arithmetic because OWNS-R propagates each resolvent-solve error through all later stages. The central diagnostic experiment in §4.1.1 uses identical auxiliary poles from the same polynomial solve and shows the accumulated recursive-product norm growing to roughly 10^40 at N=100 while the summation norm remains bounded. The method is then validated on incompressible, hypersonic, and transonic boundary layers, including a swept-wing transonic case in which the disturbance spectrum reorganizes from subsonic to supersonic topology during the march. A greedy, paired parameter-selection procedure with an analytic candidate pool is also introduced, avoiding eigen-decomposition during the march.

Significance. If the central claim holds, the paper makes a practically important contribution: it removes an unknown, flow-dependent upper bound on the approximation order N that limits OWNS-R, turns N into a convergence parameter in the ordinary sense, and exposes a parallel structure with per-station critical path reduced to 1+ceil(N/n_t) solves. The controlled comparison in §4.1.1 is a genuine strength: the poles are held fixed while only the evaluation changes, so the observed ~40-order-of-magnitude differential identifies the recursive product rather than the auxiliary poles as the source of instability. The replication of the qualitative behaviour across incompressible, hypersonic, and transonic configurations is also persuasive. However, two load-bearing points need further support before the strongest claims can be accepted: the conditioning of the weight computation in OWNS-S at large N, and the validity of constructing the projector from the first-order pencil only while the second-order C term is retained in the march.

major comments (2)
  1. [§3.1, §4.1.1] The claim that N in OWNS-S is a 'pure convergence parameter' is stronger than the evidence presented. Section 3.1 explicitly states that the residue formula (17) degrades when auxiliary poles cluster and when weights acquire large alternating signs, and that the least-squares system (18)-(19) becomes severely ill-conditioned for densely distributed parameters. The synthetic experiment in §4.1.1 measures the accumulated-stage norm (28), not the conditioning of the weight computation or the backward error of the weighted sum; it is reported to N=100, with a passing mention of N≈300. If weight computation itself develops flow-dependent ill-conditioning at larger N, OWNS-S will also have an upper bound on N. Please report condition numbers of D, weight norms, or residual/backward errors versus N across the three test configurations, or explicitly limit the 'pure convergence' claim to the ran
  2. [§2.1] The projector is constructed from the first-order pencil (B,A) only, while the marched equation retains the second-order C d^2/dx^2 term. The authors acknowledge that C does not commute with the projection operator and that the construction is therefore not an exact spectral splitting of the second-order system. This is load-bearing for the physical validity of the one-way march: if the neglected C term contributes to the upstream–downstream partition at low Reynolds number or under strong non-parallel effects, the projected march could miss or misfilter modes. The statement that 'its neglect has not been observed to affect' the separation is an empirical assertion, not a demonstrated bound. Please quantify the neglected contribution, for example by comparing against a projection built from the full quadratic pencil in a representative low-Reynolds-number or strongly non-parallel case, o
minor comments (4)
  1. [§6.5.2] The PSE reference is quoted at n_x=150 and described as under-resolved, while OWNS-S is step-converged at n_x=5000 and reproduces PSE when run at the coarse step. This is plausible, but the discussion would be clearer if the PSE reference were also shown at a step where it is converged in its own resolution study, or if the resolution limitation were documented with a separate PSE step-size test.
  2. [Captions, Figures 4, 6, 23, 24] The color/line legend entries are dense and some are distinguished only by marker style; please add a short description in each caption of which curve corresponds to which variant, especially for 'Greedy-Heuristic Nc=100/200/300'.
  3. [Abstract and §1] The abstract states that N 'becomes a pure convergence parameter' without the caveats given in §3.1. Please soften the abstract to reflect that the evidence covers the tested placement families and that weight conditioning at very large N is a separate question.
  4. [References] Some references are typeset inconsistently (e.g., 'Badcock and Mughal, 2026a,b', 'Gushchin and Fedorov, 1990' spacing). A final formatting pass would help.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: OWNS-S vs OWNS-R comparison is a controlled numerical experiment with identical inputs, and the central claim does not reduce to a fit or self-citation.

full rationale

The paper's central derivation is self-contained. The key controlled experiment in §4.1.1 supplies both OWNS-R and OWNS-S with identical auxiliary poles from the same polynomial solve, so the observed difference (≈10^40 growth in the recursive Frobenius diagnostic versus a bounded summation diagnostic) is an empirical measurement of the two evaluation schemes, not a quantity fitted to the conclusion. The algebraic equivalence of the recursive product (22) and additive sum (13) in exact arithmetic is established by the polynomial identity (14), and the failure mechanism is derived analytically via the error recursion (25)–(26); neither step presupposes the conclusion. The validation uses external references (PSE) and an independent eigenvalue-seeded greedy benchmark (Greedy-Spectrum) to check the analytical candidate pool, so the parameter-selection pipeline is not validated solely on its own heuristic inputs. The acknowledged limitations—the first-order-pencil projector (§2.1) and the conditioning of the weight calculation at large N (§3.1)—are stated open issues rather than circular moves: they do not smuggle the target result into the assumptions. The self-citations to Badcock & Mughal for heuristic placement and wave-carrier factoring are not load-bearing because the present paper reproduces the explicit formulas (Appendix C) and controls for the candidate pool; the central claim about additive versus recursive floating-point evaluation remains independently demonstrated.

Assumptions & free parameters 5 free parameters · 6 assumptions · 3 invented entities

The paper's central claim rests on a small set of modeling assumptions, the most fragile being the first-order-pencil projection (neglect of C in the projector), which is acknowledged and untested in generality. The heuristic placement relies on analytic branch estimates plus one empirical input (trapped upstream mode locations). The new numerical evaluations (summation, LS weights, paired greedy selection) are not invented physical entities; they are algorithmic alternatives supported by controlled experiments. No new particles, forces, or conserved quantities are introduced.

free parameters (5)
  • balance parameter c = 1 (subsonic, balanced), 0.01 (SWiFT supersonic), 0.1 (hypersonic), 100 (illustrative)
    Eq. (14) and Appendix C.3: c controls the weighting between retention and removal parameters and the contour shape. Values are chosen per configuration by hand, not derived.
  • heuristic placement constants (h_a, L_a, h_v, L_v, h_1, L_1, h_2, L_2, s, p, delta, etc.) = Representative values: L_a ~ 1e2, h_a/L_a ~ 1e-2, L_v=1, h_v~0.3, s=2, L_2~4, h_2~0.5, delta~1e-2
    Appendix C: parameters of the Malik-type clustering for retention/removal parameter distributions. The paper states these are 'tied to the present discretisation' and 'not intended as optimal values'. They are tuned per configuration, so they are free parameters.
  • location of upstream discrete acoustic modes (alpha_remove or the alpha = -i beta mode) = For SWiFT: alpha = -i beta, with a ladder descending the imaginary axis; for hypersonic: real part estimated, shifts wit
    Sections 5.1, 6.4.2, 6.5.1: the paper explicitly states these are 'taken from a computed spectrum at a single station' and that a WKB treatment 'would remove the one input the parameter selection currently takes from a computed spectrum.' This is a fitted/empirically supplied input to the placement.
  • greedy threshold and candidate pool size (N_c) = N_c = 100, 160, 200, 300; convergence threshold ~ 1e-14
    Section 5.2.2 and Table 3: N_c and the stopping threshold are chosen by the user. Insensitivity to N_c is shown, but these are still tunable procedural parameters.
  • PSE reference parameters (n_x = 150, n_iter, carrier correction) = n_x = 110 to 150
    Section 6: the PSE reference uses its own carrier iteration and step size; the comparison in the SWiFT case relies on the PSE result being run at n_x=150 as 'the PSE reference'. The different streamwise discretizations are a comparison setting, not a fitted constant, but they affect the plotted amplitude differences.
assumptions (6)
  • domain assumption The one-way projector can be constructed from the first-order pencil (B, A) while neglecting the C d^2/dx^2 term, and this neglect does not affect the upstream-downstream separation.
    Section 2.1: The paper states the projection is built from the first-order pencil only, that C is retained in the marching but not in the projector, and that the approximation is 'real' and 'has not been observed to affect the upstream-downstream separation.' This is an unproven modeling assumption that the entire theory rests on.
  • domain assumption The finite spectrum of the pencil and Briggs' criterion correctly partition disturbances into upstream/downstream/pinch classes for the marching problem.
    Section 2.2: The spectral separation relies on Briggs' criterion and the claim that the pinch-point category is empty in practice after discretisation. This is standard in the field but is an assumption about the relationship between discrete eigenvalues and physical wave propagation.
  • domain assumption The continuous branches of the frozen-coefficient freestream pencil adequately represent the full discretised spectrum for parameter placement.
    Appendix A: The analytic branch formulas (A.1)-(A.4) are derived from constant-coefficient analysis and are used to build the retention/removal parameters. The paper acknowledges discrete modes lie off these branches and their locations must be supplied separately.
  • standard math The Cayley transformation and the unit circle estimate of the separating contour preserve the eigenvalue partitioning and projector.
    Appendix B: The map is eigenvector-preserving and the projectors are claimed identical in both representations. This is correct mathematical manipulation of the pencil, with the usual caveat about non-normality.
  • standard math The polynomial identity (14) has N roots that interleave into a simple separating curve, with the chosen pairing.
    Section 2.4.2 and Appendix C: The identity is lifted from Zhu and Towne (2023). The properties (simple curve, no closed loops) are asserted and illustrated, not proven in generality.
  • domain assumption Per-step projection of the state after the BDF2 update is equivalent to integrating the projected evolution equation.
    Section 6.1: The paper cites Zhu and Towne (2023) for this equivalence. It is a standard result in the OWNS literature but is still an assumption about the integration scheme and the projector.
invented entities (3)
  • OWNS-Summation (OWNS-S) evaluation form independent evidence
    purpose: Evaluates the rational spectral projector as an additive partial-fraction sum instead of a recursive product, preventing multiplicative error growth and enabling parallel solves.
    OWNS-S is not a new physical entity but a new numerical evaluation. It has independent evidence in the form of the synthetic experiment (Fig. 3), the projection-quality metrics, and the three validation configurations. Since it is a mathematical reformulation of the same operator, the independent evidence is the floating-point behaviour itself, which is falsifiable.
  • Paired-symmetric greedy refinement with analytic candidate pool independent evidence
    purpose: Selects retention/removal parameter pairs from an oversampled analytic heuristic pool, avoiding eigendecomposition during the march.
    A numerical procedure, not a physical entity. Independent evidence comes from the greedy-spectrum benchmark (Section 6.3.1) showing comparable parameters to the eigenvalue-based pool, and from the seven-parameter sweep in Section 6.5.3.
  • OWNS-S (LS) least-squares weight recalibration independent evidence
    purpose: Computes partial-fraction weights by enforcing separation conditions at the computed poles via least squares, repairing pole-error degradation.
    A numerical stabilisation scheme. Independent evidence: in all configurations it matches or improves on residue weights, and the paper shows it is indistinguishable from standard OWNS-S when poles are well placed.

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Cite this review

Pith. "Pith review of Stable and Efficient One-Way Modelling of Convective Disturbances in Laminar Boundary Layers: OWNS-Summation." pith.science (2026). https://pith.science/paper/7TGUGY3X

@misc{pith2026260801155,
  author       = {Pith},
  title        = {Pith review of: Stable and Efficient One-Way Modelling of Convective Disturbances in Laminar Boundary Layers: OWNS-Summation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7TGUGY3X}},
  note         = {Machine review of arXiv:2608.01155}
}
abstract

One-way spatial marching methods separate upstream- from downstream-propagating disturbances using a rational approximation of a spectral projector. Among existing one-way Navier--Stokes (OWNS) formulations, the recursive variant, OWNS-R, is the most economical, evaluating the approximation as a product of $N$ resolvent factors. This product amplifies rounding errors multiplicatively, imposing a flow-dependent upper limit on the approximation order that cannot be known in advance. We reformulate the same approximation as an additive partial-fraction sum (OWNS-Summation, OWNS-S). In exact arithmetic, the recursive and summation evaluations are equivalent when supplied with identical weights and poles; in floating-point arithmetic, they are not. Each of the $N$ resolvent solves acts on the same input state and contributes independently to a weighted sum, preventing multiplicative error amplification. The approximation order $N$ therefore becomes a pure convergence parameter, and the solves can run in parallel. Using the same auxiliary poles as OWNS-R, OWNS-S remains accurate in every configuration tested, showing that the recursive evaluation, rather than the poles, is the dominant source of instability. A paired greedy parameter-selection procedure is also introduced, with candidates drawn from analytic estimates of the upstream and downstream spectral regions, avoiding eigen-decomposition during the numerical march. OWNS-S is validated on incompressible, hypersonic and transonic boundary layers. In the transonic case, the disturbance spectrum reorganises from a subsonic to a supersonic topology during the march. The one-way computation proceeds continuously through this transition at a streamwise resolution unattainable by the parabolised stability equations.

Figures

Figures reproduced from arXiv: 2608.01155 by the authors.

Figure 1
Figure 1. (a) Subsonic disturbance at (𝑀 = 0.5, 𝛽 = 0) showing an acoustic pinch-point and both upstream and downstream continua; the Tollmien–Schlichting discrete mode is denoted by T-S. (b) Supersonic disturbance (𝑀 = 4.5, 𝛽 = 0) with purely downstream-propagating acoustic branches. We define a spectral reference point 𝛼mid whose construction depends on the disturbance regime (Eq. (B.2), Appendix B). For subsonic disturbanc… view at source ↗
Figure 2
Figure 2. Cayley-transformed spectra for (a) a subsonic disturbance at 𝑀 = 0.5, 𝛽 = 0 and (b) a supersonic disturbance at 𝑀 = 4.5, 𝛽 = 0. Upstream modes lie inside the unit circle; downstream modes lie outside. The projector 𝐏 + is idempotent, (𝐏 +) 2 = 𝐏 +, and the complementary projector 𝐏 − = 𝐈 − 𝐏 + isolates the upstream subspace. We define the projected pencils 𝚷 ± = (𝐁𝐏± , 𝐀), (8) which retain the eigenvectors of 𝚷 whil… view at source ↗
Figure 3
Figure 3. Accumulated-stage Frobenius norms for the recursive (OWNS-R, left, Eq. (27)) and summation (OWNS-S, right, Eq. (28)) evaluations at a fixed pencil (𝐁, 𝐀) using identical auxiliary poles {𝛽 ∗ 𝑘 }, for 𝑁 ∈ {10, 20,… , 100} with the Malik￾clustered retention/removal parameters. OWNS-R develops accumulated norms of order 1040 at 𝑁 = 100, whereas OWNS-S remains bounded throughout. 4.2. Solve-count comparison The relative… view at source ↗
Figures from the paper (22 more)
Figure 4
Figure 4. Figure 4: Projection-quality metrics for the 2D incompressible boundary layer, comparing OWNS-R, OWNS-S, and OWNS￾S (LS) with greedy and non-greedy selection, against approximation order 𝑁: (a) downstream retention retain, (b) upstream leakage leak , (c) idempotency idem. OWN…
Figure 5
Figure 5. Figure 5: Convergence of max |𝑢| at 𝑅𝛿 ≈ 815 against approximation order 𝑁. OWNS-R and standard OWNS-S coincide for 𝑁 ≤ 44 at identical retention/removal parameters (OWNS-R is hidden beneath OWNS-S over this range), but OWNS-R fails drastically for 𝑁 > 44. The greedy variants re…
Figure 6
Figure 6. Figure 6: Projection-quality metrics for the hypersonic boundary layer (𝑀 = 4.5). OWNS-R diverges for 𝑁 ≥ 36; OWNS-S and OWNS-S (LS) remain stable. Greedy LS selection gives the lowest retention and leakage errors. Isolated leakage spikes in (b) are mode-pairing artefacts of the…
Figure 7
Figure 7. Figure 7: Wall-pressure amplitude |𝑝wall| for the hypersonic boundary layer (𝑀 = 4.5) at (a) 𝑁 = 30 and (b) 𝑁 = 60. At 𝑁 = 30 the three heuristic formulations (OWNS-R, OWNS-S, OWNS-S (LS)) diverge at the same 𝑅𝛿 ≈ 1080, indicating a common cause: a moving or emerging upstream di…
Figure 8
Figure 8. Figure 8: Convergence of max |𝑝wall| at the amplification peak 𝑅𝛿 ≈ 1050. OWNS-R diverges for 𝑁 ≤ 30 and 𝑁 ≥ 46; OWNS-S and OWNS-S (LS) converge for 𝑁 > 30. Greedy OWNS-S (LS) (𝑁c = 200) converges smoothly by 𝑁 ≈ 26. evident in the surface-pressure coefficient and camber-line ge…
Figure 9
Figure 9. Figure 9: (a) SWiFT model in the ARA wind-tunnel facility (Coppin et al., 2025). (b) Streamwise skin-friction coefficient showing the 2.5D slice used for the present stability analysis (red arrow). 6.5.1. Spectral Evolution and Transonic Recursion-Parameter Strategy As the flow …
Figure 10
Figure 10. Figure 10: Surface pressure distribution and camber-line geometry of the SWiFT model. The chordwise coordinate (bottom axis) is mapped to 𝑅𝛿 (top axis) for comparison with the marching domain. 200 400 600 800 1000 R/ 0.5 0.75 1 1.25 M2 D = 3 D (a) 𝑀3D (red) and 𝑀2D (blue). 200 4…
Figure 11
Figure 11. Figure 11: Streamwise variation of (a) the base-flow and disturbance Mach numbers 𝑀3D and 𝑀2D, and (b) the freestream flow-deflection angle 𝜃fs. In (a) the solid line marks where the base flow becomes supersonic (𝑀3D = 1, 𝑅𝛿 ≈ 270) and the dashed line where the continuous acoust…
Figure 12
Figure 12. Figure 12: Schematic of the spectral reorganisation through the transonic transition, in standard 𝛼-space, at five representative stages: (a) subsonic inlet, the acoustic continua separated and all remaining modes near the origin; (b) the least-damped upstream discrete mode deta…
Figure 13
Figure 13. Figure 13: The same five stages as [PITH_FULL_IMAGE:figures/full_fig_p026_13.png]
Figure 14
Figure 14. Figure 14: Computed spectra at successive stations (𝑅𝛿 = 145–527), standard 𝛼-space, showing the continuous rightward migration of the acoustic branches and the progressive peeling of discrete modes as the flow accelerates through the transition. The near-origin cluster is where…
Figure 15
Figure 15. Figure 15: Computed spectra and recursion-parameter placement at a mixed subsonic station (𝑅𝛿 = 329), in the Cayley 𝜉-plane, for (a) greedy OWNS-S (LS) and (b) OWNS-R heuristic placement at 𝑁 = 40. The freestream is supersonic but the disturbance is firmly subsonic; the pinch-po…
Figure 16
Figure 16. Figure 16: Computed spectra and recursion-parameter placement at a supersonic station (𝑅𝛿 = 461), in the Cayley 𝜉-plane, for (a) greedy OWNS-S (LS) and (b) OWNS-R heuristic placement at 𝑁 = 40. The disturbance is supersonic: upstream discrete modes are fully separated and the ac…
Figure 17
Figure 17. Figure 17: Convergence of the peak max |𝑢| with approximation order 𝑁. OWNS-R converges only within a narrow window (𝑁 ≈ 26–28) and diverges outside it; all OWNS-S variants remain stable, the greedy variant converging earliest. Dashed line: PSE reference (𝑛𝑥 = 150). (a) Wavenumb…
Figure 18
Figure 18. Figure 18: Spatial evolution at 𝑁 = 60: (a) wavenumber Re(𝛼), (b) growth rate Im(𝛼), (c) maximum streamwise velocity amplitude max |𝑢|, (d) wall pressure amplitude. All OWNS-S variants coincide. Vertical lines mark the base-flow (𝑀3D = 1) and continuous-spectrum (𝑀2D = 1) sonic …
Figure 19
Figure 19. Figure 19: OWNS-R march: (a) growth rate, (b) maximum streamwise velocity amplitude. At 𝑁 = 28 (within the stable window) OWNS-R tracks the converged result; at 𝑁 = 40 it diverges at 𝑅𝛿 ≈ 450, just after the 𝑀2D = 1 transition (vertical line) where the continuous spectrum goes s…
Figure 20
Figure 20. Figure 20: Low-𝑁 (𝑁 = 18) march zoomed on the upstream laddering peel-off: (a) growth rate, (b) wall pressure amplitude. Non-greedy OWNS-S, OWNS-S (LS), and OWNS-R, which share the same tuned parameters, carry a localised defect (𝑅𝛿 ≈ 385) just after the upstream modes detach (𝑅…
Figure 21
Figure 21. Figure 21: Real part of streamwise velocity Re(𝑢) for the SWiFT transonic boundary layer, comparing PSE (𝑛𝑥 = 150, top) and greedy OWNS-S (LS) (𝑁 = 60, 𝑁c = 200, 𝑛𝑥 = 1000, bottom). The PSE solution is under-resolved. Second, 𝑁 = 50 lies well outside the range 𝑁 ≈ 26–28 within w…
Figure 22
Figure 22. Figure 22: Seven SWiFT marches over spanwise wavenumber and frequency, all with greedy OWNS-S (LS) at 𝑁 = 50, 𝑁c = 200, and all converged in 𝑁: (a) wavenumber, (b) growth rate, (c) maximum streamwise velocity amplitude, (d) legend. Case (i), black, is the disturbance of §6.5.2; …
Figure 23
Figure 23. Figure 23: Numerical spectrum of the Cayley-transformed two-way system 𝜎(𝚷) and the parabolised downstream system 𝜎(𝚷+ ) for a 2D incompressible boundary layer (𝑀 = 0.02, 𝑅𝛿 = 400, 𝑓 = 86× 10−6 , 𝑏 = 0) with 𝑁 = 30 retention/removal parameters. Blue and red diamonds denote 𝛽 + a…
Figure 24
Figure 24. Figure 24: Retention/removal parameter distributions for the hypersonic boundary layer (𝑀 = 4.5) in Cayley space with 𝑁 = 60. Blue and red diamonds denote 𝛽 + and 𝛽 − , green diamonds the auxiliary poles 𝛽 ∗ , and contours show | + 𝑁 |. (a) Heuristic placement. (b) Greedy place…
Figure 25
Figure 25. Figure 25: Effect of the tuning parameter 𝑐 on auxiliary-pole placement in Cayley space. Blue and red diamonds show 𝛽 ± ; green diamonds show 𝛽 ∗ . (a) 𝑐 = 1: balanced retention and removal, 𝛽 ∗ contour near the unit circle. (b) 𝑐 = 0.01: contour contracted inward, favouring dow…

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