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arxiv: 2601.18017 · v1 · submitted 2026-01-25 · ⚛️ physics.flu-dyn

Beyond secondary instability: on the emergence of finite-amplitude waves in G\"ortler vortices

Runjie Song , Kengo Deguchi This is my paper

Pith reviewed 2026-05-16 11:19 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords Görtler vorticessecondary instabilityfinite-amplitude wavesParabolised Coherent Structuresboundary layer transitionvortex-wave interaction
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The pith

The Parabolised Coherent Structures method reproduces experimental wave amplitudes and displacement thicknesses in Görtler vortices.

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

Görtler vortices on concave walls develop wavelike disturbances via secondary instabilities that evolve to finite amplitudes and precede turbulence. The paper applies the Parabolised Coherent Structures method to incorporate nonlinear vortex-wave interactions into a spatial-marching computation. This yields predictions that match the wave amplitude and displacement thickness from the Swearingen and Blackwelder 1987 experiments. Such agreement matters because it offers an efficient route to forecasting transition without resolving every detail of the three-dimensional flow.

Core claim

Computations using the Parabolised Coherent Structures method successfully reproduce the wave amplitude and displacement thickness observed in the experiments of Swearingen & Blackwelder (1987) by incorporating nonlinear vortex-wave interactions.

What carries the argument

The Parabolised Coherent Structures (PCS) method that incorporates nonlinear vortex-wave interaction into a standard spatial-marching approach.

If this is right

  • Reproduces the finite-amplitude wave evolution in Görtler vortices.
  • Matches key experimental quantities without full three-dimensional resolution.
  • Provides a pathway to predict transition precursors in boundary layer flows.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach may apply to similar secondary instabilities in other wall-bounded flows.
  • Finite-amplitude saturation could be a general feature delaying full turbulence.
  • Engineering predictions for flow over curved surfaces could use this reduced-order method.

Load-bearing premise

The Parabolised Coherent Structures method from the authors' 2025 paper captures the essential nonlinear vortex-wave interactions accurately enough for quantitative agreement with experiment without requiring full three-dimensional resolution.

What would settle it

Direct comparison of PCS-predicted wave amplitudes against new measurements in a Görtler vortex setup with the same parameters as Swearingen & Blackwelder (1987) would confirm or refute the match.

Figures

Figures reproduced from arXiv: 2601.18017 by Kengo Deguchi, Runjie Song.

Figure 1
Figure 1. Figure 1: Displacement boundary layer thickness δ ∗ disp measured at the spanwise locations corresponding to the peaks (up triangle, solid lines) and valleys (down triangle, dashed lines) of the G¨ortler vortices. The symbols are the experimental results taken from figure 9 of SB87, while the lines show our computational results. The inset shows the streamwise velocity from the BRE computation at x ∗ = 90[cm] (in th… view at source ↗
Figure 2
Figure 2. Figure 2: (a) Neutral curve in the x ∗ –f ∗ plane resulting from the secondary instability analysis. The filled circles are the linear critical points used in the PCS computations in panel (b). The open circle corresponds to the analysis in figure 6-(b). (b) Local wavelength of the finite-amplitude wave obtained using the PCS method (the magenta, red, and green lines correspond to f ∗ = 157, 170, and 185 [Hz], respe… view at source ↗
Figure 3
Figure 3. Figure 3: A snapshot of the flow field computed using the PCS. The colourmap at the selected streamwise positions shows the steady streak field u. Red/blue isosurfaces are 20% maximum/minimum of the streamwise vorticity of the wave component, ∂yw˜ − ∂zv˜. z * [cm] y * [cm] 0 0.5 1 1.5 2 0 0.5 1 1.5 2 z * [cm] y * [cm] 0 0.5 1 1.5 2 0 0.5 1 1.5 2 z * [cm] y * [cm] 0 0.5 1 1.5 2 0 0.5 1 1.5 2 Peak 2 Peak 1 (a) (b) (c)… view at source ↗
Figure 4
Figure 4. Figure 4: The flow field at x ∗ = 100 [cm]. The black lines show contours of u at 0.1, 0.2, . . . , 0.9, with the thick line indicating 0.8. The red lines are contours of ˜urms = 0.01, 0.02, and 0.03, with the thick line highlighting 0.02. (a) BRE, (b) PCS, (c) experimental results from figures 11 and 16 of SB87. Conversely, if the starting point is chosen too far downstream, the linear critical point corresponding … view at source ↗
Figure 5
Figure 5. Figure 5: Downstream growth of the wave amplitude. (a) Amplitude of the wave field. Circles denote the experimental results taken from figure 17 in SB87. Lines are the PCS results shown in figure 2-(b). (b) Growth rate σ ∗ = σ/L∗ , where σ(X) = 1 u˜max du˜max dX . For a fair comparison, finite-difference approximations are applied to both the experimental (circles) and PCS (diamonds) results. The line shows the grow… view at source ↗
Figure 6
Figure 6. Figure 6: PCS results for various Reynolds numbers. Panels (a) and (c) show the same computational results, for the wave amplitude and the displacement thickness at the peak vortex location, respectively. The computations are performed from the linear critical point x ∗ ≈ 95 [cm] using Ω ≈ 0.1 (Ω = 0.1080, 0.1216, and 0.1262 for Re = 15625, 31250, and 40000, respectively). Panels (b) and (d) present results correspo… view at source ↗
Figure 7
Figure 7. Figure 7: Analysis near the linear critical point of the second odd mode at f ∗ = 170 [Hz]. (a) The red lines are contours of |uˆ| associated with the neutral eigenfunction found by the secondary instability analysis. The black lines denote the contours of the base flow ¯u at the linear critical point (the same format as figure 4). (b) The PCS computation near the linear critical point x ∗ = 94.88 [cm]. The external… view at source ↗
Figure 8
Figure 8. Figure 8: Comparison with DNS. (a) Amplitude of the wave field. Blue and green circles represent the DNS data reported by Souza (2017) for 120 and 180 [Hz], respectively. The red line shows the PCS result for f ∗ = 180 [Hz]. (b) Growth rate. The circles and diamonds are computed from the DNS and PCS data, respectively. The lines are the secondary instability analysis results. The dashed line is the first odd mode at… view at source ↗
read the original abstract

G\"ortler vortices developing over a concave wall support rapidly oscillating wavelike disturbances through secondary instabilities. Although experiments indicate that the finite-amplitude evolution of these waves acts as a precursor to turbulence transition, accurate and efficient prediction has remained out of reach. We overcome this limitation by using the Parabolised Coherent Structures (PCS) method of Song & Deguchi (2025), which incorporates the nonlinear vortex-wave interaction into a standard spatial-marching approach. Our computations successfully reproduce the wave amplitude and displacement thickness observed in the widely known experiments of Swearingen & Blackwelder (1987).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that the Parabolised Coherent Structures (PCS) method from Song & Deguchi (2025) enables efficient spatial marching of nonlinear vortex-wave interactions in Görtler vortices, successfully reproducing the wave amplitude and displacement thickness observed in the Swearingen & Blackwelder (1987) experiments.

Significance. If the quantitative reproduction holds, the work supplies a reduced-order framework that bridges secondary instability to finite-amplitude saturation without full 3-D resolution, offering a practical advance for predicting transition on concave surfaces.

major comments (2)
  1. [Abstract] Abstract and results presentation: the claim of successful reproduction of experimental wave amplitude and displacement thickness is stated without quantitative error metrics, grid-convergence data, or explicit post-processing details that would allow assessment of the match to Swearingen & Blackwelder (1987).
  2. [Method] Method section: the central computations rest on the PCS formulation developed in the authors' 2025 paper; no independent verification or sensitivity test of the omitted non-parallel and fully three-dimensional terms is supplied here, leaving open whether those terms alter the reported saturation amplitude by O(1).
minor comments (2)
  1. [Figures] Figure captions should explicitly state the streamwise stations at which amplitude and displacement thickness are compared to experiment.
  2. [Notation] Notation for the coherent-structure decomposition should be cross-referenced to the 2025 PCS paper for readers unfamiliar with the framework.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment below and indicate the revisions we intend to implement.

read point-by-point responses

  1. Referee: [Abstract] Abstract and results presentation: the claim of successful reproduction of experimental wave amplitude and displacement thickness is stated without quantitative error metrics, grid-convergence data, or explicit post-processing details that would allow assessment of the match to Swearingen & Blackwelder (1987).

    Authors: We agree that quantitative support is required for a rigorous assessment. In the revised manuscript we will add explicit relative error values for both the saturated wave amplitude and the displacement thickness with respect to the Swearingen & Blackwelder (1987) measurements. We will also include a short grid-convergence study and a clear description of the post-processing steps used to extract these quantities from the PCS fields. revision: yes

  2. Referee: [Method] Method section: the central computations rest on the PCS formulation developed in the authors' 2025 paper; no independent verification or sensitivity test of the omitted non-parallel and fully three-dimensional terms is supplied here, leaving open whether those terms alter the reported saturation amplitude by O(1).

    Authors: The PCS formulation and its validation against fully three-dimensional DNS, including tests of the parabolised approximations, are presented in Song & Deguchi (2025). We will expand the method section to summarise those validation results and to provide a brief scaling argument showing that the neglected non-parallel and three-dimensional terms remain higher-order for the Görtler-vortex parameter range examined here. In addition, we will report a limited sensitivity study with respect to marching step size to confirm that the reported saturation amplitude is robust. revision: partial

Circularity Check

0 steps flagged

No significant circularity; central result is external experimental validation

full rationale

The paper applies the PCS method (cited from Song & Deguchi 2025) to march the nonlinear vortex-wave system and reports quantitative reproduction of wave amplitude and displacement thickness from the independent Swearingen & Blackwelder 1987 experiments. No equation, parameter, or claim reduces by construction to a fitted input, self-definition, or unverified self-citation chain; the match to external data constitutes an independent test of the prior method rather than a tautology. Self-citation of the computational framework is normal and does not trigger circularity because the outcome remains externally falsifiable.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that the PCS spatial-marching framework retains the dominant nonlinear interactions; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Nonlinear vortex-wave interactions can be incorporated into a parabolised spatial-marching framework without loss of essential physics for quantitative prediction.
    This assumption underpins the PCS method's ability to reproduce finite-amplitude evolution beyond linear secondary instability.

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Reference graph

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