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Edge tracking in spatially developing boundary layer flows

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Pith's one-line read A moving box lets edge trajectories in a Blasius boundary layer reveal streak switching, Tollmien–Schlichting waves, and the edge as a divider of two transition routes.

desk verdict A genuinely useful moving-box technique and first streak switching in the Blasius edge, but the validation of the technique is too thin and the coexistence claim is slightly ahead of the data. read the letter →

arxiv 1908.11784 v1 pith:MZYVD73T submitted 2019-08-30 physics.flu-dyn

Miguel Beneitez , Yohann Duguet , Philipp Schlatter , Dan S. Henningson This is my paper · ORCID
classification physics.flu-dyn PACS 47.27.Cn47.20.Ft
keywords edgestatemanifoldBlasiusboundarylayerbypasstransitionTollmien–Schlichtingwavesstreakswitchingmovingboxtechniquebisectionmethod
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 adapts the “edge” concept from parallel shear flows to a spatially developing Blasius boundary layer, i.e. the flat-plate boundary layer whose laminar profile itself becomes linearly unstable downstream. In such a flow the edge—the manifold of states that separate laminar and turbulent behaviour—does not separate two stable attractors, because the laminar state is unstable. Using a computational box that moves with the localized flow structure, the authors track an edge trajectory to $Re_x\approx 9\times 10^5$, roughly ten times beyond the theoretical onset of Tollmien–Schlichting (TS) waves, the two-dimensional waves of the classical route. Along this trajectory the central streak switches its sign repeatedly—streak switching, observed here for the first time in a spatially developing Blasius layer—and at late times TS waves grow spontaneously ahead of the decaying streaks. The paper therefore proposes that in this regime the edge should be read as the boundary between the bypass (streak-breakdown) and classical (TS-wave) routes to turbulence, not as a basin boundary between two attracting states.

What carries the argument

The machinery is the moving box technique combined with bisection on a specially chosen observable. The box translates streamwise at piecewise-constant speed $c_{\mathrm{box}}$ (here alternating between 0 and $0.8U_\infty$), implemented by Galilean transformations of the velocity and by updating the Blasius base flow used in the fringe (damping) region at the outflow, so the localized structure remains inside a domain of length $L_x=6000$ for times that would otherwise need a box twice as long; this extends the tracking horizon roughly threefold compared with earlier Blasius edge computations. The scalar observable used in the bisection is the volume-averaged root-mean-square of the streamwise vorticity, which vanishes for the laminar profile and is exactly zero for spanwise-invariant TS waves in their linear stage, so the bisection continues to bracket the edge even after the base-flow instability has set in. These two elements produce a finite-time edge trajectory that stays spatially localized, scales with the local displacement thickness $\delta^*(x_G(t))$, and lives long enough for streak switching and the later coexistence of TS waves with the streak core.

What would settle it

Perform the same edge-bracketing bisection in a fixed domain of size $(L_x,L_y,L_z)=(12000,60,100)$ with the same resolution and compare, up to $t\approx 4700$, the observable $a(t)$, the streak-switch times, the structure position $x_G(t)$, and the TS wave crest trajectories against the moving-box run; if the trajectories differ by more than the bisection uncertainty ($\delta\lambda/\lambda^*\sim 10^{-4}$) in any of these diagnostics, the equivalence between moving and fixed boxes fails.

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Extended reading notes

Core claim

The central discovery is that both classical and bypass transition mechanisms can be reached from a single finite-time trajectory on the edge manifold, selected by bisection in the amplitude of an initial pair of counter-rotating vortices. At moderate times the trajectory is a localized self-sustained streaky structure that regenerates and switches the sign of its central streak, with no discrete symmetry imposed. At long times, varying the bisection parameter by a relative amount of order $10^{-4}$ decides the outcome: slightly above the edge the streaks break down locally, reproducing bypass transition; slightly below, the streak core decays while TS waves emerge upstream of it, travel at $c\approx 0.32U_\infty$, match the wavelength and phase speed predicted by linear stability analysis, and eventually undergo a secondary Klebanoff-type instability that forms a turbulent spot. The bisection stays meaningful until the two routes become indistinguishable to the chosen observable, which occurs at $t\approx 4700\,\delta_0^*/U_\infty$. The paper’s conclusion is a reinterpretation of the edge as the manifold separating the basins of the two transition routes, both leading to one common turbulent attractor.

Load-bearing premise

The moving box is assumed to reproduce exactly the dynamics of a much longer fixed computational domain over the whole tracking horizon; the paper states a validation against the fixed domain of length 12000 but gives no metrics, so if the frame changes or the time-updated base flow in the fringe distorts the streak core or the wake where TS waves grow, the long-time observations would change.

Editorial extensions

If this is right

  • For free-stream turbulence levels $Tu\gtrsim 2\%$, the achieved horizon $Re_x^{\max}\approx 9\times 10^5$ covers the experimentally observed intermittency range, so the computed edge trajectories can serve as an alternative, unsteady, localized base flow for stability analysis in that regime.
  • Because only one attracting turbulent state is evident, the edge here does not separate two basins of attraction; the edge-tracking observable loses discrimination once the two routes converge, so the bisection time limit is not a numerical artifact but a property of the state space.
  • Streak switching is not an artifact of parallel-flow approximations: it occurs in the spatially developing Blasius layer as well, with a switching period that grows with local boundary-layer thickness.
  • TS waves can be generated spontaneously from the wake of a decaying streak core without any external forcing, and they subsequently undergo a secondary instability and spot formation—so classical and bypass mechanisms can coexist in one unforced simulation.
  • The mixed $Tu<2\%$ regime is precisely where the one-dimensional observable cannot distinguish the routes, motivating observables that explicitly weight TS-wave growth or local methods that avoid global state-space information.

Reading between the lines

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

  • If the moving-box equivalence holds quantitatively, the same frame-tracking idea could be applied to other spatially developing shear flows with a finite-Reynolds-number linear instability—for instance adverse-pressure-gradient boundary layers or channel entrance regions—to locate the route-dividing manifold without computing enormous fixed domains.
  • The choice of observable is what postpones the bisection failure; using an observable that is sensitive to TS waves from the start would presumably make the two routes distinguishable earlier, possibly sharpening the definition of the route boundary at the cost of a shorter tracking horizon.
  • The paper’s reinterpretation suggests testable predictions for experiments: in the weak-bypass regime, turbulent spots should appear either from streak breakdown or from TS wavepackets, and the boundary between those two origins should sit on a state-space manifold whose finite-time shadow is the computed edge trajectory.
  • Because the edge trajectory is computed only for finite times, its status as a genuine invariant set in the $t\to\infty$ limit is open; tracking the rescaled switching cycle over more periods in self-similar variables would indicate whether the streak-switching recurrence is asymptotic or a transient feature.
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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 / 5 minor

Summary. This paper presents large-scale edge-tracking simulations in a spatially developing Blasius boundary layer, using a novel moving-box technique that advects the computational domain downstream at piecewise-constant speed. The authors report that edge trajectories can be bracketed for about 4700 time units, observe streak-switching dynamics (claimed as a first for the Blasius boundary layer), and find that slightly off-edge trajectories evolve either into streak breakdown (bypass route) or into Tollmien–Schlichting wave growth (classical route) at later times. The paper also compares the measured TS wave phase speed with linear stability theory and uses the results to argue for a reinterpretation of the edge as a separator of the two transition routes, with practical relevance to bypass-transition experiments.

Significance. If the moving-box method is truly equivalent to a much longer fixed-domain computation, this is a significant methodological contribution that enables long-time studies of localized coherent structures in spatially developing flows. The streak-switching observation extends earlier results from the asymptotic suction boundary layer and from symmetric Blasius computations to an unconstrained Blasius boundary layer, and the quantitative TS wave speed comparison is a convincing check. The conceptual interpretation of the edge in a flow whose base state is linearly unstable is thought-provoking and likely to stimulate further work. However, the strength of these claims is currently limited by the very brief, non-quantitative validation of the moving-box technique and by the fact that the coexistence of the two routes is demonstrated only on off-edge trajectories beyond the tracking limit.

major comments (2)
  1. [Section 2.2] The sentence 'The present computation of the edge trajectory using the moving box technique was validated against the case of a non-moving domain of size (Lx,Ly,Lz)=(12000,60,100)' is the only evidence for the central enabling assumption, yet no quantitative comparison, convergence data, or description of the validation procedure is given. Because the Blasius base flow is not Galilean-invariant and the fringe forcing becomes time-dependent through x0(t) (Appendix A), the equivalence between the moving box and a fixed long domain is a nontrivial assumption, not a routine change of reference frame. All long-time results -- the streak-switching cycles and the apparent coexistence of TS waves -- are produced with the moving box, so the paper's main claims depend on this equivalence. Please provide a quantitative validation: e.g., compare time series of the bisection observable a(t), the center-of-mass position x_G(t), and the two bracketing trajectories over t in [0,4700] between the moving-box run and the fixed-domain run, and report the level of agreement. The statement that 'different histories of cbox yield exactly the same results' should also be substantiated with actual trajectories.
  2. [Section 3.2 and abstract] The abstract states that 'At long enough times, TS waves co-exist with the coherent structure characteristic of edge trajectories,' but the TS wave growth is actually observed on an off-edge trajectory (the δλ<0 case, Fig. 6 right) after the streak structure begins to decay, and the edge trajectory itself is only tracked up to t≈4700. The paper's own discussion in Section 4 acknowledges this limitation ('we have no further information on the nature (bypass or classical) of the trajectories within these 2%'). As it stands, the evidence supports coexistence of the two routes for trajectories infinitesimally below the edge, not on the edge trajectory itself. Please either (i) provide evidence that the edge trajectory itself, tracked further with a different observable or method, exhibits simultaneous streaks and TS waves, or (ii) rephrase the conclusion (and abstract) to state that the two routes coexist in the neighbourhood of the edge, which is what the data show.
minor comments (5)
  1. [Section 2.3] The text 'In the present study we used und by bisection, the two dotted lines are the observable bounds...' contains an incomplete/garble ('und by bisection'); please restore the intended sentence.
  2. [Section 1] The phrase 'The structure of the paper is at follows' should read 'is as follows.'
  3. [Section 4] Please define how the '2% or less' relative difference in the main observable is computed; this is the operational criterion for the tracking limit and should be specified precisely.
  4. [Appendix A] The word 'substracting' should be 'subtracting.'
  5. [References] The reference 'Jordison 1970' is likely a misspelling of 'Jordinson' (J. Fluid Mech. 43, 801-811); please verify the correct spelling.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the edge trajectories, streak switching, and TS-wave coexistence are DNS observations checked against independent linear-stability data.

full rationale

The paper's central claims are numerical observations from DNS, not closed-form derivations, so there is no reduction of a prediction to an input by construction. The scalar observable a(t), defined in equation (2.1) as a normalized integral of |omega_x|^2, is explicitly chosen to ignore Tollmien-Schlichting waves in their linear stage; the paper states 'This observable is ... zero when the aforementioned profile is perturbed by TS waves in their linear stage.' That is a transparent modeling choice, not a hidden definition of the result. The TS waves themselves are identified by comparing wavelength and phase speed with independent linear stability analysis ('matching quantitatively the speed of the TS waves obtained from linear stability analysis of the frozen Blasius profile at their onset'), which is an external benchmark rather than a fitted input. The one-sentence moving-box validation ('The present computation of the edge trajectory using the moving box technique was validated against the case of a non-moving domain of size ...') is under-documented and would be a robustness concern, but it is not circular: the moving-box equivalence is an assumption about numerical domain treatment, not an equation that defines the observed edge dynamics. Self-citations to Duguet et al. (2012) and Khapko et al. (2016) provide context and methodology, but the streak-switching dynamics and the coexistence of bypass and classical transition are computed in the present simulations, not imported by citation. No fitted parameter is renamed as a prediction, and no uniqueness theorem is invoked to force the chosen interpretation. Overall, the derivation chain is self-contained with respect to the claims made, and any weaknesses are evidentiary rather than circular.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central results rest on established governing equations and linear stability theory, plus three hand-chosen numerical parameters (thresholds and box speed) and one fit used for interpretation. No new physical entities are postulated. The weakest assumption is the moving-box equivalence, listed among axioms as a domain assumption.

free parameters (3)
  • Bisection thresholds aL, aT = aL = 8.74e-5, aT = 2.68e-3
    Thresholds for the observable a(t) in eq. (2.1), chosen by hand to classify trajectories as laminar or turbulent. The paper notes different aL values were tested without significant influence, so the central claim is not sensitive to them.
  • Box advection speed cbox = 0 or 0.8 (piecewise constant)
    Galilean frame speed chosen to keep the coherent structure inside the computational domain. Different histories of cbox are claimed to give identical results (Section 2.2); it is a numerical convenience, not fitted to the physics.
  • Edge trajectory drift fit (xG0, vGx) = xG0 = 70, vGx = 0.625
    Linear fit to the computed center-of-mass position xG(t) (eq. 3.1, Fig. 8 left). Used to compare advection speed with the TS wave speed and to motivate the rescaling; it is a measured output rather than a model input.
assumptions (4)
  • standard math The incompressible Navier-Stokes equations govern the flow, solved in velocity-vorticity form with pseudo-spectral SIMSON code.
    Governing equations and solver in Section 2.1 and Appendix A (eqs. A1-A2).
  • domain assumption The Blasius profile is an excellent approximation to the steady base flow for x >> 1, and is used as the reference and fringe-inflow profile.
    Section 2.1 and Appendix A; standard boundary-layer approximation.
  • domain assumption The Blasius base flow is linearly unstable to Tollmien-Schlichting waves with onset at Re_delta* about 520 (Jordison 1970), and the wave speed/wavelength from frozen-profile linear stability is used to identify the simulated waves as TS waves.
    Section 3.2.1 and Fig. 8 (right); identification relies on external linear-stability results.
  • domain assumption A codimension-one edge manifold separates two dynamical fates in this spatially developing flow, at least on finite time horizons, and can be approximated by bisection on the single observable a(t).
    Section 2.3 and Section 4; the paper itself revises this picture at long times because the base flow is linearly unstable.

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Pith. "Pith review of Edge tracking in spatially developing boundary layer flows." pith.science (2026). https://pith.science/paper/MZYVD73T

@misc{pith2026190811784,
  author       = {Pith},
  title        = {Pith review of: Edge tracking in spatially developing boundary layer flows},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MZYVD73T}},
  note         = {Machine review of arXiv:1908.11784}
}
read the original abstract

Recent progress in understanding subcritical transition to turbulence is based on the concept of the edge, the manifold separating the basins of attraction of the laminar and the turbulent state. Originally developed in numerical studies of parallel shear flows with a linearly stable base flow, this concept is adapted here to the case of a spatially developing Blasius boundary layer. Longer time horizons fundamentally change the nature of the problem due to the loss of stability of the base flow due to Tollmien--Schlichting (TS) waves. We demonstrate, using a moving box technique, that efficient long-time tracking of edge trajectories is possible for the parameter range relevant to bypass transition, even if the asymptotic state itself remains out of reach. The flow along the edge trajectory features streak switching observed for the first time in the Blasius boundary layer. At long enough times, TS waves co-exist with the coherent structure characteristic of edge trajectories. In this situation we suggest a reinterpretation of the edge as a manifold dividing the state space between the two main types of boundary layer transition, i.e. bypass transition and classical transition.

Figures

Figures reproduced from arXiv: 1908.11784 by the authors.

Figure 1
Figure 1. Sketch of the flow geometry including the moving box. condition is used in the free-stream (y = Ly) for the three velocity components. The local resolution is comparable to the one used in Duguet et al. (2012). 2.2. The moving box technique Numerical simulation of localized coherent structures usually require computational domains at least one order of magnitude larger than their typical size (Duguet et al. 2009). W… view at source ↗
Figure 2
Figure 2. Observable a(t) vs. t during application of the bisection algorithm over moderate time horizons. The thick line (blue online) represents the edge trajectory found by bisection, the two dotted lines are the observable bounds a = aL = 8.74 × 10−5 and aT = 2.68 × 10−3 (see text). instance the initial and final frames in figure 4. Visualisation of the λ2 criterion (Jeong & Hussain 1995) in figure 3 shows the presence of… view at source ↗
Figure 3
Figure 3. Three-dimensional perspective view from above of the edge trajectory at times t = 1700, 2750 and 3550 (from top to bottom), showing isosurfaces of streamwise perturbation velocity with respect to spanwise mean with values 0.06 and −0.08 (red and blue, respectively), together with vortical structures λ2 = −1.5 · 10−5 (green). Flow from left to right. The black lines are separated by a distance of 200 in units of δ ∗ … view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Illustration of a streak switching event along the edge trajectory at t = 3550. (y, z) cross-sections of ux. From left to right x = 2140, 2420 and 2980. location. The local displacement thickness δ = δ ∗ (xG), evaluated at this time-dependent location, provides the sca…
Figure 5
Figure 5. Figure 5: (z, t) space-time diagram of ux(x, y, z, t) along the edge trajectory, in a frame moving with the center of mass located at x = xG(t) and y = yp, of the perturbation streamwise velocity. Both z and t are rescaled by the local boundary layer thickness δ/δ∗ 0 , causing t…
Figure 6
Figure 6. Figure 6: (z, t) space–time diagrams of ux(x, y, z, t) evaluated at x = xG(t) and y = yp, for two initially nearby trajectories bracketing the edge trajectory. Both z and t are rescaled by the local boundary layer thickness δ/δ∗ 0 . Left: bypass transition route. Right: streak d…
Figure 7
Figure 7. Figure 7: Space-time diagram of ux(x, y, z, t) for z = −10 and y = yp for the trajectory away from the edge manifold, as displayed in figure 6 (right). The streaks decay while the TS wavepacket grows in amplitude, forming a turbulent spot. The red cross indicates the initial pos…
Figure 8
Figure 8. Figure 8: Streamwise propagation velocities. Left: position of the center of mass of the coherent structure xG(t) for a trajectory below the edge versus time. The vertical line indicates the time limitation of edge tracking (see text). Circles: data, solid line (blue online): li…
Figure 9
Figure 9. Figure 9: Three-dimensional phase portrait using global variables Ωx, Ωy and W. The initial condition associated lies at approximately (Ωx, Ωy, W) = (1.72, 6.62, 1.80)×10−4 but the first 500 time units are not shown. The blue dot at the (0, 0, 0) is the laminar state. The red tr…
Figure 10
Figure 10. Figure 10: Zoom on the edge region using only Ωx and Ωy (same data as [PITH_FULL_IMAGE:figures/full_fig_p012_10.png]
Figure 11
Figure 11. Figure 11: Observable a(t) vs. t during application of the bisection algorithm. The green vertical line marks the maximum time for the edge tracking algorithm using that observable, and the two dotted lines stand for the observable bounds a = aL and aT (see text). trajectories w…
Figure 12
Figure 12. Figure 12: T u(%) vs. Rex for experimental bypass transition data (Shahinfar & Fransson 2011), showing intermittency of 10% and 90% (thick lines) and 50% (dashed line). The arrows mark the Rex-limitation of the bisection algorithm. For larger times and Rex, mixed transition is e…

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