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Direct numerical simulation of complete transition to turbulence with a fluid at supercritical pressure

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A single two-dimensional wave, amplified by the Mode-II instability around the Widom line, can carry a supercritical boundary layer through complete transition to turbulence, with turbulent heat transfer still predictable from the…

desk verdict First full DNS of transition at supercritical pressure; the noise-seeded K-type claim needs calibration, but the core billow pathway is solid and worth refereeing. read the letter →

arxiv 2506.06703 v1 pith:7QGMU2EI submitted 2025-06-07 physics.flu-dyn physics.comp-ph

Pietro Carlo Boldini , Benjamin Bugeat , Jurriaan W.R. Peeters , Markus Kloker , Rene Pecnik This is my paper
classification physics.flu-dynphysics.comp-ph MSC 76F0676F4076F6576N15 PACS 47.20.Ft47.27.Cn47.27.nb
keywords supercriticalfluidsMode-IIinstabilitylaminar-turbulenttransitionpseudo-boilingWidomlinedirectnumericalsimulationvariable-propertyscalingReynoldsanalogy
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 uses direct numerical simulation to follow a heated flat-plate boundary layer at supercritical pressure all the way from laminar flow to full turbulence. In the transcritical regime, where the boundary layer crosses the pseudo-critical (Widom) line, the Mode-II instability amplifies a single two-dimensional wave far more strongly than the usual low-speed instability mode. The nonlinear stage is dominated by spanwise billows and travelling near-wall flow reversals, and the central claim is that one 2-D wave can then amplify the background noise floor enough to trigger a naturally selected K-type breakdown to turbulence, without any intentional oblique-wave forcing. In the fully turbulent state, the paper shows that variable-property velocity scaling collapses the profiles and that a Reynolds-analogy-based estimate predicts skin friction and heat transfer. If true, this gives a practical route for predicting transition and heat transfer in supercritical-fluid systems.

What carries the argument

The load-bearing object is the Mode-II instability of a boundary layer at transcritical supercritical pressure: an inviscid mode tied to a minimum of kinematic viscosity at the pseudo-critical (Widom) line and to a generalised inflection point in the density-weighted velocity profile. In the linear stage it appears as two phase-locked, out-of-phase vorticity waves produced by shear and baroclinic effects; nonlinearly it produces billow roll-ups that periodically displace the Widom line, driving near-wall flow reversal and strong local shear layers. That shear-layer mechanism converts a single 2-D wave plus background noise into a full K-type breakdown, and the same flow then serves as a test case for variable-property turbulent scaling.

What would settle it

Run the same noise-initiated transcritical case with the solver's numerical noise reduced or reshaped, and with controlled broadband disturbances of measured amplitude injected instead; if transition moves downstream beyond the computational domain or does not occur when the noise amplitude is lowered, the claim that a single 2-D wave triggers transition by amplifying background noise is falsified.

Watch

Extended reading notes

Core claim

The central discovery is that Mode-II instability converts a single, moderate-amplitude two-dimensional disturbance into a complete laminar-to-turbulent transition, provided the wall temperature crosses the Widom line. The mechanism proceeds through a Kelvin-Helmholtz-like train of billows that displaces the Widom line, creating travelling near-wall flow reversals; these reversals generate localised shear layers that are inviscidly unstable, so three-dimensional modes grow from numerical background noise without any forced oblique wave. The resulting breakdown follows a fundamental (K-type) scenario with peak-valley splitting rather than the classic subharmonic H-type route. After transition, the turbulent boundary layer obeys semi-local variable-property scaling, and a variable-Prandtl enthalpy relation together with the Reynolds analogy gives a predictive estimate of skin friction and Stanton number.

Load-bearing premise

The transcritical breakdown claimed to start from background noise depends on the numerical noise floor in the DNS: its amplitude and spectral content are not quantified or matched to any physical disturbance environment, so the early transition could shift downstream or vanish under a different noise level.

Editorial extensions

If this is right

  • Transition in transcritical supercritical boundary layers can be triggered by a single 2-D wave, with no need for controlled oblique-wave forcing.
  • The natural path to turbulence is fundamental (K-type) breakdown, with aligned lambda-like vortices forming inside billow roll-ups rather than the staggered H-type pattern.
  • Subharmonic H-type forcing still works, but it is delayed, and the resulting breakdown is driven by flow-reversal shear layers rather than classic secondary instability.
  • The semi-local velocity transformation collapses the transcritical turbulent profiles, so variable-property scaling applies when density and viscosity fluctuations are moderate.
  • Skin friction and heat transfer in the fully turbulent boundary layer can be estimated from the variable-Prandtl enthalpy relation combined with the Reynolds analogy, avoiding the need for costly DNS in engineering prediction.

Reading between the lines

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

  • Because the early transition in the noise-initiated case is attributed to an unquantified numerical background noise floor, a testable extension would be to add controlled broadband noise of measured amplitude and spectrum and observe whether transition moves downstream or disappears.
  • If the Mode-II mechanism is tied to a kinematic-viscosity minimum, so other non-polar supercritical fluids with the same property topology should exhibit the same single-wave transition route; this could be checked by linear stability analysis and DNS for fluids beyond the Van der Waals model used here.
  • If the Reynolds analogy indeed holds across transcritical conditions, engineering heat-transfer correlations for supercritical heat exchangers and power cycles could be simplified substantially, though the present evidence is limited to one Reynolds number and one wall-temperature ratio.
  • The travelling flow-reversal zones behave like unsteady separation bubbles, which suggests that experimental techniques developed for laminar separation bubbles, such as time-resolved wall-temperature or velocity imaging, could be adapted to detect this transition route in supercritical flows.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper presents direct numerical simulations of laminar-to-turbulent transition in a heated zero-pressure-gradient flat-plate boundary layer with a fluid at supercritical pressure, comparing a liquid-like subcritical case (Tw095) with a transcritical case (Tw110) that crosses the pseudo-critical (Widom) line. The authors analyze the linear and nonlinear evolution of the newly identified Mode-II instability, show that nonlinear 2-D forcing produces billow-like structures and near-wall flow reversal, and examine 3-D breakdown with finite and infinitesimal oblique-wave forcing. They report a K-type breakdown triggered by numerical background noise in the Tw110-IA case and argue that a single 2-D wave can therefore promote transition in the transcritical regime. In the turbulent region, they show that the Patel-Trettel-Larsson velocity transformation collapses the profiles and develop an estimation model for skin friction and Stanton number based on van Driest's variable-Prandtl-number enthalpy relation.

Significance. If the main claims hold, this is a substantial contribution to the stability and transition literature for supercritical fluids. The paper provides the first complete DNS of transition with a highly non-ideal fluid, demonstrates the nonlinear consequences of Mode-II instability, identifies an APG-like breakdown mechanism caused by pseudo-boiling, and offers a practical heat-transfer estimation route. The work is strengthened by careful LST-DNS comparisons of eigenfunctions, growth rates, and phase speeds, a grid-resolution study for the transcritical case (Appendix A), detailed modal and structural analyses of K- and H-type routes, and publicly available source code for the engineering estimation model. The central transition pathway for finite-amplitude 3-D forcing is well supported by the DNS evidence. However, the stronger claim that transition can be triggered by a single 2-D wave amplifying background noise rests on a case whose noise floor is not quantified, and the engineering model is partly calibrated against the same DNS it is asked to predict.

major comments (3)
  1. [§5.1, Fig. 11(d), Table 2] The abstract and §7 claim that in the transcritical regime a single 2-D wave can 'strongly amplify background noise' and trigger transition. This claim rests on case Tw110-IA, where the 3-D forcing amplitude is A3-D = 1.0e-8 and the text states that mode (1,1) 'arises initially due to numerical background noise'. The amplitude and spectral content of that noise are never measured, and no controlled variation of the noise level or composition is reported. Because the spanwise domain is restricted to one fundamental wavelength and the grid-resolution study in Appendix A is performed for Tw110-LA rather than for Tw110-IA, the observed K-type breakdown could in principle be a grid, round-off, or scheme-dependent artifact. Quantifying the noise floor, preferably with a resolution study and a small-amplitude sweep of 3-D disturbances, is necessary before the 'transition from noise alone' pathway is established as a physical receptivity result.
  2. [§6.2, Eqs. (6.2)-(6.5), Figs. 21-25] The 'predictive model' for skin friction and heat transfer is presented as a tool, but its inputs include the DNS-computed mixed Prandtl number P r_m and the DNS-informed assumption P r_m ≈ P r. The Stanton number prediction is over-predicted by about 30% at R e_theta = 881, and the close agreement obtained via the Reynolds analogy uses C'_f from the same estimation. For a reader, it is unclear which quantities are truly predicted from free-stream and wall conditions alone and which are fitted or extracted from the DNS. Please state this limitation explicitly and distinguish the consistency-check nature of the comparison from a fully independent prediction.
  3. [Appendix A, Table 3] The grid-resolution sensitivity analysis is reported only for Tw110-LA, whose 3-D forcing is finite and large (A3-D = 8.5e-5). The case most sensitive to numerical background noise, Tw110-IA, has a 3-D forcing amplitude four orders of magnitude smaller, so a coarsening test for that case would be the relevant check for the central 'noise-amplification' claim. Without it, the robustness of the Tw110-IA breakdown to resolution and numerical scheme remains unverified.
minor comments (5)
  1. [§4.1, Eq. (4.1)] The typeset vorticity-perturbation equation contains stray markup symbols in the term annotations (e.g., '⌟⟨⟨⟪rl⟫⌟⟨l⟪2'), which obscure the definitions of S_xi, C_xi, and B_xi. Please re-typeset the equation and its labels.
  2. [§6.2, Fig. 23] The statement that Jensen's inequality implies 'χ ≠ χ(h,p)' is imprecise; the point is presumably that the Reynolds average of the thermophysical property evaluated at the Favre-averaged enthalpy differs from the property evaluated at the mean enthalpy due to fluctuation correlations. Please rephrase for clarity.
  3. [§4.2, Figs. 9, 15, and 17] The green shading used to indicate the Widom-line region is defined as 'between 98% max{c_p} and max{c_p}' in Fig. 9 but as 'between 95% max{c_p} and max{c_p}' in Figs. 15 and 17. Please make the threshold consistent and justify the choice.
  4. [§6.1, Fig. 20] The statement that the Patel et al. (2016) transformation 'collapses well the velocity profiles' should be rephrased as 'collapses the velocity profiles well' or similar, and the collapse should be quantified (e.g., maximum deviation from the log law) rather than stated qualitatively.
  5. [§3.2, Eq. (3.1)] The disturbance function f(x) is given without an indication of its peak value; stating that the integral or maximum of f(x) is normalized to unity would help the reader connect the specified A2-D and A3-D values to the actual forcing amplitude.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: transition claims are DNS observations, and the §6 estimation model is a consistency check; minor self-citations are not load-bearing.

full rationale

The paper's central transition and breakdown findings are derived from direct numerical simulation and are not circular. Case Tw110-IA uses an infinitesimal 3-D forcing amplitude (A3-D = 1.0e-8, Table 2), and the authors state that mode (1,1) 'arises initially due to numerical background noise' (§5.1, Fig. 11d). This noise floor is not quantified or calibrated to a physical disturbance environment, which is a missing external support for the 'transition from noise alone' claim, but it is not a reduction of the result to its inputs: no parameter is fitted to the observed breakdown location, and the breakdown is observed rather than derived from the noise assumption. The §6 estimation model is a consistency check against the same DNS rather than a fitted prediction: the skin-friction prediction uses a Johnson-King/Coles mixing-length formulation with standard constants and the VdW/JST property models; the assumption Prm ≈ Pr is tested against the same DNS, and the Reynolds-analogy-based Stanton estimate uses the classical law s ≈ 1 as verified in Fig. 24, not a parameter fitted to the DNS Stanton number. Self-citations (e.g., Boldini et al. 2024a for K-type mode (0,1) dominance, and Bugeat et al. 2024 for the vorticity mechanism) support interpretation and context, but the load-bearing observations are made and compared within the present DNS/LST framework. No circular step meets the quote-and-reduction standard, so the score reflects only minor, non-load-bearing self-citations.

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

No new physical entities are introduced. The load-bearing assumptions are the representativeness of the Van der Waals EoS, the use of numerical noise as a physical disturbance proxy, and the standard boundary-layer idealizations. The free-parameter list is empty because the forcing amplitudes are simulation controls, not fitted parameters, and the turbulence-model constants come from prior literature.

assumptions (5)
  • domain assumption The Van der Waals cubic equation of state with JST transport models accurately represents the relevant supercritical thermodynamics and transport for the instability and breakdown mechanisms.
    The paper uses this reduced EoS throughout and claims generality, but provides no comparison to a real-fluid EoS (e.g., GERG-2008) for the simulated conditions.
  • ad hoc to paper Numerical background noise in the DNS acts as a proxy for physical free-stream turbulence or other 3-D disturbances in the IA case.
    Section 5.1 states that mode (1,1) arises from numerical background noise; the noise amplitude and spectrum are not calibrated to any measured disturbance environment.
  • domain assumption The flow is two-dimensional in the mean, spanwise-periodic, and the initial boundary layer is a ZPG self-similar solution.
    Standard canonical setup; the paper validates the steady state in Appendix B, but the ZPG and periodicity restrict the conclusions to flat-plate conditions.
  • standard math Stokes hypothesis (zero bulk viscosity) and Fourier's law of heat conduction hold for the supercritical fluid.
    Methodology §2.1 states these assumptions; they are conventional for this class of DNS.
  • domain assumption The recovery factor for the adiabatic wall enthalpy is r = Pr^(1/3) = 1.
    Section 5.3 states this assumption, asserting it has been verified for the laminar boundary layer, but it is applied into the turbulent regime.

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Pith. "Pith review of Direct numerical simulation of complete transition to turbulence with a fluid at supercritical pressure." pith.science (2026). https://pith.science/paper/7QGMU2EI

@misc{pith2026250606703,
  author       = {Pith},
  title        = {Pith review of: Direct numerical simulation of complete transition to turbulence with a fluid at supercritical pressure},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7QGMU2EI}},
  note         = {Machine review of arXiv:2506.06703}
}
abstract

The objective of this work is to investigate the unexplored laminar-to-turbulent transition of a heated flat-plate boundary layer with a fluid at supercritical pressure. Two temperature ranges are considered: a subcritical case, where the fluid remains entirely in the liquid-like regime, and a transcritical case, where the pseudo-critical (Widom) line is crossed and pseudo-boiling occurs. Fully compressible direct numerical simulations are used to study (i) the linear and nonlinear instabilities, (ii) the breakdown to turbulence, and (iii) the fully developed turbulent boundary layer. In the transcritical regime, two-dimensional forcing generates not only a train of billow-like structures around the Widom line, resembling Kelvin-Helmholtz instability, but also near-wall travelling regions of flow reversal. These spanwise-oriented billows dominate the early nonlinear stage. When high subharmonic three-dimensional forcing is applied, staggered $\Lambda$-vortices emerge more abruptly than in the subcritical case. However, unlike the classic H-type breakdown under zero pressure gradient observed in ideal-gas and subcritical regimes, the H-type breakdown is triggered by strong shear layers caused by flow reversals -- similar to that observed in adverse-pressure-gradient boundary layers. Without oblique wave forcing, transition is only slightly delayed and follows a naturally selected fundamental breakdown (K-type) scenario. Hence, in the transcritical regime, it is possible to trigger nonlinearities and achieve transition to turbulence relatively early using only a single two-dimensional wave that strongly amplifies background noise. In the fully turbulent region, we demonstrate that variable-property scaling accurately predicts turbulent skin-friction and heat-transfer coefficients.

Figures

Figures reproduced from arXiv: 2506.06703 by the authors.

Figure 1
Figure 1. Reduced thermodynamic and transport properties at [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Reduced temperature-pressure (𝑇r – 𝑝r) diagram with isolines of reduced density 𝜌r: isobar at 𝑝r,∞ = 1.10 with cases at supercritical pressure of table 1, i.e. Tw095 ( ) and Tw110 ( ). The saturation line and pseudo-critical (Widom) line, i.e. locus of the maxima of the isobaric specific heat capacity, follow the approximate generalised equation 𝑝r = exp{(𝑇r − 1)𝐴VdW /min(𝑇r, 1)} with 𝐴VdW = 4 (Banuti 2015). Non-ide… view at source ↗
Figure 3
Figure 3. Laminar profiles for the considered cases: (a) temperature [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (26 more)
Figure 4
Figure 4. Figure 4: Growth-rate (−𝛼i) contours in the 𝑅𝑒–𝐹 stability diagram: (a) TadIG, (b) Tw095, and (c) Tw110 (Mode I and II). The dotted blue lines in (b,c) represent the ideal-gas neutral stability at equal 𝑇 ∗ w /𝑇 ∗ ∞-ratios. In the inset of (c), the wide frequency band of Mode I …
Figure 5
Figure 5. Figure 5: Case Tw095 (a,b) and Tw110 (c,d): (a,c) wall-normal eigenfunctions (lines DNS [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: Case Tw110. Terms of the vorticity perturbation equation ( [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: Case Tw095 with 𝐴 (1,0) 2-D = 7.5 × 10−3 : (a) maximum wall-normal mass-flux amplitude for mode (1,0) (solid line), (2,0) (dash-dotted line), and (3,0) (dashed line); streamwise velocity (b) and density (c) perturbations as a function of the wall-normal coordinate 𝑦/𝛿9…
Figure 8
Figure 8. Figure 8: Case Tw110: (a,c) 𝐴 (1,0) 2-D = 7.5 × 10−4 and (b,d) 𝐴 (1,0) 2-D = 7.5 × 10−3 ; (a,b) maximum wall-normal mass-flux amplitude for mode (1, 0) (solid line), (2, 0) (dash￾dotted line), (3, 0) (dashed line), (4, 0) (dotted line), (5, 0) (solid line with ▿), and (6, 0) (so…
Figure 9
Figure 9. Figure 9: Case Tw110. Instantaneous contours at 𝑇/𝑇0 = 0, where 𝑇0 = 2𝜋/𝜔0 (fundamental frequency 𝜔0), with 𝐴 (1,0) 2-D = 7.5 × 10−3 : (a) reduced pressure fluctuation 𝑝 ′ 𝑟 = 𝑝 ∗′/𝑝 ∗ c , (b) density fluctuation 𝜌 ′ , (c) vorticity Ω, (d) streamwise velocity 𝑢, with boundary￾la…
Figure 10
Figure 10. Figure 10: Case Tw110 for 𝐴 (1,0) 2-D = 7.5 × 10−3 . Wall-normal slice at 𝑅𝑒 = 802 showing: (a,b) instantaneous streamwise velocity 𝑢 and reduced specific heat at constant pressure 𝑐p,r/𝑐p,r,∞ at time periods 𝑡/𝑇0 = [0, 0.25, 0.5, 0.95], where 𝑇0 = 2𝜋/𝜔0 is the fundamental forci…
Figure 11
Figure 11. Figure 11: Streamwise evolution of the 𝑦-maximum of (𝜌𝑢) ′ for the most relevant modes (𝜔/𝜔2-D, 𝛽/𝛽0) for case: (a) Tw095-IA, (b) Tw095-LA, (c) Tw110-LA, and (d) Tw110- IA. The minimum and maximum values of the time- and spanwise-averaged skin-friction coefficient are indicated …
Figure 12
Figure 12. Figure 12: Instantaneous isosurfaces of the 𝑄-criterion, coloured by the streamwise velocity magnitude: (a) case Tw095-LA (𝑄 = 0.015) at 𝑡/𝑇0 = 0, (b) case Tw110-LA (𝑄 = 0.020) at 𝑡/𝑇0 = 0.5, and (c) case Tw110-IA (𝑄 = 0.020) at 𝑡/𝑇0 = 0.5. 𝑇0 is the period of the fundamental wa…
Figure 13
Figure 13. Figure 13: Contours of instantaneous streamwise velocity ( [PITH_FULL_IMAGE:figures/full_fig_p022_13.png]
Figure 14
Figure 14. Figure 14: Case Tw110-LA. Instantaneous isosurfaces of (a) spanwise vorticity [PITH_FULL_IMAGE:figures/full_fig_p023_14.png]
Figure 15
Figure 15. Figure 15: Case Tw110-LA. Instantaneous contours of spanwise vorticity [PITH_FULL_IMAGE:figures/full_fig_p024_15.png]
Figure 16
Figure 16. Figure 16: Case Tw110-IA. Instantaneous isosurfaces of (a) spanwise vorticity [PITH_FULL_IMAGE:figures/full_fig_p025_16.png]
Figure 17
Figure 17. Figure 17: Case Tw110-IA. Instantaneous contours of streamwise vorticity [PITH_FULL_IMAGE:figures/full_fig_p026_17.png]
Figure 18
Figure 18. Figure 18: Contours of the time- and spanwise-averaged (a,b) streamwise velocity [PITH_FULL_IMAGE:figures/full_fig_p027_18.png]
Figure 19
Figure 19. Figure 19: Time- and spanwise-averaged: (a) skin-friction coefficient [PITH_FULL_IMAGE:figures/full_fig_p028_19.png]
Figure 20
Figure 20. Figure 20: Wall-normal profiles of the transformed streamwise velocity using (a) [PITH_FULL_IMAGE:figures/full_fig_p030_20.png]
Figure 21
Figure 21. Figure 21: Case Tw095-LA (𝑅𝑒 𝜃 = 1387) in orange and Tw110-LA (𝑅𝑒 𝜃 = 881) in red: (a) mixed Prandtl number 𝑃𝑟m, (b) turbulent Prandtl number 𝑃𝑟t, and (c) mean molecular Prandtl number 𝑃𝑟 and 𝑐p/𝑐p,∞ (red dash-dotted line) for case Tw110-LA. Next, we focus on estimating the mean…
Figure 22
Figure 22. Figure 22: Reynolds-averaged mean enthalpy from van Driest (1955) in black for (a) Tw095- LA at 𝑅𝑒 𝜃 = 1387 (DNS profile in orange) and (b) Tw110-LA at 𝑅𝑒 𝜃 = 881 (DNS profile in red). In grey, the relation of Walz (1969) as ℎ/ℎ∞ = ℎw/ℎ∞ + (ℎaw − ℎw)(𝑢/𝑢∞)/ℎ∞ − 𝑟𝑢 2 ∞(𝑢/𝑢∞) 2 /(…
Figure 23
Figure 23. Figure 23: Estimated mean (a,b) temperature 𝑇/𝑇∞, (c,d) density 𝜌/𝜌∞, (e,f) viscosity 𝜇/𝜇∞, and (g,h) streamwise velocity 𝑢 + profiles (dashed grey lines) compared to DNS results (solid lines, case Tw095-LA (𝑅𝑒 𝜃 = 1387) in orange and case Tw110-LA (𝑅𝑒 𝜃 = 881) in red). eddy vis…
Figure 24
Figure 24. Figure 24: Reynolds analogy factor 𝑠 = 2𝑆𝑡/𝐶f as a function of the momentum￾thickness Reynolds number 𝑅𝑒 𝜃 : case Tw095-LA (orange) and Tw110-LA (red). Solid lines correspond to the DNS results, while dashed lines represent the turbulent Reynolds analogy factor, 𝑠 = S∞, accordin…
Figure 25
Figure 25. Figure 25: Case Tw095-LA (orange) and Tw110-LA (red): (a) skin-friction coefficient [PITH_FULL_IMAGE:figures/full_fig_p035_25.png]
Figure 26
Figure 26. Figure 26: Case Tw110-LA: streamwise development of the [PITH_FULL_IMAGE:figures/full_fig_p038_26.png]
Figure 27
Figure 27. Figure 27: Case Tw110-LA: time- and spanwise-averaged (a) skin-friction coefficient and [PITH_FULL_IMAGE:figures/full_fig_p039_27.png]
Figure 28
Figure 28. Figure 28: 2-D DNS laminar profiles for cases Tw095 and Tw110: (a) streamwise velocity, [PITH_FULL_IMAGE:figures/full_fig_p039_28.png]
Figure 29
Figure 29. Figure 29: Comparison between low-amplitude DNS (lines) and LST (symbols) for a 2-D [PITH_FULL_IMAGE:figures/full_fig_p040_29.png]

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