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Free-stream turbulence is a governing parameter in adverse-pressure-gradient airfoil boundary layers: stronger free-stream turbulence reduces shape factor from ~1.5 to ~1.4, suppresses the APG wake, and raises skin friction toward ZPG value

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2026-08-01 08:37 UTC pith:YGVTQVPC

load-bearing objection First controlled FST×APG airfoil boundary-layer database; the real risk is that uτ and δ both come from the same profile fit, so the headline mean-flow numbers need an independent cross-check. the 4 major comments →

arxiv 2607.21047 v1 pith:YGVTQVPC submitted 2026-07-23 physics.flu-dyn

Effect of free-stream turbulence on a moderate adverse pressure gradient turbulent boundary layer developing over an airfoil

Tomek Jaroslawski , Francesco Scarano This is my paper
classification physics.flu-dyn
keywords free-stream turbulenceadverse pressure gradientturbulent boundary layerairfoil boundary layershape factorskin frictionlarge-scale motionsamplitude modulation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper tries to establish that free-stream turbulence (FST) is not a minor disturbance but a governing parameter in a spatially developing adverse-pressure-gradient (APG) turbulent boundary layer over an airfoil. Using wind-tunnel measurements on a NACA 0015 airfoil at fixed chord Reynolds number, the authors show that raising FST from about 0.2% to 6% systematically thickens the boundary layer, lowers the shape factor from roughly 1.5 toward 1.4, partially suppresses the APG-induced wake, and raises skin friction toward zero-pressure-gradient values. The mechanism they identify is the penetration of energetic large-scale free-stream motions, with wavelength around 13 times the boundary-layer thickness, which add energy to the outer layer, leave a near-wall footprint, and modulate near-wall small scales rather than simply superposing. A sympathetic reader would care because real airfoils in wind-turbine wakes, turbomachinery, urban and atmospheric flows experience exactly this combination, and the paper argues that APG makes boundary layers more receptive to external turbulence, so FST must be accounted for when interpreting mean-flow evolution, turbulence statistics, and scale interactions.

Core claim

The authors claim that in a developing APG turbulent boundary layer over an airfoil, increasing free-stream turbulence systematically alters the mean flow and turbulence structure in ways that counteract canonical APG effects. At the highest FST level (Tu ≈ 6%), the shape factor drops to about 1.4, a value typical of zero-pressure-gradient boundary layers, despite the imposed adverse pressure gradient; the wake region of the mean velocity profile is partially suppressed; and the skin-friction coefficient rises toward the Coles–Fernholz ZPG correlation. Spectral analysis shows that the extra energy comes from large-scale free-stream motions with characteristic wavelength λx/δ ≈ 13, which pene

What carries the argument

The central object is the set of large-scale free-stream motions with characteristic wavelength λx/δ ≈ 13 that penetrate the turbulent boundary layer and modulate near-wall small scales. The analysis machinery consists of three linked tools: (1) a mean-velocity-profile fit (Rodríguez-López et al.) that simultaneously returns friction velocity uτ and boundary-layer thickness δ, since direct edge detection was unusable under FST; (2) premultiplied spectra and a scale decomposition with a deliberately chosen threshold at λx = 8δ to isolate the free-stream-dominated large scales from the boundary-layer small scales; and (3) a skewness decomposition whose modulation term 3u_L^+ u_S^{+2} quantifie

Load-bearing premise

The load-bearing premise is that the two fitted quantities from one mean-velocity-profile fit, friction velocity uτ and boundary-layer thickness δ, are accurate enough that every headline result—shape factor, Clauser parameter, skin friction, λx/δ ≈ 13, and the λx = 8δ split—faithfully reflects the flow; if the assumed profile form mislocates the edge or misattributes the FST-modified outer region, the reported wake suppression and skin-friction increase could be partly artif

What would settle it

Measure skin friction on the same airfoil, at the same Tu and angle of attack, using an independent technique such as oil-film interferometry or a micro force balance, and measure boundary-layer thickness independently with PIV or a dual-probe total-pressure method; if the independently obtained Cf and δ no longer show the reported increase in Cf and decrease in H with Tu, the central claim collapses. Alternatively, a DNS of the same airfoil flow with clean versus FST inflow conditions at matching Rec and β would settle whether the λx/δ ≈ 13 mode is the actual agent.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If FST is a governing parameter in APG airfoil boundary layers, then wind-tunnel measurements and numerical simulations of airfoil flows must report and prescribe the free-stream turbulence level and its length scale, not just the pressure gradient and Reynolds number.
  • Because the APG amplifies FST effects, stronger adverse pressure gradients will make airfoil boundary layers more sensitive to incoming turbulence, meaning separation and skin-friction predictions in turbulent environments cannot rely on clean-inflow data alone.
  • The large-scale free-stream motions do not merely add energy; they modulate near-wall small scales, so simple additive corrections to turbulence statistics will be insufficient and scale-interaction models are needed.
  • Matching only Reτ or only β is not enough to characterize the boundary-layer state when FST varies, since profiles with nominally matched parameters differ substantially.
  • Combined APG and FST can produce features usually associated with higher-Reynolds-number wall turbulence at moderate friction Reynolds numbers, which may affect how flow-control strategies are designed and tested.

Where Pith is reading between the lines

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

  • The paper leaves implicit a testable extension: varying the free-stream integral length scale Lu/δ independently of Tu would show whether the reported penetration and near-wall effect scale with Lu/δ and whether the APG amplification systematically increases with that ratio.
  • A direct consequence the authors do not pursue is that FST, by raising skin friction and lowering shape factor, may delay APG separation on airfoils at otherwise identical pressure distributions; this could be tested by measuring separation location or using surface pressure signatures.
  • The λx ≈ 13δ scale is close to the grid-generated turbulence's energy-containing scale; in numerical simulations of airfoil flows, inflow boundary conditions would need to reproduce such large-scale modes to capture the physics described here.
  • The saturation of the large-scale near-wall peak at high Tu and β, which the paper flags as needing further work, suggests there may be a limit to how much additional external energy can penetrate the near-wall region; identifying that limit would sharpen the claimed governing-parameter role.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 8 minor

Summary. Jaroslawski and Scarano report wind-tunnel measurements of a spatially developing turbulent boundary layer on the suction side of a NACA 0015 airfoil at chord Reynolds number ≈2.5×10^5, two angles of attack (2° and 4°), and four freestream turbulence levels (Tu≈0.2%, 1.4%, 2.5%, 6%), at chordwise positions x/c=0.400–0.625. The central claim is that FST is a governing parameter in APG airfoil boundary layers: increasing Tu thickens the boundary layer, lowers the shape factor from ≈1.5 toward ≈1.4, partially suppresses the APG wake, and raises the skin-friction coefficient toward the ZPG correlation. The paper further claims that energetic freestream large-scale motions with λx/δ≈13 penetrate into the boundary layer, amplify inner- and outer-region streamwise variance, and modulate near-wall small scales, with the effect amplified by the adverse pressure gradient. Skin friction and boundary-layer thickness are not measured directly but are obtained together from a single Rodríguez-López et al. (2015) mean-velocity-profile fit; all integral quantities, Reynolds numbers, spectral normalizations, and the λx=8δ scale-split threshold depend on those two fitted parameters.

Significance. If substantiated, this is a valuable contribution: the parametric coverage of Tu and β on a spatially developing airfoil boundary layer is rare, and the combination of mean-flow, variance, spectral, scale-decomposition, and skewness diagnostics provides a rich dataset. The qualitative trends are large, internally consistent, and broadly consistent with earlier ZPG-FST studies and APG-TBL studies. The main weakness is that the headline quantitative results are expressed in inner/outer scaling built on uτ and δ from a single profile fit, with no direct wall-shear measurement and no independent usable check of δ. The qualitative conclusion that FST matters is not in doubt; however, the quantitative strength of the wake suppression, the Cf increase, and the APG-amplification effect is less secure than the text implies. This is fixable with uncertainty quantification and/or an independent skin-friction validation.

major comments (4)
  1. [§2.1.3, Appendix A] The two parameters on which nearly every quantitative statement rests — uτ and δ — are both free parameters of the same Rodríguez-López et al. (2015) profile fit; Appendix A explicitly states that δ is treated as an open fitting parameter together with the friction velocity. Consequently, H (Eq. 11), β (Eq. 6), Cf, Reτ, λx/δ≈13, and the λx=8δ filter threshold are all derived from those fits. Because the wake strength in U+ and Cf are inner-scaled quantities, an FST-dependent overestimate of uτ would by itself produce exactly the reported 'wake suppression' and 'Cf toward ZPG' signatures; a shifted δ would similarly move the spectral peak and the 8δ split. The one independent δ diagnostic attempted, the Vinuesa et al. (2016) method, is explicitly unusable under FST (Fig. A.1). I am not claiming the trends are artifacts, but the load-bearing mean-flow and spectral claims need an independen
  2. [§3.1, §3.2, Figs. 6 and 8] The headline integral quantities are presented without uncertainty bars or repeatability estimates. The text itself states that for low Tu the shape-factor variation is 'within the uncertainty of the measurements' (§3.1), while the high-Tu reduction from ≈1.5 to ≈1.4 is used to support the central mean-flow claim. Similarly, the Cf increase toward the ZPG curve in Fig. 8 has no error bars, even though Cf is obtained from a fitted uτ. Please provide quantified uncertainties for uτ, δ, H, and Cf for each condition — for example, bootstrap over profile points and fit parameters, or repeated traverses — and state explicitly whether the Tu=1.4% and Tu=2.5% cases are distinguishable from the clean case. Without this, the reader cannot separate genuine FST-driven shifts from fit uncertainty.
  3. [§3.4, §3.5, Figs. 12–15] The spectral and scale-decomposition evidence for the proposed mechanism is sensitive to the fitted δ. The wavelength normalization λx/δ and the filter threshold λx=8δ both use the fitted δ; Fig. 5 shows that δ itself increases with Tu. Thus the same physical wavelength is reported as a larger λx/δ, and more energy is assigned to the 'large-scale' band as Tu increases. The claims that the freestream peak is at λx/δ≈13 and that the λx<8δ variance collapses across Tu should therefore be tested for sensitivity to δ. Please show the premultiplied spectra and the filtered variance profiles for δ±10% (or using an independently estimated δ99 where feasible) and report the resulting uncertainty in the peak wavelength and in the large-scale variance amplification. This is a necessary check before the scale-penetration mechanism can be regarded as established.
  4. [§3.1, §3.5, Figs. 10 and 15; Eq. (6)] The conclusion that 'the influence of FST is amplified by the adverse pressure gradient' is presented using β as the abscissa, but β is computed from fitted uτ and δ via Eq. (6). Since increasing Tu changes both δ* and τw, part of the observed β shift with Tu is a restatement of the fit outputs rather than an independent measure of the pressure-gradient strength. To make the amplification claim robust, please additionally report the inner- and outer-peak trends against directly measured quantities such as x/c, local edge velocity Ue, or dPe/dx, and note explicitly that β is not an independent control parameter in this experiment. The physical interpretation may be correct, but the current presentation conflates the fitted quantities with the physical pressure-gradient forcing.
minor comments (8)
  1. [§2.1.1] The sentence describing the boundary-layer trip is duplicated ('The boundary layer on the suction side of the airfoil was tripped upstream of the measurement region...'). Remove the repetition.
  2. [Throughout] Please proofread for typos and grammatical errors: 'usefull', 'despide', 'pronunced', 'thet', 'highligths', 'simular with', 'Figure figure 11', and similar. Several sentences are unfinished or ungrammatical, especially in §3.1.
  3. [§3.1, Fig. 6] The figure appears to contain repeated panels with duplicated axes; check the final figure assembly so that each panel is distinct and labeled consistently.
  4. [§3.5, Eq. (9)] The notation 'where the double overbar denotes normalization by u+23/2' is obscure. Define the normalization explicitly and state how each term is computed from the filtered signals.
  5. [§3.3] The text refers to x/c=0.6235, while all other locations are given as x/c=0.625. Make the reported coordinate consistent.
  6. [Table 1, §2.1.2] Table 1 reports Tu=0.3% at x/c=0.400 for α=2°, while §2.1.2 quotes the clean tunnel condition as Tu≈0.2%. Clarify whether these are local values and indicate the precision of the Tu values.
  7. [Fig. 13] The caption says the spectral slices are taken 'along the dashed white lines', but the dashed lines are not visible in the rendered figure. Label the row/column positions directly on the figure.
  8. [Fig. 16] There appears to be a mismatch between the panel labels in the figure and the descriptions in the text: the text calls panel (a) the total skewness, but the figure panel (a) appears to show the modulation term. Reorder the panels/caption so they match.

Circularity Check

0 steps flagged

No significant circularity; all central claims rest on independent experimental measurements, and the disclosed fitting assumptions are methodological caveats, not circular inputs.

full rationale

This experimental paper does not contain a derivation chain that reduces to its inputs. The central finding—that increasing FST reduces shape factor, suppresses the APG wake, and raises Cf toward ZPG—is based on direct hot-wire measurements of mean and fluctuating velocity profiles. The friction velocity uτ and boundary-layer thickness δ are obtained from the externally published Rodríguez-López et al. (2015) profile-fitting method, which is not a self-citation and whose use under FST is independently supported by oil-film interferometry comparisons (Esteban et al., 2017) cited in the paper. The paper explicitly discloses in Appendix A that δ is an open fitting parameter alongside uτ, and that the alternative Vinuesa et al. (2016) diagnostic method was unusable under FST. This is an honest methodological limitation that could affect the accuracy of derived quantities (e.g., Cf, H, λx/δ), but it does not make those quantities equivalent to the fit by construction: the fit is grounded in measured profile shapes, and the trends are consistent across multiple stations, angles of attack, and turbulence levels. The characteristic length scale λx/δ ≈ 13 is measured independently in the free stream and inside the boundary layer, and the 8δ scale-split threshold is chosen from a local minimum in the measured spectra, not from a self-referential prediction. The only self-citations (Jaroslawski et al., 2023a,b) concern the experimental facility, grids, and length-scale evaluation procedure, which are not load-bearing for the physical conclusions. No step of the analysis is self-definitional, no fitted parameter is relabeled as a prediction, and no unique-solution claim is imported from the authors' prior work. The paper is therefore self-contained in its evidentiary logic, with residual concerns belonging to measurement uncertainty rather than circularity.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

The central claims rest on two fitted quantities (uτ, δ) from a single profile fit, a hand-chosen scale threshold (λx=8δ), and standard assumptions (Taylor hypothesis, sensor-length insensitivity of large scales, iso-Tu across angles of attack). No new physical entities are introduced. The counts reflect measurement-reduction choices rather than theoretical postulates.

free parameters (3)
  • friction velocity uτ = 0.61–0.86 m/s per profile
    Obtained simultaneously with δ from the Rodríguez-López et al. (2015) mean-profile fit (§2.1.3, App. A); enters Cf, β, wall scaling, and all inner-peak quantities.
  • boundary-layer thickness δ = ≈6–10 mm per profile
    Open fitting parameter in the same fit (App. A) because the Vinuesa diagnostic-plot method fails under FST; feeds δ*, θ, H, β, Reτ, λx/δ normalization, and the 8δ scale-separation threshold.
  • scale-separation threshold λx = 8δ = 8δ for all cases
    Chosen deliberately in §3.5 at the spectral minimum of the same measured premultiplied spectra; defines the large/small-scale split on which the 'FST penetration' variance attribution rests.
axioms (5)
  • domain assumption The Rodríguez-López et al. (2015) mean-profile fit (composite log-law + wake with uτ and δ as free parameters) remains identifiable and valid for boundary layers simultaneously subjected to APG and FST.
    Invoked in §2.1.3 and App. A; all integral parameters used in the claims (H, β, Cf) inherit the fit. If the assumed profile form misattributes FST effects, the Cf/wake trends could be artifacts.
  • standard math Taylor's frozen-turbulence hypothesis converts time series to spatial wavenumber and length scales.
    Used to compute Lu (Eq. 7), all spectra (§3.4), and the scale decomposition (§3.5). Standard for this facility's hot-wire speeds.
  • domain assumption Finite hot-wire length (ℓ+ ≈ 48–68) attenuates small scales but does not affect the large-scale motions and outer-layer trends discussed.
    Stated in §2.1.3; underpins the interpretation of near-wall variance and the scale decomposition, though the inner peak is partly spatially filtered.
  • domain assumption The two angles of attack constitute approximately iso-Tu configurations over the measurement region.
    Stated in §2.2.2; the APG-vs-Tu attribution requires Tu to be matched while β differs, with only 'moderate deviations' at the highest turbulence level.
  • domain assumption FST does not substantially modify the outer inviscid pressure distribution except at Tu ≈ 6%.
    Based on Fig. 2 (§2.2.1); used to argue pressure-gradient changes are controlled by angle of attack and streamwise position rather than by FST, except at the highest level.

pith-pipeline@v1.3.0-alltime-deepseek · 23058 in / 18590 out tokens · 193035 ms · 2026-08-01T08:37:22.170195+00:00 · methodology

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read the original abstract

Turbulent boundary layers (TBLs) subjected to adverse pressure gradients (APGs) are common to industrial aerodynamic applications, yet the effect of freestream turbulence (FST) on TBLs developing under moderate APGs remains insufficiently understood. Wind-tunnel experiments were conducted to investigate the effects of FST on a developing TBL over a NACA 0015 airfoil. Varying the angle of attack (2 and 4$^\circ$) adjusted the pressure gradient, and hotwire anemometry measured boundary layer properties at different chordwise positions ($x/c$ = 0.400--0.625, with $\beta = \delta^*/\tau_0 dP/dx$ = 0.2-1.5). The FST level was increased using static grids, resulting in levels ranging from 0.15 to 6$\%$. The chord-based Reynolds number was kept constant at around 250,000 for all configurations. The results show that increasing FST systematically modifies the mean-flow development of the APG boundary layer. Higher FST levels reduce the shape factor and partially suppress the APG-induced wake in the mean velocity profile, while increasing the skin-friction coefficient towards values closer to canonical ZPG behaviour. The streamwise velocity variance is amplified in both the inner and outer regions, and spectral analysis shows that this increase is associated with energetic large-scale motions introduced by the freestream turbulence, with characteristic wavelengths of order $\lambda_x/\delta \approx 13$. These large scale structures penetrate into the boundary layer and contribute to the near-wall variance, with a stronger effect observed as the adverse pressure gradient increases. The results show that FST is a governing parameter in developing APG TBLs over airfoils and that its influence is amplified by the pressure gradient. It must therefore be considered when interpreting mean-flow evolution, turbulence statistics, and scale interactions in realistic aerodynamic environments.

Figures

Figures reproduced from arXiv: 2607.21047 by Francesco Scarano, Tomek Jaroslawski.

Figure 1
Figure 1. Figure 1: Wind tunnel experimental setup (a) and grid samples used to generate free-stream [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Pressure gradient parameter β for the four measurement location over the airfoil, for the four Tu tested, open marks represent AoA of 2 ◦ , solid marks represent AoA of 4 ◦ . 9 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Turbulence intensity evolution (a) and integral length-scale (b) in the wind tunnel [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Boundary layer thickness as function of the streamwise coordinate [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Shape factor as function of the streamwise coordinate [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Shape factor as function of momentum thickness Reynolds number (a) and pressure [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Skin friction coefficient as function of momentum thickness Reynolds number, [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Profiles of mean (a,c) and variance (b,d) of the streamwise velocity in wall units [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Variance (a) inner and (b) outer peak as function of the adverse pressure gradient [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Profiles of mean (a,c) and variance (b,d) of the streamwise velocity in wall units; [PITH_FULL_IMAGE:figures/full_fig_p022_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Pre-multiplied energy spectra at [PITH_FULL_IMAGE:figures/full_fig_p024_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Slices of the spectra reported in figure [PITH_FULL_IMAGE:figures/full_fig_p025_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Profiles of the variance of the streamwise velocity in wall units at [PITH_FULL_IMAGE:figures/full_fig_p027_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Inner peak of the streamwise velocity variance computed using (a) scales with [PITH_FULL_IMAGE:figures/full_fig_p028_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Skewness decomposition of the streamwise velocity fluctuations at [PITH_FULL_IMAGE:figures/full_fig_p030_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Schematic of the interaction between freestream turbulence and an APG air [PITH_FULL_IMAGE:figures/full_fig_p032_17.png] view at source ↗

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