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arxiv: 2606.03555 · v1 · pith:FXE73T2Pnew · submitted 2026-06-02 · ⚛️ physics.flu-dyn

Passive transverse forcing of turbulent boundary-layer flow using sinusoidal surface grooves

Max W. Knoop , Bas W. van Oudheusden , Luuk Pelkmans , Ferry F. J. Schrijer This is my paper

Pith reviewed 2026-06-28 08:14 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords turbulent boundary layerpassive forcingsurface groovesStokes layerdrag reductiontransverse flowpressure gradient
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The pith

Sinusoidal surface grooves induce a passive transverse flow pattern in turbulent boundary layers through pressure-driven convergence and divergence, but yield at most a few percent frictional drag reduction after pressure losses.

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

The paper examines parallel meandering streamwise grooves as a passive way to force spanwise flow in a turbulent boundary layer. Instead of a uniform undulation, the grooves produce a converging-diverging pattern driven by the periodic lateral pressure gradient. This pattern forms a Passive Stokes Layer consisting of an inviscid outer flow from the surface displacement and a viscous inner layer at the wall. An inviscid model ties the forcing strength to groove geometry and matches the measurements. Near-wall turbulence drops, yet the estimated frictional savings remain small and are likely canceled by added pressure drag.

Core claim

The grooves generate a Passive Stokes Layer (PSL) composed of an inertial pressure-driven outer solution from the surface displacement effect and a viscous inner solution satisfying no-slip at the wall. The spanwise periodicity of the lateral pressure gradient produces a converging-diverging flow rather than spanwise-uniform undulation. The forcing increases with groove amplitude until the slope becomes too steep and saturates. An inviscid model relates the induced transverse velocity directly to the surface geometric properties and agrees with experiment. Turbulence intensity near the wall decreases, but no net drag reduction is measured; a comparison to an equivalent active spatial Stokes

What carries the argument

Passive Stokes Layer (PSL): the composite flow consisting of an inviscid outer solution generated by surface displacement and a viscous inner solution at the wall, driven by the spanwise-periodic lateral pressure gradient.

If this is right

  • The induced transverse velocity saturates once groove slope exceeds a threshold set by the surface geometry.
  • Near-wall turbulence intensity decreases over the grooved surface.
  • Any frictional savings are offset by pressure drag, limiting net drag reduction to a few percent at most.
  • The mechanism is controlled by the spanwise periodicity of the lateral pressure gradient rather than by a uniform wall-normal displacement.

Where Pith is reading between the lines

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

  • Groove wavelength and amplitude could be optimized to maximize the pressure gradient while keeping the slope below the saturation limit.
  • The same pressure-gradient mechanism might appear on other streamwise-periodic roughness elements whose lateral variation creates spanwise pressure differences.
  • Direct particle-image velocimetry of the inner viscous layer would confirm whether the PSL thickness scales with the active Stokes layer thickness.

Load-bearing premise

Frictional drag reduction can be estimated from a tentative match to an equivalent active spatial Stokes layer even though net drag on the grooved surface was never measured directly.

What would settle it

A direct force-balance measurement of net drag (friction plus pressure) on a flat plate with and without the sinusoidal grooves that shows whether total drag rises or falls by more than a few percent.

Figures

Figures reproduced from arXiv: 2606.03555 by Bas W. van Oudheusden, Ferry F. J. Schrijer, Luuk Pelkmans, Max W. Knoop.

Figure 1
Figure 1. Figure 1: Photograph of a SU test surface. Overlayed are [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) Schematic of wind-tunnel configuration and (b) detail of the particle image velocmetry (PIV) experiments over [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Schematic of the SU geometry, (a) top-view, (b) groove cross-section. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Mean flow in the wall-parallel plane at y + s ≈ 15 ± 10 for SU model A3-D10-d050 at Reτ = 1020. (a) Streamwise velocity u +, (b) spanwise velocity w+, and (c) spanwise w+ profiles located at x/λx = [0.25, 0.75] as indicated by the vertical lines in (b) with the same linestyle. Grey-shaded regions represent the flat wall between grooves; grooves are unshaded. A=6x = 0:06 x=6x 0 0.25 0.5 0.75 1 z=6x -0.2 0 0… view at source ↗
Figure 5
Figure 5. Figure 5: Effect of increasing groove amplitude A on w+ in the wall-parallel plane for SU models (a) A3-D10-d050, (b) A5- D10-d050, and (c) A7-D10-d050. Colour scales are the same as in figure 4. is steepest (i.e., αmax in (4)), are shown in fig￾ure 4(c). In this top-view of the flow, the trans￾parent grey shading indicates the flat wall regions between the grooves, whereas the grooves are un￾shaded. The streamwise … view at source ↗
Figure 6
Figure 6. Figure 6: Mean flow in the streamwise-wall-normal plane along the centerline of a groove for SU model A3-D10-d050 at [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Wall-normal turbulence statistics profiles for in [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Two-point correlation of the streamwise veloc [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of wall-parallel flow for (a,c) measurements over A3-D10-d050 and (b,d) potential flow. (a,b) Mean [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Conceptual sketch for the mechanism of spanwise flow generation. (a) Spanwise pressure gradient, where grey ‘ [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Quantification of the spanwise forcing param [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Comparison of the active spatial Stokes layer (SSL) and passive Stokes layer (PSL) for model A3-D10-d050. The first [PITH_FULL_IMAGE:figures/full_fig_p016_12.png] view at source ↗
read the original abstract

A surface geometry consisting of parallel, meandering streamwise grooves has been experimentally studied as an alternative means of passive transverse forcing of turbulent boundary-layer flow. Contrary to the original expectation, the flow does not exhibit a spanwise-uniform undulation aligned with the grooves; instead, a converging-diverging flow pattern results. This flow pattern can be attributed to the spanwise periodicity of the lateral pressure gradient. The forcing effect is found to initially increase with the groove amplitude, but it saturates when the groove slope becomes too steep. The observed induced flow, referred to as a Passive Stokes Layer (PSL), can be considered as being composed of an inertial (pressure-driven) outer solution generated by the displacement effect of the non-smooth surface geometry, and a viscous inner solution to accommodate the no-slip condition at the wall. The mechanism of transverse flow generation is elucidated by an inviscid flow model that relates the forcing to the surface geometric properties, with predictions in good agreement with the experimental results. Although a reduction in the near-wall turbulence levels over the groove surfaces is observed, no direct evidence for (mean) drag reduction is evident from the data. Instead, an estimate of the frictional drag potential is based on establishing a tentative relation to an equivalent spatial Stokes layer (SSL) induced by active wall forcing. This theoretical comparison indicates that the induced passive forcing is sufficient to act on the (active) spanwise forcing mechanism, but produces at most a few per cent of frictional drag reduction. Any potential savings are likely offset by pressure drag and other losses, so that, similar to active forcing, its potential for net drag reduction in practical applications is limited.

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 / 1 minor

Summary. The manuscript experimentally studies parallel meandering streamwise sinusoidal grooves as a means of passive transverse forcing in turbulent boundary-layer flow. Contrary to expectation, a converging-diverging flow pattern emerges due to spanwise-periodic lateral pressure gradients. The induced flow is decomposed into an inertial outer solution and viscous inner solution, termed a Passive Stokes Layer (PSL). An inviscid model relating forcing to surface geometry agrees well with experiments. Near-wall turbulence levels are reduced, but no direct mean drag reduction is measured. Frictional drag reduction is instead estimated at most a few percent via a tentative equivalence to an active spatial Stokes layer (SSL), leading to the conclusion that net savings are likely offset by pressure drag and other losses, limiting practical potential similarly to active forcing.

Significance. The experimental characterization of the PSL and the inviscid model's agreement with data provide a clear mechanistic explanation for passive transverse forcing and its saturation with groove amplitude. These elements would remain valuable contributions even if the drag-reduction estimate is reframed. The work highlights that passive geometries can induce spanwise forcing comparable in mechanism to active methods, but the tentative nature of the drag bound limits implications for net drag reduction applications.

major comments (2)
  1. [Abstract] Abstract: the central quantitative claim that the passive forcing 'produces at most a few per cent of frictional drag reduction' rests entirely on an unvalidated mapping to prior active SSL results. No direct wall-shear integration, skin-friction coefficient, or Reynolds-stress profile comparison between the grooved (passive) and active cases is reported to support the equivalence, and the paper itself notes the absence of direct drag evidence.
  2. [Abstract] Abstract / implied discussion: the conclusion that 'any potential savings are likely offset by pressure drag and other losses' is stated without quantification of the pressure-drag penalty or an integrated force balance on the grooved surface, making the net-drag assessment load-bearing for the practical-limitation claim yet unsupported by new data.
minor comments (1)
  1. [Abstract] The introduction of the acronym PSL (Passive Stokes Layer) occurs without an explicit definition at first use in the abstract; a parenthetical expansion on first appearance would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable feedback. We address each major comment below, proposing revisions where appropriate to clarify the tentative nature of our drag estimates.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central quantitative claim that the passive forcing 'produces at most a few per cent of frictional drag reduction' rests entirely on an unvalidated mapping to prior active SSL results. No direct wall-shear integration, skin-friction coefficient, or Reynolds-stress profile comparison between the grooved (passive) and active cases is reported to support the equivalence, and the paper itself notes the absence of direct drag evidence.

    Authors: We agree that the frictional drag reduction bound is derived from a tentative analogy to prior active SSL work rather than direct validation. The manuscript already qualifies the relation as tentative and notes the lack of direct drag evidence. The analogy is motivated by the mechanistic similarity between the observed PSL and active spanwise forcing (matching velocity profiles and turbulence attenuation), but we accept that this does not constitute equivalence. We will revise the abstract to remove the specific quantitative phrasing and instead state that the induced forcing is comparable in scale to levels previously shown to produce modest frictional drag reduction in active cases, while explicitly underscoring the limitations of the mapping. revision: partial

  2. Referee: [Abstract] Abstract / implied discussion: the conclusion that 'any potential savings are likely offset by pressure drag and other losses' is stated without quantification of the pressure-drag penalty or an integrated force balance on the grooved surface, making the net-drag assessment load-bearing for the practical-limitation claim yet unsupported by new data.

    Authors: The statement on pressure drag offsetting savings is qualitative and not supported by a measured force balance or separate pressure-drag quantification in the present experiments. We will revise the abstract and discussion to present this as an expected outcome based on the presence of form drag in non-streamlined geometries, rather than a firm conclusion, and will note it as a limitation requiring future integrated measurements. The core contributions—the experimental characterization of the PSL and the inviscid model—remain independent of this net-drag assessment. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central claims derive from experiment and model without reduction to inputs by construction.

full rationale

The paper derives the converging-diverging flow pattern and Passive Stokes Layer from experimental observations and an inviscid flow model relating forcing to surface geometry, with direct agreement to data. The frictional drag estimate is explicitly labeled tentative and based on external comparison to active SSL literature rather than any internal fit, self-definition, or equation that reduces the output to the paper's own inputs. No self-citation chains, ansatz smuggling, or renaming of known results appear as load-bearing steps. The derivation chain for the flow mechanism and saturation behavior remains self-contained against the reported measurements.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on experimental observations of pressure-driven flow and an inviscid outer-solution model; the drag conclusion depends on analogy to active forcing. No explicit free parameters are fitted in the reported model. The Passive Stokes Layer is a descriptive label rather than a new postulated physical entity.

axioms (1)
  • domain assumption Inviscid flow approximation suffices for the outer solution relating forcing to surface geometry
    Invoked to derive the model predictions that agree with experiments
invented entities (1)
  • Passive Stokes Layer (PSL) no independent evidence
    purpose: Descriptive term for the observed inertial outer plus viscous inner transverse flow induced by grooves
    New nomenclature for the composed flow pattern; no independent falsifiable prediction beyond the experiments

pith-pipeline@v0.9.1-grok · 5850 in / 1609 out tokens · 39030 ms · 2026-06-28T08:14:41.222657+00:00 · methodology

discussion (0)

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Reference graph

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