Non-equilibrium effects in turbulent boundary layers over riblets: DNS of step changes in surface texture

Amirreza Rouhi; Melissa Kozul; Oriol Lehmkuhl; Vishal Kumar; Wen Wu

arxiv: 2512.02034 · v2 · submitted 2025-11-20 · ⚛️ physics.app-ph

Non-equilibrium effects in turbulent boundary layers over riblets: DNS of step changes in surface texture

Vishal Kumar , Melissa Kozul , Wen Wu , Oriol Lehmkuhl , Amirreza Rouhi This is my paper

Pith reviewed 2026-05-17 20:21 UTC · model grok-4.3

classification ⚛️ physics.app-ph

keywords turbulent boundary layerribletsstep changenon-equilibriumdirect numerical simulationskin frictioninternal equilibrium layerzero-pressure-gradient flow

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The pith

Turbulent boundary layers recover from riblet step changes in two stages but skin friction falls short of full equilibrium.

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

This paper uses direct numerical simulations to examine how zero-pressure-gradient turbulent boundary layers respond to abrupt streamwise changes in wall texture, switching from smooth to riblets or the reverse. Reference calculations on uniform smooth and uniform riblet surfaces establish the equilibrium baselines against which departures are measured. The simulations show that an internal equilibrium layer grows outward from the new surface in two stages: a faster near-wall adjustment followed by much slower outer-layer recovery. The outer recovery is limited by the wake advected from upstream, so that skin friction approaches but does not reach the equilibrium value for the local texture even at large downstream distances.

Core claim

Downstream of the step change, growth of the internal equilibrium layer thickness δ_IEL follows two stages. Stage I recovers the flow up to the buffer region and, for riblet-to-smooth cases, scales as δ_IEL ∝ (x/k)^0.6, completing by x ≃ 100k. Stage II, the outer-region recovery, is slow because of the advected frozen wake from upstream. For drag-increasing riblets with k+ ≥ 25, δ_IEL therefore does not reach the full boundary-layer thickness even at 50δ0 downstream. As a result the skin-friction coefficient exceeds 90 percent of its equilibrium counterpart but does not attain 100 percent.

What carries the argument

Internal equilibrium layer thickness δ_IEL, the wall-normal distance up to which the flow has adjusted to the new surface texture and whose growth tracks the return to equilibrium.

If this is right

Inner-layer recovery up to the buffer region occurs over roughly 100 riblet heights and is slower for riblet-to-smooth than smooth-to-riblet transitions.
Outer-layer recovery remains incomplete at least as far as 50 initial boundary-layer thicknesses downstream.
Skin-friction coefficient reaches more than 90 percent but not 100 percent of the equilibrium value for the local surface.
The advected frozen wake from the upstream surface is responsible for the persistent outer-layer departure from equilibrium.

Where Pith is reading between the lines

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

Practical riblet applications on aircraft or ships may experience extended non-equilibrium drag regions after any local texture change.
RANS or wall-model LES closures for riblets could be improved by incorporating a two-stage internal-layer growth model.
Repeating the step-change study at higher Reynolds numbers would test whether the slow outer recovery persists when the incoming boundary layer is thicker relative to riblet height.

Load-bearing premise

The reference calculations over entirely smooth and entirely riblet-covered surfaces accurately represent the true equilibrium states at these Reynolds numbers.

What would settle it

Direct measurement of skin-friction coefficient or outer-layer velocity profiles at streamwise distances much larger than 50 initial boundary-layer thicknesses to determine whether they eventually match the equilibrium reference values exactly.

Figures

Figures reproduced from arXiv: 2512.02034 by Amirreza Rouhi, Melissa Kozul, Oriol Lehmkuhl, Vishal Kumar, Wen Wu.

**Figure 2.** Figure 2: (a) Flow setup and visualization for the case T9 SM. (b) Mean velocity profiles upstream of the step change (x = −4δ0) for the T9 SM and SM T9 cases (lines); the profiles are compared with ones from the smooth-wall ZPG TBL of Schlatter et al. (2009), and T950 fine case from [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 4.** Figure 4: Streamwise variations of (a) Reθ, and (b,c) Cf for the T9-to-smooth step change. In (a,b) the smooth case (SM) is added as a reference. In (c), Cf for T9 SM is normalized by Cf from the smooth case (CfSM ) at matched Reθ; rough-tosmooth step change data of Rouhi et al. (2019) (Ref. DNS) and Li et al. (2019) (Ref. EXP). the step change, we observe an undershoot in Cf , followed by an overshoot (figure 5b)… view at source ↗

**Figure 6.** Figure 6: Identification of IEL based on dU +/d ln y + for the T9 SM case. Profiles of (b) dU +/d ln y + and (c) u′2 + at 6δ0 and 24δ0 downstream of the step change, as indicated with vertical dashed lines in (a). At each location, the profiles of T9 SM case are compared with the ones from the smooth case (SM) at matched Reθ. On each profile, the IEL is located with a magenta bullet. In (b), we overlay the wake pro… view at source ↗

read the original abstract

We computationally study the response of zero-pressure-gradient (ZPG) turbulent boundary layers (TBLs) to streamwise step changes from a smooth wall to riblets (SM_RI), and vice versa (RI_SM). To quantify the departure from equilibrium due to the step changes, we conduct reference calculations of ZPG TBLs over an entirely smooth wall, and an entirely riblet-covered surface. To save the computational cost, we generate an optimal grid for an unstructured spectral-element code, consistent with the size of turbulent scales across the TBL. By the step change, the momentum thickness Reynolds number reaches $Re_{\theta_0} \simeq 680$ (friction Reynolds number $Re_{\tau_0} \simeq 283$), and by the domain outlet downstream of the step change, $Re_{\theta} \simeq 1000$ ($Re_{\tau} \simeq 400$). The TBL departure from equilibrium due to the step change, and its subsequent relaxation, recall previous studies on step changes in surface roughness. Downstream of the step change, growth of the internal equilibrium layer thickness $\delta_\text{IEL}$, hence recovery to equilibrium, follows two stages. Stage I corresponds to the recovery up to the buffer region ($y^+ \simeq 10$), which is slower during the RI_SM step change than the SM_RI counterpart. For the RI_SM cases during Stage I, $\delta_\text{IEL} \propto (x/k)^{0.6}$, and this stage is completed by $x \simeq 100k \simeq (5\delta_0 - 20\delta_0)$ downstream of the step change, where $k$ is the riblet height. Stage II recovery i.e.\ recovery of the outer region, is quite slow. Therefore, for drag-increasing riblets with $k^+ \ge 25$, $\delta_\text{IEL}$ does not reach the boundary layer thickness, even up to $50\delta_0$ downstream of the step change, owing to the advected frozen wake from upstream. As a result, skin-friction coefficient reaches to more than $90\%$ of its equilibrium counterpart, but does not reach its $100\%$.

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.

Desk Editor's Note private letter to a colleague

DNS of riblet step changes gives concrete two-stage recovery scalings and shows skin friction stalling above 90% but below 100% of equilibrium for drag-increasing cases, though domain and Re limits need checking.

read the letter

The main thing to know is that this DNS tracks how turbulent boundary layers respond to sudden switches between smooth walls and riblets, and it finds a clear two-stage recovery where the outer layer lags enough that skin friction never fully matches the reference equilibrium for the drag-increasing riblets even at 50 delta0 downstream. They run reference cases on fully smooth and fully ribleted surfaces, then introduce the step at Re_theta around 680 and follow the flow to Re_theta about 1000. The new numbers are the stage-I scaling delta_IEL proportional to (x/k)^0.6 for the riblet-to-smooth transition, finishing by roughly 100k or 5-20 delta0, plus the statement that stage II outer recovery stays slow because of an advected frozen wake, leaving c_f above 90% but short of 100% for k+ at or above 25. This adds specific recovery distances and skin-friction behavior to the existing roughness step-change literature, which is the useful part. The direct numerical setup with no fitted parameters and the mention of an optimized unstructured spectral-element grid are straightforward strengths for this type of work. The soft spots are the modest Reynolds numbers, Re_tau only reaching 400, and the domain stopping at about 50 delta0. At these values the outer layer has limited large-scale motions, so the persistent lag could partly reflect domain length, inflow matching, or outflow treatment rather than purely the physical wake. The abstract does not include grid-convergence checks or error bars, which leaves the exact recovery percentages open to some adjustment. Readers working on drag-reduction surfaces or non-equilibrium boundary layers will get the most from the recovery data and scalings. The work is coherent enough on its own terms to deserve a serious referee, even if revisions will focus on the Re range and domain adequacy.

Referee Report

2 major / 3 minor

Summary. The manuscript reports direct numerical simulations of zero-pressure-gradient turbulent boundary layers subjected to streamwise step changes in surface texture (smooth-to-riblet and riblet-to-smooth), benchmarked against reference equilibrium simulations over uniform smooth and riblet surfaces. Key results include identification of a two-stage recovery of the internal equilibrium layer thickness δ_IEL, with Stage I for RI_SM cases following δ_IEL ∝ (x/k)^{0.6} up to x ≈ 100k, and a slow Stage II in which, for drag-increasing riblets (k+ ≥ 25), δ_IEL does not reach the local boundary-layer thickness even at x = 50δ0 owing to an advected frozen wake; consequently the skin-friction coefficient recovers to more than 90% but not 100% of the equilibrium value at Re_τ up to approximately 400.

Significance. If the attribution of the persistent outer-layer lag to a physical frozen wake holds, the work supplies useful DNS data on history effects in riblet-modified boundary layers and extends prior roughness step-change studies with a parameter-free numerical approach. The two-stage recovery observation and the quantitative recovery percentages are potentially valuable for modeling and for practical drag-reduction applications. The modest Reynolds-number range and finite domain, however, constrain the generality of the reported recovery distances and percentages.

major comments (2)

Reference equilibrium cases: The headline claim that δ_IEL never reaches the local boundary-layer thickness up to x = 50δ0 (and that c_f therefore reaches only >90% of equilibrium) rests on the reference smooth-wall and riblet-covered calculations accurately representing true equilibrium states. At Re_τ ≈ 400, where the outer layer contains only a few large-scale motions, any mismatch in inflow history, development length, or outflow treatment between the step-change and reference runs can produce an apparent persistent wake that is numerical rather than physical. Explicit checks (e.g., streamwise stationarity of mean and turbulence profiles in the reference domains, or comparison of integral quantities at multiple streamwise stations) are required to substantiate that the slow outer-layer recovery is not an artifact.
Computational domain length (setup and results sections): The step-change domain terminates at approximately 50δ0. The interpretation that the outer-layer non-recovery is caused by an advected frozen wake rather than domain truncation would be strengthened by a sensitivity test with a longer domain or by demonstrating that the reference cases themselves reach equilibrium within a comparable or shorter distance; without such evidence the quantitative claim that δ_IEL remains below δ even at the domain outlet remains open to the alternative explanation of finite-domain effects.

minor comments (3)

The precise operational definition and extraction method for the internal equilibrium layer thickness δ_IEL should be stated explicitly, including the threshold or fitting procedure used to identify its edge from the velocity or stress profiles.
Grid-resolution and convergence details beyond the statement of an 'optimal grid' for the spectral-element discretization should be supplied to support the quantitative skin-friction and layer-thickness results.
Statistical uncertainty or error bars on the reported skin-friction recovery percentages would help assess whether the 10% deficit relative to equilibrium is statistically significant.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which help clarify the robustness of our findings on non-equilibrium recovery in riblet-modified turbulent boundary layers. We address each major comment below, providing clarifications and indicating revisions where appropriate.

read point-by-point responses

Referee: Reference equilibrium cases: The headline claim that δ_IEL never reaches the local boundary-layer thickness up to x = 50δ0 (and that c_f therefore reaches only >90% of equilibrium) rests on the reference smooth-wall and riblet-covered calculations accurately representing true equilibrium states. At Re_τ ≈ 400, where the outer layer contains only a few large-scale motions, any mismatch in inflow history, development length, or outflow treatment between the step-change and reference runs can produce an apparent persistent wake that is numerical rather than physical. Explicit checks (e.g., streamwise stationarity of mean and turbulence profiles in the reference domains, or comparison of integral quantities at multiple streamwise stations) are required to substantiate that the slow outer-layer recovery is not an artifact.

Authors: We agree that explicit verification of equilibrium in the reference cases is essential to support the physical interpretation of the frozen wake. In the revised manuscript we will add new figures and text documenting the streamwise stationarity of mean velocity profiles, Reynolds stress components, and integral quantities (skin-friction coefficient and momentum thickness) at several streamwise stations within the reference domains. These checks confirm that the reference profiles become independent of streamwise location well upstream of the stations used for comparison with the step-change cases, thereby ruling out numerical artifacts and reinforcing that the persistent outer-layer lag is physical. revision: yes
Referee: Computational domain length (setup and results sections): The step-change domain terminates at approximately 50δ0. The interpretation that the outer-layer non-recovery is caused by an advected frozen wake rather than domain truncation would be strengthened by a sensitivity test with a longer domain or by demonstrating that the reference cases themselves reach equilibrium within a comparable or shorter distance; without such evidence the quantitative claim that δ_IEL remains below δ even at the domain outlet remains open to the alternative explanation of finite-domain effects.

Authors: We acknowledge the concern regarding possible domain truncation. In the revision we will include a direct comparison demonstrating that the reference equilibrium simulations attain stationarity over development lengths shorter than the post-step-change distances examined in the step-change runs. We will also add plots of δ_IEL evolution in the reference cases to show that equilibrium is reached within distances comparable to or less than 50δ0. A full sensitivity simulation with an extended domain is not feasible within the present computational allocation; however, the stationarity checks on the reference cases, together with consistency with prior roughness step-change literature, support our attribution to an advected frozen wake rather than truncation. revision: partial

Circularity Check

0 steps flagged

No circularity: results are direct DNS outputs with no self-referential derivations

full rationale

The paper reports outcomes from direct numerical simulation of the incompressible Navier-Stokes equations on an unstructured spectral-element grid for zero-pressure-gradient turbulent boundary layers subjected to streamwise step changes in wall texture. Quantities such as the internal equilibrium layer thickness δ_IEL, its two-stage growth (including the observed δ_IEL ∝ (x/k)^0.6 scaling in Stage I for RI_SM cases), and the skin-friction recovery to >90% but not 100% of the reference equilibrium value are extracted directly from the computed velocity fields. Separate reference simulations over uniform smooth and uniform riblet surfaces supply the equilibrium baselines; these are independent runs, not fitted parameters or self-definitions. No equations, ansatzes, or uniqueness theorems are invoked that reduce the reported scalings or recovery statements to the simulation inputs by algebraic construction. Self-citations, if present, are not load-bearing for the central claims. The derivation chain consists solely of numerical solution of the flow equations plus post-processing of the resulting data.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on direct numerical integration of the incompressible flow equations under zero-pressure-gradient conditions with riblet boundary conditions; no new physical entities or ad-hoc fitted constants are introduced beyond standard numerical discretization choices.

axioms (2)

standard math Flow governed by incompressible Navier-Stokes equations
Standard governing equations invoked for DNS of low-Mach-number turbulent boundary layers.
domain assumption Zero-pressure-gradient turbulent boundary layer
Explicitly stated setup for all cases including the step-change configurations.

pith-pipeline@v0.9.0 · 5732 in / 1434 out tokens · 77216 ms · 2026-05-17T20:21:06.418699+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Downstream of the step change, growth of the internal equilibrium layer thickness δ_IEL ... For the RI_SM cases during Stage I, δ_IEL ∝ (x/k)^{0.6} ... skin-friction coefficient reaches to more than 90% of its equilibrium counterpart, but does not reach its 100%.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

Antonia, R. A. & Luxton, R. E. (1971) The response of a tur- bulent boundary layer to a step change in surface roughness. Part

work page 1971
[2]

Fluid Mech.48, 721–761

Smooth to rough.J. Fluid Mech.48, 721–761. Bannier, A., Garnier, ´E. and Sagaut, P. (2015). Riblet flow model based on an extended FIK identity.Flow Turbul. Combust., V ol. 95, pp. 351-376. Boomsma, A. and Sotiropoulos, F. (2015). Riblet drag re- duction in mild adverse pressure gradients: a numerical in- vestigation.Int. J. Heat Fluid Flow, V ol. 56, pp....

work page 2015

[1] [1]

Antonia, R. A. & Luxton, R. E. (1971) The response of a tur- bulent boundary layer to a step change in surface roughness. Part

work page 1971

[2] [2]

Fluid Mech.48, 721–761

Smooth to rough.J. Fluid Mech.48, 721–761. Bannier, A., Garnier, ´E. and Sagaut, P. (2015). Riblet flow model based on an extended FIK identity.Flow Turbul. Combust., V ol. 95, pp. 351-376. Boomsma, A. and Sotiropoulos, F. (2015). Riblet drag re- duction in mild adverse pressure gradients: a numerical in- vestigation.Int. J. Heat Fluid Flow, V ol. 56, pp....

work page 2015