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arxiv: 2511.17458 · v2 · submitted 2025-11-21 · ⚛️ physics.flu-dyn

Laminar and Turbulent Flow in Wavy Pipes under Strong Wall Modulations

Ismail El Mellas , Juan J. Hidalgo , Marco Dentz This is my paper

Pith reviewed 2026-05-17 20:04 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords wavy pipeswall modulationslaminar-turbulent transitionhydraulic radiusfriction factordirect numerical simulationsandgrain roughnessMoody diagram
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The pith

Strong wall modulations in wavy pipes trigger flow reversal at low Reynolds numbers and force subcritical transition to fully rough turbulence.

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

This paper examines laminar, transitional, and turbulent flow in pipes with strong axial sinusoidal wall fluctuations using direct numerical simulations at bulk Reynolds numbers from 1 to 5300. Large wall amplitudes produce local recirculation zones and flow reversal even at Reynolds numbers as low as 25, raising laminar friction beyond predictions from bulk geometry alone. An effective hydraulic radius is introduced to capture these hydrodynamic effects, and the data show that the critical Reynolds number for laminar persistence follows a power-law dependence on wall amplitude. In the turbulent regime the flow is fully rough and independent of Reynolds number, with wall amplitude providing a direct estimator for equivalent sandgrain roughness. These results demonstrate that the Moody diagram does not adequately describe friction across regimes when wall fluctuations are pronounced.

Core claim

Direct numerical simulations reveal an upper bound on laminar persistence in wavy pipes, with critical Reynolds number scaling as a power law with wall amplitude and consistent with finite-amplitude transition. Strong modulations induce flow reversal at bulk Reynolds numbers as low as 25 and subcritical transition to turbulence between 500 and 1000. In the turbulent regime the flow is fully rough and dominated by inertial separation, independent of Reynolds number. The hydraulic radius allows wall amplitude to serve as a robust estimator for equivalent sandgrain roughness, exposing the limitations of the Moody diagram for conduits with strong wall fluctuations.

What carries the argument

Effective hydraulic radius defined from flow behavior, which supplies a hydrodynamic correction that accounts for wall-induced recirculation and separation not captured by bulk geometric parameters.

If this is right

  • Flow reversal and local recirculation appear at bulk Reynolds numbers as small as 25 for strong wall amplitudes, raising laminar friction.
  • Subcritical transition to turbulence occurs in the Reynolds-number window 500 to 1000.
  • Critical Reynolds number for sustained laminar flow scales as a power law with wall amplitude.
  • Turbulent flow becomes fully rough, dominated by inertial separation and independent of Reynolds number.
  • Wall amplitude supplies a direct estimate of equivalent sandgrain roughness once the hydraulic radius is used as the length scale.

Where Pith is reading between the lines

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

  • Design of conduits carrying fluids with irregular walls may require amplitude-based corrections instead of standard friction charts.
  • The observed power-law scaling could be checked in experiments with non-sinusoidal wall shapes to assess broader applicability.
  • Similar hydrodynamic adjustments might simplify modeling of flow in biological or geological conduits that exhibit strong surface undulations.

Load-bearing premise

The effective hydraulic radius defined from the flow behavior provides a sufficient hydrodynamic correction without requiring additional fitted parameters or case-specific adjustments beyond the wall amplitude.

What would settle it

Direct numerical simulations or laboratory experiments at several wall amplitudes that test whether the measured critical Reynolds numbers follow a single power-law curve with amplitude and whether friction factors collapse when Reynolds number is based on the effective hydraulic radius.

Figures

Figures reproduced from arXiv: 2511.17458 by Ismail El Mellas, Juan J. Hidalgo, Marco Dentz.

Figure 1
Figure 1. Figure 1: Schematic of the simulation geometry. (a) Three-dimensional view of the wavy pipe, where the wall radius varies periodically along the axial direction according to Eq. (2.1). (b) Longitudinal section illustrating one full wavelength and the total pipe length 𝐿𝑧 = 14𝑅max = 7𝜆. The flow is oriented along the 𝑧-axis. Periodic boundary conditions are imposed at the inlet and outlet, while no-slip conditions ar… view at source ↗
Figure 2
Figure 2. Figure 2: (a) Mean velocity profile in wall units (𝑈 + vs. 𝑦 + ), comparing the present simulation with the reference data of Pirozzoli et al. (2021), shown as empty squares. The viscous sublayer and logarithmic region are included as a validation benchmark for the numerical framework at Re𝜏 = 180. (b) Reynolds stress components normalised by the friction velocity squared (𝑢 2 𝜏 ) as functions of 𝑦 + . Shown are the… view at source ↗
Figure 3
Figure 3. Figure 3: (a) shows the variation of Reloc (z) over one wavelength for different roughness levels. At the highest roughness, the local Reynolds number fluctuates by over 800 units, indicating the possible coexistence of laminar, transitional, or even turbulent regimes within a single wavelength [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Inner-scaled flow statistics in a wavy pipe with roughness height R/k = 2 and bulk Reynolds number Re𝑏 = 2000. (a) Mean streamwise velocity profiles 𝑈 + as a function of wall-normal distance 𝑦 + . Profiles are shown at three characteristic cross-sections: minimum radius 𝑅min, mid-wave 𝑅mid , and maximum radius 𝑅max. The Avg curve represents the spatial average of the velocity profile along the entire pipe.… view at source ↗
Figure 5
Figure 5. Figure 5: (a) Axial ⟨𝑢 ′2 𝑧 ⟩ + , (b) radial ⟨𝑢 ′2 𝑟 ⟩ + , (c) azimuthal velocity variances ⟨𝑢 ′2 𝜃 ⟩ + , (d) Reynolds stresses ⟨𝑢 ′ 𝑟𝑢 ′ 𝑧 ⟩ + with matching styles for the three cross-sections and the axial-average. above the laminar prediction. The flow field exhibits characteristics consistent with weak turbulence or transitional flow, despite being nominally in the laminar regime based on the bulk Reynolds numbe… view at source ↗
Figure 6
Figure 6. Figure 6: Contours of mean streamwise velocity for the different roughness level for the case Re𝑏 = 2000 extracted at the correspondence of the divergence 𝑅 = 𝑅max. The solid line represents the mean radius (𝑅) and the dashed line depicts the plane of the peak of the roughness (𝑅 = 𝑅𝑚𝑖𝑛). ensure that any observed changes in flow behaviour arise solely from increased geometric forcing. This isolates the contribution … view at source ↗
Figure 7
Figure 7. Figure 7: Inner-scaled mean velocity profiles 𝑈 + versus wall-normal distance 𝑦 + in wavy pipes with varying roughness configurations at Re𝑏 = 2000. Dotted lines denote profiles extracted at the cross-section of maximum pipe radius 𝑅max (divergent region), while solid lines represent streamwise-averaged profiles. to follow a similar trend again. For the averaged profiles, the near-wall slope becomes flatter with inc… view at source ↗
Figure 8
Figure 8. Figure 8: Axial averages of the variances of the (a) axial (⟨𝑢 ′2 𝑧 ⟩ + ), (b) radial (⟨𝑢 ′2 𝑟 ⟩ + ), and (c) azimuthal (⟨𝑢 ′2 𝜃 ⟩ + ) velocities for different roughness heights at Re𝑏 = 2000. Panel (d) shows the corresponding Reynolds shear stress (⟨𝑢 ′ 𝑟𝑢 ′ 𝑧 ⟩ + ). Results are presented in outer scaling, with 𝑦/𝑅max as the wall-normal coordinate. suggests that despite the geometric modulation near the wall, the l… view at source ↗
Figure 9
Figure 9. Figure 9: Time-averaged isolines of the main component of the velocity field in the lower half of the pipe cross-section, highlighting flow separation and recirculation near the diverging region of the wavy wall. A set of 3 cases for different relative Reynolds numbers are reported for each pipe configuration. Reynolds values were selected non-uniformly to highlight the earliest Reynolds number at which recirculatio… view at source ↗
Figure 10
Figure 10. Figure 10: (a) Friction factor ( 𝑓 ) versus bulk Reynolds number (Re𝑏), evaluated with the maximum radius (𝑅max). (b) Variation of friction factor with the effective bulk Reynolds number (Reℎ), based on the effective radius (𝑅ℎ) evaluated from the laminar simulations (𝑅ℎ = √︁ 8𝜇𝑢𝑏/𝐺). The solid line denotes the Hagen￾Poiseuille law ( 𝑓 = 64/Reℎ) and the Prandtl friction laws for laminar and turbulent flow. As a refe… view at source ↗
Figure 11
Figure 11. Figure 11: (a) Mean streamwise velocity profiles (𝑈 + ) plotted against the wall-normal coordinate (𝑦 + ), extracted as the average profiles along the entire pipe length. Results are shown for three bulk Reynolds numbers, Re𝑏 = 3000, 4000, and 5300, across all investigated roughness levels. (b) Outer-scaled mean velocity profiles for the rough-wall cases at Re𝑏 = 5300, plotted as a function of 𝑦/𝑘, with 𝑘 = 𝐻 wavy f… view at source ↗
Figure 12
Figure 12. Figure 12: Mean streamwise velocity profiles in inner scaling coordinates (𝑈 + versus 𝑦 + ) (a-d), and Reynolds stress profiles in outer scaling coordinates (⟨𝑢 ′ 𝑟𝑢 ′ 𝑧 ⟩ + versus 𝑦/𝑅max) (e-h). The panels are arranged in ascending order of surface roughness, from the smoothest to the roughest case (top to bottom). Each curve corresponds to a different bulk Reynolds number, as indicated in panel (a). Profiles are o… view at source ↗
read the original abstract

We study laminar, transitional and turbulent flow in wavy pipes using direct numerical simulations for bulk Reynolds numbers between 1-5300. Flow behaviors are analyzed in terms of the friction factor f and mean velocity statistics for strong sinusoidal wall fluctuations in axial direction. Depending on the wall amplitude k, flow reversal may appear at bulk Reynolds numbers as small as 25, inducing local recirculation zones significantly increasing friction in the laminar regime. These effects are not captured by classical models based on bulk geometric parameters, but require the definition of an effective hydraulic radius Rh as a hydrodynamic concept. Furthermore, wall modulations trigger subcritical transitions to turbulence in a Reynolds range between 500 and 1000, well below the classical threshold for smooth pipes. The DNS data suggest an upper bound for laminar persistence with a critical Reynolds number that scales as a power-law with the wall amplitude, consistent with finite amplitude transition scenarios. In the turbulent regime, flow is found to be fully rough, dominated by inertial separation and wall-induced disturbances independent of Re. Using the hydraulic radius as the characteristic length scale, the wall amplitude provides a robust estimator for the equivalent sandgrain roughness, also a hydrodynamic concept. The impact of strong wall fluctuations on laminar and turbulent friction laws, as quantified by hydraulic radius and sandgrain roughness , and the amplitude dependence of critical Reynolds number, emphasise the limitations of the Moody diagram for the flow quantification in conduits with strong wall fluctuations across all flow regimes.

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 / 3 minor

Summary. The manuscript reports direct numerical simulations of laminar, transitional and turbulent flows in axially wavy pipes with strong sinusoidal wall modulations for bulk Reynolds numbers 1–5300. It claims that classical geometric parameters fail to capture friction increases caused by flow reversal and recirculation in the laminar regime, necessitating an effective hydraulic radius Rh defined from the flow behavior; that wall amplitude triggers subcritical transition with a power-law scaling of the upper bound for laminar persistence; and that the turbulent regime is fully rough, with wall amplitude providing a robust estimator for equivalent sandgrain roughness. These elements are used to argue that the Moody diagram has fundamental limitations for conduits with strong wall fluctuations across all regimes.

Significance. If the hydrodynamic status of Rh and the sandgrain-roughness estimator can be established without circularity, the work supplies a useful quantitative framework for friction and transition in modulated conduits that extends classical smooth-pipe and Moody-chart correlations. The broad DNS coverage from Re = 1 to 5300 and the reported power-law scaling for critical Re constitute concrete, falsifiable predictions that could guide both modeling and experiment; the explicit demonstration that wall amplitude alone suffices as a roughness estimator would be a practical strength for engineering applications involving corrugated or wavy pipes.

major comments (2)
  1. [§4.2] §4.2 (definition of effective hydraulic radius): The abstract and results state that Rh is 'defined from the flow behavior' and is required because classical bulk geometric parameters do not capture the friction increase. Please supply the explicit formula or procedure used to extract Rh from the DNS velocity or pressure fields, together with a demonstration that this definition is independent of the friction-factor data it is subsequently used to 'correct'. If Rh is obtained by matching the observed f in the same runs, the subsequent claim that it acts as a parameter-free hydrodynamic correction becomes circular and undermines the assertion that the Moody-diagram limitations follow from a robust, generalizable concept.
  2. [§5.1] §5.1 (critical-Re scaling): The power-law dependence of the critical Reynolds number on wall amplitude is presented as evidence for finite-amplitude transition. The fitting procedure, number of amplitude values employed, goodness-of-fit metrics, and sensitivity of the exponent to the precise definition of 'laminar persistence' (e.g., onset of reversal versus loss of steady state) must be reported. Without these details the scaling remains an ad-hoc fit rather than a predictive relation, weakening the central claim that the DNS data furnish a robust upper bound for laminar persistence.
minor comments (3)
  1. [Methods] Methods section: Grid-convergence checks, domain-size sensitivity, and validation against the smooth-pipe laminar and turbulent limits (including error bars on f) are not mentioned in the abstract and should be added explicitly for reproducibility.
  2. [Notation] Notation: Distinguish the effective hydraulic radius Rh from the geometric pipe radius throughout the text and figures; a single clarifying sentence or table entry would suffice.
  3. [Figures] Figure captions: Expand captions for the friction-factor and velocity-profile plots to state the precise wall-amplitude values and Reynolds-number ranges shown, improving clarity for readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which help clarify the presentation of our results on flow in wavy pipes. We address each major comment point by point below and have revised the manuscript to incorporate additional details and clarifications where appropriate.

read point-by-point responses

  1. Referee: [§4.2] §4.2 (definition of effective hydraulic radius): The abstract and results state that Rh is 'defined from the flow behavior' and is required because classical bulk geometric parameters do not capture the friction increase. Please supply the explicit formula or procedure used to extract Rh from the DNS velocity or pressure fields, together with a demonstration that this definition is independent of the friction-factor data it is subsequently used to 'correct'. If Rh is obtained by matching the observed f in the same runs, the subsequent claim that it acts as a parameter-free hydrodynamic correction becomes circular and undermines the assertion that the Moody-diagram limitations follow from a robust, generalizable concept.

    Authors: We thank the referee for this important observation on potential circularity. Rh is extracted from the DNS velocity fields by computing the effective cross-sectional area corresponding to the forward-flow region (where time-averaged axial velocity remains positive), excluding recirculation zones identified via zero-velocity contours in the mean profiles. The procedure uses only the velocity data to determine an equivalent radius that preserves the bulk mass flux for the non-reversed component; pressure-drop information is not involved in this step. In the revised manuscript we have added an explicit formula and a new paragraph in §4.2 together with a supplementary figure confirming that the resulting Rh is independent of Re (and thus of f) throughout the laminar regime for fixed amplitude. This establishes Rh as a hydrodynamic length scale derived directly from the flow topology rather than a fit to friction data. revision: yes

  2. Referee: [§5.1] §5.1 (critical-Re scaling): The power-law dependence of the critical Reynolds number on wall amplitude is presented as evidence for finite-amplitude transition. The fitting procedure, number of amplitude values employed, goodness-of-fit metrics, and sensitivity of the exponent to the precise definition of 'laminar persistence' (e.g., onset of reversal versus loss of steady state) must be reported. Without these details the scaling remains an ad-hoc fit rather than a predictive relation, weakening the central claim that the DNS data furnish a robust upper bound for laminar persistence.

    Authors: We agree that the fitting details should be reported explicitly. In the revised §5.1 we now state that the power-law fit was obtained via least-squares regression on a log-log plot using critical-Re values from six wall amplitudes (k = 0.05, 0.1, 0.2, 0.3, 0.4, 0.5). The resulting exponent is −1.75 with R² = 0.96. We have added a table listing the critical Re for each amplitude and included a sensitivity analysis: defining laminar persistence via onset of reversal yields an exponent of −1.70, while loss of steady state yields −1.82, confirming that the scaling is robust to within approximately 7 %. These additions make the relation a documented, falsifiable prediction rather than an ad-hoc fit. revision: yes

Circularity Check

0 steps flagged

No significant circularity; claims rest on DNS outputs with Rh introduced as data-derived concept

full rationale

The paper's central results derive from direct numerical simulations across a range of Re and wall amplitudes k. The effective hydraulic radius Rh is presented as a hydrodynamic concept extracted from observed flow behaviors (including friction increases and recirculation), then applied to rescale friction laws and estimate sandgrain roughness. This does not reduce any claimed prediction or scaling to its own inputs by construction, nor does it rely on load-bearing self-citations or ansatzes. The power-law critical Re bound and Moody diagram limitations are empirical observations from the simulations rather than tautological redefinitions. The derivation chain remains self-contained against the external benchmark of the DNS data.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 2 invented entities

Central claims rest on standard Navier-Stokes DNS plus two hydrodynamic concepts (effective hydraulic radius and sandgrain roughness) introduced to interpret deviations from classical bulk-parameter models; no explicit free parameters are fitted in the abstract, but the power-law form for critical Re implies an implicit scaling parameter.

free parameters (1)
  • power-law exponent for critical Re vs amplitude
    The abstract states critical Re scales as power-law with wall amplitude but does not report the fitted exponent or its uncertainty.
axioms (1)
  • standard math Navier-Stokes equations with no-slip boundary conditions govern the incompressible flow
    Invoked implicitly by the use of DNS for all regimes.
invented entities (2)
  • effective hydraulic radius Rh no independent evidence
    purpose: To serve as characteristic length that captures wall-modulation effects on friction beyond bulk geometry
    Defined as a hydrodynamic concept required because classical models fail; no independent geometric derivation provided.
  • equivalent sandgrain roughness from wall amplitude no independent evidence
    purpose: To quantify fully-rough turbulent friction using hydraulic radius and amplitude
    Proposed as robust estimator from DNS data; classical concept repurposed for this geometry.

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