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arxiv: 2604.11414 · v1 · submitted 2026-04-13 · ⚛️ physics.flu-dyn

Compressible turbulent boundary layers over two-dimensional square-rib roughness

Youtian Su , Wei-Xi Huang , Chunxiao Xu This is my paper

Pith reviewed 2026-05-10 16:15 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords compressible turbulent boundary layersquare-rib roughnessgeneralized Reynolds analogyvirtual originwall heat transferturbulent Prandtl numberdirect numerical simulationReynolds analogy
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The pith

A modified rough-wall generalized Reynolds analogy reconstructs the temperature-velocity relationship over square-rib roughness using equivalent slip-plane boundary conditions.

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

Direct numerical simulations of Mach 2.5 turbulent boundary layers over square ribs show that roughness breaks the classical Reynolds analogy because form drag has no matching heat transfer mechanism. This causes the effective turbulent Prandtl number to deviate from unity inside the roughness sublayer. A fitting procedure locates a kinematic virtual origin that restores the logarithmic velocity profile and allows the Griffin-Fu-Moin transformation to recover outer-layer similarity. The central advance is a modified rough-wall generalized Reynolds analogy that imposes equivalent slip-plane boundary conditions to bypass near-wall thermal heterogeneity and recover the temperature-velocity relation.

Core claim

Direct numerical simulations at Mach 2.5 demonstrate that a fitting-based optimization procedure determines a kinematic virtual origin restoring logarithmic behavior in the velocity profiles over square-rib roughness. With this origin, the Griffin-Fu-Moin transformation recovers outer-layer similarity better than van Driest. Thermodynamically, the classical generalized Reynolds analogy breaks down due to the disparity between momentum drag and absent heat transfer, but a modified rough-wall GRA using equivalent slip-plane boundary conditions accurately reconstructs the temperature-velocity relationship by bypassing near-wall thermal heterogeneity.

What carries the argument

Modified rough-wall generalized Reynolds analogy (rGRA) with equivalent slip-plane or reference-point boundary conditions that bypass near-wall thermal heterogeneity.

If this is right

  • The fitting optimization yields a consistent kinematic virtual origin that restores the logarithmic velocity profile.
  • The Griffin-Fu-Moin transformation outperforms van Driest in recovering outer-layer similarity for the velocity defect.
  • The modified rGRA accurately reconstructs the temperature-velocity relationship despite classical GRA breakdown.
  • The refined strong Reynolds analogy maintains accuracy for outer-layer fluctuation intensities.

Where Pith is reading between the lines

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

  • The slip-plane approach could be built into wall models for large-eddy simulations of high-speed rough-wall flows.
  • Momentum and thermal virtual origins must be treated separately when roughness and heat transfer are both present.
  • The method should be tested on three-dimensional roughness elements or a wider range of Mach numbers to check generality.

Load-bearing premise

The fitting-based optimization procedure determines a physically consistent kinematic virtual origin, and the equivalent slip-plane boundary conditions capture thermal effects without new inconsistencies in the roughness sublayer.

What would settle it

A new DNS run with changed roughness spacing or wall temperature ratio in which the rGRA-predicted temperature-velocity curve deviates from the simulated data would falsify the claim.

Figures

Figures reproduced from arXiv: 2604.11414 by Chunxiao Xu, Wei-Xi Huang, Youtian Su.

Figure 1
Figure 1. Figure 1: Schematic of the computational domain configurations, comprising the [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Contours of Reynolds-averaged density in the rough-wall boundary layers: (a) [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Streamwise evolution of (a) boundary layer thickness [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Schematic of the two-dimensional rough-wall geometry and the decomposition [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Van Driest (VD) transformation of mean streamwise velocity profiles. Gray [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: GFM transformation of mean streamwise velocity profiles. Light gray lines: [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Evolution of (a) the inner-scaled zero-plane displacement [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 6
Figure 6. Figure 6: Upon correcting for this deficit, the adiabatic rough-wall profile exhibits an [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: Velocity deficit curves: (a) VD transformation; (b) GFM transformation. [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Turbulent velocity fluctuations of smooth-wall cases scaled according to [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Turbulent velocity fluctuations scaled according to equation (3.2) as a function [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Mean temperature-velocity relations in outer scaling: (a) Smooth-wall cases; [PITH_FULL_IMAGE:figures/full_fig_p018_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Mean temperature-velocity relations in outer scaling: (a) Adiabatic rough [PITH_FULL_IMAGE:figures/full_fig_p020_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Consequently, the underlying assumption of Equation (3.6) is invalidated at [PITH_FULL_IMAGE:figures/full_fig_p020_13.png] view at source ↗
Figure 13
Figure 13. Figure 13: Spatial distributions of the time-averaged temperature: (a) Adiabatic rough [PITH_FULL_IMAGE:figures/full_fig_p021_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Prediction errors for the three strong Reynolds analogy (SRA) methods in the [PITH_FULL_IMAGE:figures/full_fig_p021_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Wall-normal distributions of the Prandtl numbers: (a) Turbulent Prandtl [PITH_FULL_IMAGE:figures/full_fig_p023_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Wall-normal distributions of the effective turbulent Prandtl number [PITH_FULL_IMAGE:figures/full_fig_p023_16.png] view at source ↗
read the original abstract

Direct numerical simulations are performed to investigate the combined effects of surface roughness and wall heat transfer on spatially developing compressible turbulent boundary layers at $Ma=2.5$. The roughness consists of transverse square bars with $\lambda_x/k=8$ and $k^+ \approx 35$, under adiabatic and wall-cooling ($T_w/T_r = 0.5$) conditions. Dynamically, the conventional zero-moment method fails to yield a consistent zero-plane displacement for the present cavity-type roughness. Instead, a fitting-based optimization procedure is proposed to determine the kinematic virtual origin, which successfully restores the logarithmic behavior. Based on this displacement, Griffin--Fu--Moin (GFM) transformation outperforms the classical van Driest transformation in recovering outer-layer similarity for the velocity defect. Thermodynamically, the physical disparity between momentum form drag and the absence of a corresponding heat transfer mechanism disrupts the classical Reynolds analogy. The effective turbulent Prandtl number ($Pr_e$) deviates severely from unity within the roughness sublayer, leading to the breakdown of the classical Generalized Reynolds Analogy (GRA). To address this, a modified rough-wall GRA (rGRA) is formulated by introducing an equivalent slip-plane or reference-point boundary conditions, which accurately reconstructs the temperature-velocity relationship by bypassing the near-wall thermal heterogeneity. Finally, the refined strong Reynolds analogy (RSRA) is shown to maintain predictive accuracy for fluctuation intensities in the outer layer despite near-wall modulation by roughness and cooling.

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

3 major / 3 minor

Summary. The manuscript reports direct numerical simulations of spatially developing compressible turbulent boundary layers at Mach 2.5 over two-dimensional square-rib roughness (λ_x/k=8, k^+≈35) under adiabatic and wall-cooling (T_w/T_r=0.5) conditions. It shows that the conventional zero-moment method fails to yield a consistent zero-plane displacement for this cavity-type roughness and proposes a fitting-based optimization procedure to determine the kinematic virtual origin, which restores logarithmic behavior. With this displacement, the Griffin-Fu-Moin (GFM) transformation is reported to outperform the classical van Driest transformation in recovering outer-layer similarity for the velocity defect. Thermodynamically, the mismatch between momentum form drag and heat transfer causes the effective turbulent Prandtl number to deviate from unity in the roughness sublayer, breaking the classical Generalized Reynolds Analogy (GRA); a modified rough-wall GRA (rGRA) is introduced via equivalent slip-plane or reference-point boundary conditions to reconstruct the temperature-velocity relationship. The refined strong Reynolds analogy (RSRA) is shown to hold for outer-layer fluctuation intensities despite near-wall modulation.

Significance. If the proposed procedures prove robust and generalizable, this work would contribute useful diagnostics and modeling approaches for compressible rough-wall turbulence, where standard transformations and analogies break down due to form-drag/heat-transfer disparities. The DNS dataset for this specific roughness geometry and cooling ratio adds to the limited high-fidelity data available in the compressible regime. Explicit comparisons between GFM and van Driest transformations, together with the rGRA reconstruction, provide a concrete assessment of their relative performance.

major comments (3)
  1. [§3] §3 (kinematic virtual origin determination): The fitting-based optimization procedure is explicitly constructed to restore logarithmic behavior after the zero-moment method fails. This choice makes the subsequent claim that GFM recovers outer-layer similarity dependent on the fitting target rather than an independent test; the manuscript should demonstrate that the same origin yields consistent results when validated against alternative metrics (e.g., momentum-integral balance or different roughness spacings) instead of being tuned to the log-law assumption.
  2. [§4.3] §4.3 (formulation of rGRA): The modified rough-wall GRA introduces an equivalent slip-plane/reference-point boundary condition to bypass near-wall thermal heterogeneity and reconstruct the temperature-velocity relation. While the reconstruction is shown to be accurate for the present k^+≈35, λ_x/k=8, T_w/T_r=0.5 case, the physical derivation of the reference location from the compressible Navier-Stokes equations (rather than empirical adjustment) is not provided; the manuscript must show that this location is unique, insensitive to the specific roughness parameters, and does not introduce new inconsistencies in the roughness sublayer.
  3. [Abstract and §4] Abstract and §4 (quantitative validation): The abstract and results state that GFM 'outperforms' van Driest and that rGRA 'accurately reconstructs' the temperature-velocity relationship, yet no error norms, grid-convergence data, or direct quantitative comparisons (e.g., L2 deviations from the target log-law or from DNS temperature profiles) are supplied. These omissions leave the central claims on transformation superiority and analogy restoration without the numerical support required for load-bearing conclusions.
minor comments (3)
  1. [Abstract] The abstract would be strengthened by including at least one quantitative measure (e.g., reduction in profile deviation or correlation coefficient) for the claimed improvements of GFM over van Driest and of rGRA over classical GRA.
  2. [§4] Notation for the effective turbulent Prandtl number Pr_e and the slip-plane location should be defined explicitly at first use and kept consistent between text and figures.
  3. [Figures] Figure captions should clearly distinguish symbols and line styles for the adiabatic versus cooled-wall cases and for the different transformations being compared.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major point below, indicating where revisions will be made to improve the manuscript.

read point-by-point responses

  1. Referee: [§3] §3 (kinematic virtual origin determination): The fitting-based optimization procedure is explicitly constructed to restore logarithmic behavior after the zero-moment method fails. This choice makes the subsequent claim that GFM recovers outer-layer similarity dependent on the fitting target rather than an independent test; the manuscript should demonstrate that the same origin yields consistent results when validated against alternative metrics (e.g., momentum-integral balance or different roughness spacings) instead of being tuned to the log-law assumption.

    Authors: The optimization is required because the zero-moment method yields inconsistent results for this cavity-type roughness, as documented in the manuscript. The procedure identifies the origin that best aligns the mean velocity with the expected log-law in the overlap region. To address dependence concerns, we will add a cross-check in the revision using the momentum-integral balance to confirm consistency with the global force balance. Additional simulations at different roughness spacings are beyond the present scope, but we will expand the discussion on the physical basis for broader applicability. revision: partial

  2. Referee: [§4.3] §4.3 (formulation of rGRA): The modified rough-wall GRA introduces an equivalent slip-plane/reference-point boundary condition to bypass near-wall thermal heterogeneity and reconstruct the temperature-velocity relation. While the reconstruction is shown to be accurate for the present k^+≈35, λ_x/k=8, T_w/T_r=0.5 case, the physical derivation of the reference location from the compressible Navier-Stokes equations (rather than empirical adjustment) is not provided; the manuscript must show that this location is unique, insensitive to the specific roughness parameters, and does not introduce new inconsistencies in the roughness sublayer.

    Authors: We will revise §4.3 to include an explicit derivation obtained by integrating the Reynolds-averaged compressible Navier-Stokes equations across the roughness sublayer, yielding the equivalent slip-plane location from the balance between form-drag momentum deficit and the adjusted heat flux. In the revision we will also present a sensitivity study showing that small shifts in the reference location (±0.1k) produce negligible changes in the reconstructed temperature-velocity relation, and we will verify that no new inconsistencies appear in the sublayer budgets for the parameters studied. revision: yes

  3. Referee: [Abstract and §4] Abstract and §4 (quantitative validation): The abstract and results state that GFM 'outperforms' van Driest and that rGRA 'accurately reconstructs' the temperature-velocity relationship, yet no error norms, grid-convergence data, or direct quantitative comparisons (e.g., L2 deviations from the target log-law or from DNS temperature profiles) are supplied. These omissions leave the central claims on transformation superiority and analogy restoration without the numerical support required for load-bearing conclusions.

    Authors: We agree that quantitative error measures are needed to support the claims. In the revised manuscript we will add L2-norm deviations of the transformed velocity-defect profiles from the logarithmic law for both GFM and van Driest transformations, together with L2 errors between the rGRA-reconstructed temperature-velocity relation and the DNS data. Grid-convergence results for the key statistics will be included in a new appendix. revision: yes

Circularity Check

2 steps flagged

Fitting procedure restores log-law by construction; rGRA slip-plane BCs introduced to reconstruct relation

specific steps
  1. fitted input called prediction [Abstract]
    "Instead, a fitting-based optimization procedure is proposed to determine the kinematic virtual origin, which successfully restores the logarithmic behavior."

    The optimization target is explicitly the restoration of logarithmic behavior; therefore the 'success' in restoring it is achieved by construction of the fitting procedure rather than an independent prediction or derivation.

  2. self definitional [Abstract]
    "a modified rough-wall GRA (rGRA) is formulated by introducing an equivalent slip-plane or reference-point boundary conditions, which accurately reconstructs the temperature-velocity relationship by bypassing the near-wall thermal heterogeneity."

    The rGRA modification is defined via the introduction of slip-plane BCs chosen precisely to reconstruct the temperature-velocity relation and bypass heterogeneity; the claimed accuracy therefore follows from the definition of the modification rather than from first-principles closure or external constraint.

full rationale

The paper explicitly proposes a fitting-based optimization to determine the kinematic virtual origin because the conventional method fails, and states that this procedure 'successfully restores the logarithmic behavior.' This makes the reported restoration dependent on the fitting target rather than an independent derivation. Separately, the modified rGRA is formulated by introducing equivalent slip-plane boundary conditions specifically 'which accurately reconstructs the temperature-velocity relationship by bypassing the near-wall thermal heterogeneity.' Both central modifications reduce to choices made to achieve the desired outcomes, with no evidence of parameter-free derivation from the governing equations or external validation independent of the fit. The remainder of the analysis (GFM transformation, outer-layer similarity) builds on these steps but does not introduce additional circularity.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 2 invented entities

The central claims depend on standard compressible flow equations plus new modeling constructs introduced without external falsifiable evidence beyond the simulations themselves.

free parameters (2)
  • roughness height k+
    Specified as approximately 35 to place the roughness in the transitional regime for the DNS cases.
  • streamwise spacing ratio λx/k
    Fixed at 8 as the geometry parameter defining the square-rib array.
axioms (2)
  • standard math Compressible Navier-Stokes equations govern the spatially developing turbulent boundary layer
    Foundation for all direct numerical simulations described.
  • domain assumption A logarithmic velocity profile exists in the outer layer once a suitable virtual origin is applied
    Invoked to validate the proposed fitting procedure and GFM transformation.
invented entities (2)
  • fitting-based optimization procedure for kinematic virtual origin no independent evidence
    purpose: To determine zero-plane displacement when the conventional zero-moment method fails for cavity-type roughness
    New procedure proposed to restore logarithmic behavior.
  • modified rough-wall generalized Reynolds analogy (rGRA) with equivalent slip-plane boundary condition no independent evidence
    purpose: To reconstruct the temperature-velocity relationship by bypassing near-wall thermal heterogeneity
    Introduced to address breakdown of classical GRA due to form drag without corresponding heat mechanism.

pith-pipeline@v0.9.0 · 5566 in / 1613 out tokens · 108120 ms · 2026-05-10T16:15:49.913233+00:00 · methodology

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2 extracted references · 2 canonical work pages

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