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arxiv: 1907.02504 · v1 · pith:XR6FPMTNnew · submitted 2019-07-04 · ⚛️ physics.flu-dyn

Heat transfer in rough-wall turbulent thermal convection in the ultimate regime

Michael MacDonald , Nicholas Hutchins , Detlef Lohse , Daniel Chung This is my paper

Pith reviewed 2026-05-25 08:52 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords rough wallturbulent convectionNusselt numberRayleigh numberlogarithmic profilesheat transfer scalingultimate regime
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The pith

Rough-wall heat transfer follows Nu ~ Ra^{0.42} scaling and never reaches the ultimate regime at finite Ra.

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

The authors extend earlier scaling analyses to rough walls by assuming that temperature profiles stay logarithmic but are shifted by the roughness. This assumption produces a predicted heat transfer scaling of Nu proportional to Ra to the power 0.42, or equivalently a Stanton number that decreases slowly with Reynolds number. The key difference from momentum transfer arises because the temperature equation has no counterpart to the pressure drag that dominates skin friction in the fully rough regime. As a result the ultimate regime, in which Nu would scale as the square root of Ra, is not expected to appear for heat transfer.

Core claim

Using logarithmic temperature profiles with a roughness-induced shift in analyses of forced convection over rough walls predicts an effective scaling Nu ∼ Ra^{0.42} corresponding to Ch ∼ Re^{-0.16}. Unlike the skin-friction coefficient which becomes independent of Re due to pressure drag, the absence of an analog to pressure drag in the temperature advection equation produces this different scaling. Consequently the asymptotic ultimate regime where Nu ∼ Ra^{1/2} will never be reached for heat transfer at finite Ra.

What carries the argument

Logarithmic temperature profiles with a roughness-induced shift, applied to extend prior ultimate-regime analyses to rough-wall data and derive the effective scaling.

If this is right

  • Heat transfer scales differently from momentum transfer in the fully rough regime.
  • The effective exponent for Nu versus Ra is 0.42 rather than approaching 0.5.
  • The ultimate regime is not attained for heat transfer even as Ra increases.
  • Stanton number decreases as Re to the power -0.16.

Where Pith is reading between the lines

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

  • The analysis implies that increasing Ra further will not change the 0.42 exponent toward the ultimate value.
  • Similar logarithmic assumptions might be tested in other scalar transport problems with rough boundaries.
  • Experiments could check whether temperature profiles stay logarithmic at the highest accessible Ra.

Load-bearing premise

The temperature profiles stay logarithmic with a fixed roughness-induced shift even in the ultimate regime.

What would settle it

A simulation or experiment at sufficiently high Ra showing either non-logarithmic temperature profiles or a Nu-Ra exponent approaching 0.5 would contradict the prediction.

Figures

Figures reproduced from arXiv: 1907.02504 by Daniel Chung, Detlef Lohse, Michael MacDonald, Nicholas Hutchins.

Figure 1
Figure 1. Figure 1: FIG. 1. Instantaneous steamwise velocity (top) and temperature (bottom) for smooth-wall (left), [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Sketch of the Rayleigh–B´enard system with roughness. Under fully rough conditions, [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Skin-friction coefficient and (b) heat-transfer coefficient (Stanton number) against [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (a) shows the Nusselt number as a function of the Rayleigh number. The inset highlights just the DNS data, where the smooth-wall forced convection data (circles) have an effective scaling exponent of γef f ≈ 0.38 (solid black line), matching that observed in ultimate RB convection [9], as expected. Note that the mechanically supplied shear of forced convection corresponds to a very strong wind in RB convec… view at source ↗
read the original abstract

Heat and momentum transfer in wall-bounded turbulent flow, coupled with the effects of wall-roughness, is one of the outstanding questions in turbulence research. In the standard Rayleigh-B\'enard problem for natural thermal convection, it is notoriously difficult to reach the so-called ultimate regime in which the near-wall boundary layers are turbulent. Following the analyses proposed by Kraichnan [Phys. Fluids vol 5., pp. 1374-1389 (1962)] and Grossmann & Lohse [Phys. Fluids vol. 23, pp. 045108 (2011)], we instead utilize recent direct numerical simulations of forced convection over a rough wall in a minimal channel [MacDonald, Hutchins & Chung, J. Fluid Mech. vol. 861, pp. 138--162 (2019)] to directly study these turbulent boundary layers. We focus on the heat transport (in dimensionless form, the Nusselt number $Nu$) or equivalently the heat transfer coefficient (the Stanton number $C_h$). Extending the analyses of Kraichnan and Grossmann & Lohse, we assume logarithmic temperature profiles with a roughness-induced shift to predict an effective scaling of $Nu \sim Ra^{0.42}$, where $Ra$ is the dimensionless temperature difference, corresponding to $C_h \sim Re^{-0.16}$, where $Re$ is the centerline Reynolds number. This is pronouncedly different from the skin-friction coefficient $C_f$, which in the fully rough turbulent regime is independent of $Re$, due to the dominant pressure drag. In rough-wall turbulence the absence of the analog to pressure drag in the temperature advection equation is the origin for the very different scaling properties of the heat transfer as compared to the momentum transfer. This analysis suggests that, unlike momentum transfer, the asymptotic ultimate regime, where $Nu\sim Ra^{1/2}$, will never be reached for heat transfer at finite $Ra$.

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 extends Kraichnan and Grossmann-Lohse analyses of ultimate-regime thermal convection by using forced-convection rough-wall DNS data to assume logarithmic temperature profiles with a fixed roughness-induced shift. This yields the effective scaling Nu ∼ Ra^{0.42} (equivalently C_h ∼ Re^{-0.16}) and the conclusion that, unlike momentum transfer (where C_f becomes Re-independent due to pressure drag), the asymptotic ultimate regime Nu ∼ Ra^{1/2} is never reached for heat transfer at finite Ra because the temperature equation lacks an analog to pressure drag.

Significance. If the assumed logarithmic form with constant shift persists, the work supplies a concrete, assumption-driven extrapolation that explains the distinct scaling behaviors of heat versus momentum transfer in rough-wall turbulence. The approach leverages existing DNS to address a regime that remains inaccessible in natural-convection simulations, and the explicit contrast with pressure-drag effects is a useful conceptual contribution.

major comments (2)
  1. [Abstract and analysis section] Abstract and analysis section: the exponent 0.42 is obtained by direct substitution of the logarithmic temperature profile (with roughness shift taken from the authors' prior DNS) into the Grossmann-Lohse framework; the numerical value is therefore fixed by the profile assumption rather than by an independent constraint from the natural-convection equations.
  2. [Analysis and conclusion] Analysis and conclusion: the claim that Nu ∼ Ra^{1/2} is unreachable at finite Ra rests entirely on the premise that the temperature profiles remain logarithmic with a Ra-independent roughness shift even when the boundary layers become fully turbulent. No verification, sensitivity test, or argument is supplied showing why the shift remains constant once the near-wall region is in the ultimate regime.
minor comments (1)
  1. The manuscript would benefit from an explicit statement of the Reynolds-number range of the underlying DNS data and the precise functional form used for the roughness shift when inserted into the scaling relations.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their detailed review and valuable feedback on our manuscript. Below we provide point-by-point responses to the major comments. We have made revisions to clarify certain aspects of our analysis.

read point-by-point responses

  1. Referee: [Abstract and analysis section] Abstract and analysis section: the exponent 0.42 is obtained by direct substitution of the logarithmic temperature profile (with roughness shift taken from the authors' prior DNS) into the Grossmann-Lohse framework; the numerical value is therefore fixed by the profile assumption rather than by an independent constraint from the natural-convection equations.

    Authors: We concur that the specific exponent of 0.42 is determined by the substitution of the logarithmic temperature profile and the fixed roughness shift (obtained from our prior DNS) into the Grossmann-Lohse framework. This is the core of our method: to leverage high-fidelity forced-convection data to inform the profile assumptions in the natural-convection analysis. The manuscript does not claim an independent derivation from the natural-convection equations but rather an effective scaling based on this informed assumption. To address this comment, we have revised the abstract and the analysis section to more explicitly state that the exponent follows from the profile assumption. revision: yes

  2. Referee: [Analysis and conclusion] Analysis and conclusion: the claim that Nu ∼ Ra^{1/2} is unreachable at finite Ra rests entirely on the premise that the temperature profiles remain logarithmic with a Ra-independent roughness shift even when the boundary layers become fully turbulent. No verification, sensitivity test, or argument is supplied showing why the shift remains constant once the near-wall region is in the ultimate regime.

    Authors: The referee is correct that our conclusion depends on the persistence of the logarithmic profile with a constant roughness shift into the ultimate regime. This premise is based on the Reynolds-number independence of the shift observed in the forced-convection DNS over a range of Re in the fully rough regime. We argue that the analogy holds because the near-wall turbulent boundary layers in the ultimate regime of natural convection are expected to behave similarly to those in forced convection. However, we acknowledge the lack of direct verification or sensitivity tests, which is due to the inaccessibility of the ultimate regime in current natural-convection simulations. We have partially revised the conclusion to include a more detailed discussion of this assumption and its justification via the forced-convection analogy, while noting the limitation. revision: partial

standing simulated objections not resolved
  • The absence of direct numerical verification or sensitivity analysis for the constancy of the roughness shift in the ultimate regime of Rayleigh-Bénard convection, as this regime is not yet accessible to simulation.

Circularity Check

0 steps flagged

No significant circularity; scaling follows explicitly from stated assumption

full rationale

The paper states its central premise as an explicit assumption ('we assume logarithmic temperature profiles with a roughness-induced shift') and then performs the algebraic extension of the Kraichnan/GL framework to obtain Nu ∼ Ra^{0.42}. This is a forward derivation from the chosen functional form rather than a self-referential loop in which the output is used to justify the input. Prior DNS data from the authors' own 2019 JFM paper supplies the roughness shift value but is external observational input, not a fitted parameter renamed as a prediction. No equation reduces the claimed exponent to the assumption by construction, and the 'never reaches 1/2' conclusion is presented as a logical consequence of the maintained log-law premise rather than an independent verification. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The load-bearing premise is the persistence of logarithmic temperature profiles with a fixed roughness shift; no new free parameters are introduced beyond those already present in the cited log-law literature, and no new physical entities are postulated.

axioms (1)
  • domain assumption Temperature profiles are logarithmic with a roughness-induced additive shift even in the ultimate regime
    Invoked when extending Kraichnan/Grossmann-Lohse analysis to the rough-wall data (abstract).

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Reference graph

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