Transition from classical to ultimate melting

Christopher J. Howland; Detlef Lohse; Edoardo Bellincioni; Kevin Zhong; Roberto Verzicco; Sander G. Huisman; Yiyu Zhou

REVIEW 4 major objections 5 minor 56 references

Transition from classical to ultimate melting

T0 review · 4 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper shows that melting objects in turbulent flow exhibit a true transition from slow classical melting at small scales to fast ultimate melting at large scales, with the crossover near a size-based Reynolds number of 4,000.

desk verdict Real and plausible claim—melting crosses from a Re^{1/2} to a Re^{0.8–1} regime near Re_D0 ≈ 4000—but the transition location rests on a visual collapse without error bars, so treat it as a strong hypothesis, not a calibrated law. read the letter →

arxiv 2601.06517 v2 pith:LDW7DUDV submitted 2026-01-10 physics.flu-dyn

Edoardo Bellincioni , Kevin Zhong , Christopher J. Howland , Yiyu Zhou , Sander G. Huisman , Roberto Verzicco , Detlef Lohse This is my paper

classification physics.flu-dyn

keywords meltingturbulenceNusseltnumberscalingboundary-layertransitionclassical-to-ultimatehomogeneousisotropicStefanproblemlatentheattransfer

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to settle whether one scaling law can describe melting in turbulence across all object sizes. By combining new experiments and direct numerical simulations with re-analysed literature data over four orders of magnitude in the object Reynolds number, it claims to find two distinct regimes: the Nusselt number grows as Re^{1/2} for small objects (classical melting) and as Re^{0.8–1} for large objects (ultimate melting), with a transition around Re≈4000. The cause is the boundary layer around the melting object switching from laminar-type to turbulent-type. If true, this means melt-rate predictions cannot rely on a single power law; the size of the melting object determines which regime applies. That matters for extrapolating laboratory melt rates to icebergs, ice shelves, and industrial phase-change systems.

What carries the argument

The key object is the Nusselt number Nu, defined from the Stefan condition and the measured radius decay rate, which converts a melt-rate measurement into a dimensionless heat-transfer coefficient. The argument runs on the two boundary-layer scaling laws for Pr>1: Nu∝Re^{1/2}Pr^{1/3} for a laminar thermal boundary layer (Prandtl–Blasius–Pohlhausen), and Nu∝Re·L(Re), indistinguishable from an effective power law Nu∝Re^{0.8}, for a turbulent Prandtl–von Kármán boundary layer. The crossover is set by the shear Reynolds number Re_s≈Re_{D0}^{1/2}≈60, lower than the canonical flat-plate range of 100–500, which the paper attributes to roughness, freestream turbulence, unsteadiness, and three-dimens

What would settle it

A single well-resolved experiment or simulation tracking one melting object continuously while Re_{D0} is varied from below 1,000 to above 20,000 (by changing object size or turbulence intensity) would settle it: if the local exponent stays at 1/2 across the whole range, or if the crossover moves by an order of magnitude when freestream turbulence intensity is changed, the universal transition claim fails.

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Extended reading notes

Core claim

Across experiments, DNS, and prior datasets, the dimensionless melt rate Nu (compensated by Pr^{1/3}) collapses onto one curve versus Re_{D0}, the Reynolds number based on the object's initial diameter and the turbulent velocity at that scale. Below Re_{D0}≈4000 the collapse follows Nu∝Re_{D0}^{1/2}, the Prandtl–Blasius–Pohlhausen laminar boundary-layer prediction. Above it, the scaling steepens to Nu∝Re_{D0}^{γ} with 0.8≤γ≤1, the signature of a turbulent boundary layer with logarithmic corrections. The authors interpret the crossover as the melting analogue of the classical-to-ultimate transition in thermal convection: the boundary layer around the melting object becomes turbulent, sharply

Load-bearing premise

All conclusions hinge on the assumption that melt-rate data from many different experiments and simulations, with different temperatures, ice sizes, and turbulence levels, line up on one master curve when plotted properly; if that alignment breaks down, the claimed crossover could be an illusion.

Editorial extensions

If this is right

Melt-rate parametrizations that assume one power law across all scales will mispredict large-object melting; the exponent must switch from approximately 1/2 to 0.8–1 near Re_{D0}≈4000.
For geophysical applications, icebergs and glacier fronts typically sit far above the transition, so they should be modelled with the ultimate-melting scaling rather than the small-scale classical one.
Because Nu depends on Re_{D0} rather than Re_λ, a predictive scheme can be built from the local dissipation rate and object size alone, without needing the full turbulent spectrum.
The near-insensitivity to Eulerian versus Lagrangian advection (apart from a slight DNS-visible offset) means the same scaling applies to fixed and freely drifting melting objects.
The transition is non-normal and noise-sensitive, so the onset Reynolds number should be treated as a range that can shift with environmental disturbance levels, not as a sharp universal constant.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: if the collapse is universal, the same two-regime structure should appear in turbulent heat transfer from heated spheres without melting, since the boundary-layer mechanism is the same; a targeted reanalysis of heat-transfer data could test this independently.
Editorial inference: the absence of error bars and the wide parameter scatter in the master curve mean the reported crossover position near 4,000 might shift when more precise data at intermediate Re_{D0} become available; the qualitative two-regime claim is more secure than the exact threshold.
Editorial inference: the Re_{D0}-only dependence suggests a testable extension—melting objects of identical diameter in flows with different Taylor–Reynolds numbers but the same Re_{D0} should melt at the same rate.
Editorial inference: the early transition (Re_s≈60) points to a practical design lever: surface roughness or freestream turbulence could be tuned to deliberately trigger ultimate melting in latent-heat storage systems.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This paper's central claim—that melting crosses from a Nu ~ Re^{1/2} classical regime to a Nu ~ Re^{0.8–1} ultimate regime around Re_D0 ≈ 4000—is plausible and worth taking seriously, but the quantitative position of the transition rests on a visual data collapse, not on a statistical analysis. As a result, the paper is a well-argued hypothesis backed by a broad meta-data set, not yet a precisely calibrated law.

What is genuinely new: the authors combine controlled experiments, DNS, and three literature data sets spanning a wide Re_D0 range, and show that melt-rate data from very different conditions (Eulerian vs Lagrangian, different Pr, Ste, Re_lambda) collapse onto a single two-branch curve. That is a useful empirical contribution. The derivation of Nu from the measured melt rate via the Stefan condition is clean and does not involve fitted parameters. The low-Re scaling is anchored to independent boundary-layer theory, and the high-Re branch to standard turbulent heat-transfer correlations, so the circularity concern is minor. I also found the weak dependence on Re_lambda interesting—if correct, it supports the idea that only eddies near the melting-object scale matter.

Soft spots: Fig. 3 has no error bars. The exponent γ = 0.8–1.0 and the crossover near Re_D0 ≈ 4000 are read by eye, and the paper does not test whether a single power law or a smooth crossover is statistically rejected. The stress-test worry about pooling heterogeneous datasets is real: with 20–30% per-dataset scatter, the crossover could shift by a factor of two or three. Still, the fact that several independent datasets with different parameters land on the same branches makes me doubt the transition is an artifact. The boundary-layer state is never directly measured; the inferred transition is supported only by the scaling change, which is a legitimate but indirect inference. The abstract's "four orders of magnitude" is overstated—the present melting experiments cover about two orders, and the four-order range comes only by combining with DNS and literature—easily fixed. Some methods are deferred to future papers; that is acceptable for a letter, but reviewers will want the uncertainty analysis before accepting the transition as a quantitative result.

Verdict: this deserves a serious referee, and I expect the core claim to survive, but the revision needs error bars, a statistical breakpoint fit, and clearer reporting of what the present data alone establish. I'd bring it to a reading group if we had one—it's a good example of a meta-study that asks the right question and mostly delivers.

Referee Report

4 major / 5 minor

Summary. The paper reports a meta-study combining new experiments, DNS, and reanalyzed literature data for ice balls melting in homogeneous isotropic turbulence, covering roughly four orders of magnitude in the Reynolds number Re_D0 based on the initial ball diameter. The central claim is that the dimensionless melt rate (Nusselt number Nu) obeys two distinct scaling regimes: Nu ∝ Re_D0^{1/2} Pr^{1/3} at low Re_D0 ('classical melting', laminar-type boundary layer) and Nu ∝ Re_D0^γ with 0.8 ≤ γ ≤ 1 at high Re_D0 ('ultimate melting', turbulent-type boundary layer), with a transition near Re_D0 ≈ 4000. The authors interpret this as the melting analog of the classical-to-ultimate turbulence transition in thermal convection and argue that single-power-law melt parametrizations are inadequate for geophysical extrapolation.

Significance. If correct, the claimed transition would have direct implications for ice-ocean parameterizations, latent-heat storage, and other melting applications, because it would require knowing the melting regime before extrapolating laboratory measurements to geophysical scales. The paper's strengths are its combination of experiments and two-way-coupled DNS, the use of the Stefan condition to derive the Nusselt number from measured melt rates rather than from a fitted correlation, and the inclusion of external datasets spanning a wide range of control parameters. However, the central claim rests on a visually inferred collapse in Fig. 3 and on scaling arguments rather than on direct boundary-layer measurements, so the statistical and mechanistic support is currently incomplete.

major comments (4)

[Fig. 3] The identification of the two scaling regimes and the transition at Re_D0≈4000 is made by eye from scatter without error bars, goodness-of-fit statistics, or model selection. Given that the datasets differ in Re_λ, Pr, Ste, roughness, and Eulerian/Lagrangian condition, the collapse onto a single master curve needs quantitative support: per-dataset uncertainties, a breakpoint or regression analysis with confidence intervals for the exponents, and a test of whether the two-power-law model significantly outperforms a single power law over the full range. Without this, the claimed transition and even the collapse itself remain plausible but underdetermined.
[Eq. (1) and time averaging] Equation (1) uses the initial diameter D0 to define both Re_D0 and Nu, while ⟨Ṙ⟩ is the time-averaged melt rate over the melting trajectory. For large Re_D0 cases, D(t) decreases substantially during the experiment (e.g., snapshots show melting to a fraction of the initial volume), so a time average with fixed D0 mixes states with different instantaneous Reynolds numbers. This can blur or shift the apparent transition relative to a steady-state boundary-layer criterion. The authors should quantify how much D(t) varies over the averaging window and, if needed, test whether using an instantaneous Re_D(t) changes the inferred exponents or transition location.
[Data collapse heterogeneity] The master-curve collapse in Fig. 3 combines melting experiments, DNS, and the non-melting heat-transfer correlation of Guo et al. [35], spanning Stefan numbers, Prandtl numbers, roughness levels, and forcing conditions. The text states that Eulerian and Lagrangian results agree within experimental scatter, but no quantitative assessment of dataset-dependent offsets is provided. A meaningful check would be to fit the scaling exponents separately for each dataset and report the spread, or to test whether the residuals from the master curve correlate with the other control parameters. If the collapse is not statistically robust, the apparent 'universal' transition could be an artifact of pooling heterogeneous data.
[Boundary-layer interpretation] The central physical interpretation — laminar-to-turbulent boundary-layer transition — is inferred solely from the macro-scale Nusselt-number scaling. The paper does not directly measure or visualize the boundary-layer state in either experiments or DNS, and the early transition at Re_s≈60 is explained qualitatively by roughness, freestream turbulence, unsteadiness, and 3D effects. While this is a plausible interpretation consistent with the data, it is not a direct test. The DNS could provide the interface-adjacent temperature or shear fields to confirm the boundary-layer state; at minimum, the authors should state explicitly that the boundary-layer transition is an interpretation, not a measurement, and discuss what alternative mechanisms (e.g., changing controlling eddy scale) could produce the observed exponent change.

minor comments (5)

[Abstract vs. Fig. 3] The abstract states 'four orders of magnitude in scale,' but the Re_D0 range in Fig. 3 is about 10^1 to 10^5, with the melting experiments themselves covering roughly two orders. Clarify what exactly spans four orders of magnitude (or qualify as 'covered by the combined datasets').
[Methods/Experiment] Two key methodological details are deferred to 'a forthcoming publication': the spherical fit method and the full set of simulation runs. For a meta-study whose conclusion depends on data quality, provide at least a summary of the measurement uncertainty and the parameter ranges in this paper, or cite a preprint if available.
[Fig. 3 legend] The legend in Fig. 3 lists the varying parameters but does not clearly identify which symbol types correspond to the present experiments, present DNS, Machicoane et al., McCutchan et al., and Guo et al. Make the dataset mapping explicit in the caption or legend.
[Text near Fig. 2] The text says 'the largest Re_λ available from DNS is about 200' and Fig. 3 shows DNS data with Re_λ ≈ 50; the flow conditions are not fully consistent between the main text and figures. Unify the notation and ensure the reported Re_λ values are precise for each dataset.
[Acknowledgements] Typo: 'Acknlowledgements' should be 'Acknowledgements'.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: the two-regime scaling transition is extracted from measured melt rates and compared with independent theory and correlations; self-citations are present but not load-bearing.

full rationale

The paper's derivation chain is not circular. The Nusselt number is obtained from measured melt rates via the Stefan condition (Eqs. 3-6), not from a fitted parameter, and the low-Re_D0 scaling Nu ~ Re_D0^{1/2} Pr^{1/3} is anchored to independent Prandtl–Blasius–Pohlhausen theory. The high-Re_D0 behavior is compared with standard engineering correlations and external no-melting heat-transfer data (Guo et al. [35]), and the transition near Re_D0≈4000 is read from the combined data collapse rather than imposed by the definitions of Nu and Re_D0. The self-citations to the DNS framework [34] and to the authors' ultimate-turbulence reviews [28,29] are references to tools and analogies, not load-bearing premises; the same scaling is supported by new controlled experiments, DNS, and re-analysed independent literature data [32,33]. The skeptic's concern about the absence of error bars and goodness-of-fit statistics for Fig. 3 is an evidence-quality/correctness issue, not circularity: even a visually assessed collapse is not a logical reduction of the prediction to its inputs. Because minor self-citation exists, the rubric gives a score of 2 rather than 0, but there is no significant circularity.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central result rests on standard heat-transfer scalings and the choice of Re_D0 as the governing parameter. The main non-standard ingredients are the assumed master-curve collapse across heterogeneous data sets and the ad hoc explanation of the early transition onset. No new physical entities are introduced.

free parameters (2)

High-Reynolds exponent γ = 0.8–1.0
The exponent in Nu∝Re^γ for the 'ultimate melting' regime is fitted to the combined experimental data; it is not derived from theory. This exponent is part of the central claim.
Transition Reynolds number Re_D0 (or Re_s) = ≈4000 (Re_s≈60)
The crossover between the two power laws is inferred by eye from the log-log plot and then rationalized by a list of qualitative effects (roughness, freestream turbulence, unsteadiness, 3D effects). No uncertainty or statistical fitting is given.

assumptions (6)

standard math Stefan condition plus neglect of ice-side conduction (Nu ≫ Bi)
Equation (6) is reduced to Eq. (1) by dropping the Biot term; this is standard for ice-ocean heat transfer but becomes less accurate at very low convection.
standard math Prandtl–Blasius–Pohlhausen scaling Nu∝Re^{1/2}Pr^{1/3} for a laminar thermal boundary layer
Used to identify the low-Re regime and to justify the Pr^{1/3} compensation in Fig. 3.
standard math Turbulent boundary-layer heat-transfer correlation Nu∝Re^0.8 (or Re·L(Re))
Used to interpret the high-Re regime; the paper notes logarithmic corrections make this indistinguishable from an effective power law.
domain assumption U_D0=(εD0)^{1/3} is the relevant velocity scale; only eddies of size D0 and smaller matter
This defines Re_D0 and underpins the whole data collapse. The authors justify it by analogy to Kolmogorov–Hinze fragmentation, but it is not directly measured for melting.
domain assumption Datasets with different Ste, Pr, Reλ, Eulerian/Lagrangian conditions, and roughness can be combined on one master curve
Fig. 3 shows these parameters vary across data sets; the collapse is assumed and no statistical test is provided. If this assumption fails, the transition may be an artifact.
ad hoc to paper The observed early transition (Re_s≈60 instead of 100–500) is caused by roughness, freestream turbulence, unsteadiness, and 3D effects
The authors list qualitative receptivity mechanisms to reconcile the data with the flat-plate boundary-layer expectation, but no direct measurement of the boundary-layer state or transition mechanism is presented.

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Cite this review

Pith. "Pith review of Transition from classical to ultimate melting." pith.science (2026). https://pith.science/paper/LDW7DUDV

@misc{pith2026260106517,
  author       = {Pith},
  title        = {Pith review of: Transition from classical to ultimate melting},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LDW7DUDV}},
  note         = {Machine review of arXiv:2601.06517}
}

read the original abstract

Melting is omnipresent in nature and technology, with applications ranging from metallurgy, biology, food science, and latent thermal energy storage to oceanography, geophysics, and climate science, and occurring on all scales from sub-millimeter to global scales. The key objective is to understand the rate at which an object melts as a function of its size and of the ambient conditions. To achieve this it is important to be able to extrapolate from small scale experiments and observations to large or even global scales. This is done by scaling laws. However, these are only meaningful if there is no transition from one scaling relation to another one. Here we show, however, that for both fixed and freely-advected melting objects immersed in a turbulent flow a melting transition does exist, namely from slow melting at the small scales to fast melting at the large scales. We do so by controlled melting experiments and corresponding direct numerical simulations, covering four orders of magnitude in scale. The transition corresponds to the transition from a laminar-type boundary layer around the melting object to a turbulent-type boundary layer, i.e., from so-called classical turbulence to ultimate turbulence, with its enhanced transport properties. Our results thus provide a quantitative understanding of the flow physics of the melting process and thereby enable a better extrapolation and prediction of melt rates on large scales such as relevant in geophysics, oceanography, and climate science.

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