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arxiv: 2606.13505 · v1 · pith:4L4O6O3Gnew · submitted 2026-06-11 · ⚛️ physics.flu-dyn

Ultimate regime in Rayleigh-Darcy Convection

Garima Varshney , Anikesh Pal , Narsimha Reddy Rapaka This is my paper

Pith reviewed 2026-06-27 05:35 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords rayleigh-darcy convectionultimate regimeheat transfer scalingdirect numerical simulationporous domainthermal boundary layersplume dynamics
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The pith

DNS reveals onset of ultimate regime in Rayleigh-Darcy convection at Ra ≈ 4×10^5 via Nu-Ra slope change

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

The paper uses direct numerical simulations to study Rayleigh-Darcy convection in a 3D porous domain over Ra from 10^3 to 10^6. It establishes that the Nusselt number depends linearly on the Rayleigh number, but finds a clear change in the slope of this dependence at Ra ≈ 4×10^5. This slope change is taken as evidence for the start of the ultimate regime. Supporting observations include the evolution of thermal structures into finer protoplumes that merge into megaplumes and a shift of thermal dissipation into the bulk. Readers would care if this confirms how heat transport behaves at extreme Rayleigh numbers in porous materials.

Core claim

The central discovery is that the ultimate regime in Rayleigh-Darcy convection onsets at Ra ≈ 4×10^5, where the linear Nu ~ Ra scaling persists but with a different slope, accompanied by finer near-wall protoplumes, increased protoplume numbers, and thermal dissipation moving from the boundary layers to the bulk.

What carries the argument

The scaling of the Nusselt number with Rayleigh number and the mean wavenumber of thermal structures, which quantify the transition through changes in slope and linear variation with higher slope in the ultimate regime.

If this is right

  • Results for Ra ≥ 4×10^5 match extrapolated ultimate-regime predictions within 1.24%.
  • Thermal boundary-layer thickness scales as Ra^{-1} and Nu^{-1} in the ultimate regime.
  • Protoplume size decreases and their number increases with Ra, enhancing heat transport.
  • Mean wavenumber at the mid-plane shows weaker but positive scaling with Ra in the ultimate regime.

Where Pith is reading between the lines

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

  • The ultimate regime appears accessible at Ra values around 10^5 to 10^6 in porous convection, enabling direct study of its properties.
  • Linear heat transfer scaling may continue indefinitely in Darcy flows even as structures refine, unlike potential saturation in other systems.
  • The transition Rayleigh number being lower than in classical Rayleigh-Bénard convection points to the Darcy drag altering the route to ultimate-regime dynamics.

Load-bearing premise

The DNS results at Ra ≥ 4×10^5 accurately capture the physical transition without resolution or domain-size artifacts.

What would settle it

A simulation at Ra = 5×10^5 using twice the current resolution that yields a Nusselt number consistent with the lower-slope extrapolation would falsify the onset claim.

Figures

Figures reproduced from arXiv: 2606.13505 by Anikesh Pal, Garima Varshney, Narsimha Reddy Rapaka.

Figure 1
Figure 1. Figure 1: (a) Schematic representation of the cuboidal porous domain characterized by constant domain property 𝜙, 𝜅, saturated with a mixture of 𝑠𝑐𝐶𝑂2 and brine, featuring cold (𝜃 = 0 at 𝐿𝑧 = 1) and warm boundaries (𝜃 = 1 at 𝐿𝑧 = 0) at the top (blue) and bottom (red), respectively. The domain length in the 𝑧 direction remains constant across all the cases, while optimized lengths are utilized for the horizontal dime… view at source ↗
Figure 2
Figure 2. Figure 2: (a) The variation of the time-averaged Nusselt number (𝑁𝑢) in relation to the Rayleigh number (𝑅𝑎) for the current numerical simulation (shown by filled blue circles) within the range of 1 × 103 ⩽ 𝑅𝑎 ⩽ 1 × 106 . The best fit is 𝑁𝑢 = 0.009𝑅𝑎 + 8.27 (𝑅 2=0.9987) for 𝑅𝑎 ⩽ 2.5 × 105 and 𝑁𝑢 = 0.008𝑅𝑎-805.94 (𝑅 2=0.9898) for 𝑅𝑎 ⩾ 4 × 105 . Results obtained in previous studies are shown by different open symbols … view at source ↗
Figure 3
Figure 3. Figure 3: The time-averaged Nusselt number (𝑁𝑢) is presented for various 𝑅𝑎 values across different domain sizes, along with a comparison to the results reported in Pirozzoli et al. (2021). 3.2. Flow Dynamics The change in 𝑁𝑢 with 𝑅𝑎 implies the corresponding change in the flow structure—ranging from organized convective rolls to chaotic, boundary-driven plume shedding. This behavior arises from the relative strengt… view at source ↗
Figure 4
Figure 4. Figure 4: The temperature distribution at the midplane of the horizontal direction shows the protoplume region near the boundary (captured by black dotted boxes near the upper and lower boundary) and extensive megaplumes across the domain for (a)𝑅𝑎=1 × 104 (b) 𝑅𝑎=2.5 × 105 (c)1 × 106 . similar to the prediction of the three-dimensional ’heat-exchanger’ solution of Hewitt et al. (2014). We did not include the low 𝑅𝑎 … view at source ↗
Figure 5
Figure 5. Figure 5: Instantaneous dimensionless temperature distribution at the near-wall plane (𝑧 = 30/𝑅𝑎) for (a) 𝑅𝑎 = 1×104 , (b) 𝑅𝑎 = 8×104 , (c) 𝑅𝑎 = 2×105 , (d) 𝑅𝑎 = 6×105 , (d) 𝑅𝑎 = 8×105 , and (e) 𝑅𝑎 = 1×106 [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The time-averaged and horizontally averaged temperature ⟨𝜃⟩ distribution in the vertical direction (𝑧) shows a relatively weak linear temperature gradient across the domain. Inset (a) shows the boundary region by the intersection of the linear profile fitting the bulk and near-wall regions; (b) zoomed view of the near-wall region showing the boundary layer thickness with respect to 𝑅𝑎 for 𝑅𝑎 ⩾ 1 × 105 . (a… view at source ↗
Figure 7
Figure 7. Figure 7: (a) Time- and horizontally averaged variance of temperature, (b) variation of the the thermal boundary layer thickness (𝛿𝜃 ) with 𝑅𝑎 and 𝑁𝑢. for 𝑅𝑎 ⩾ 2 × 105 compared to 𝑅𝑎 = 1 × 105 . The near-wall temperature profile shrinks towards the wall for 𝑅𝑎 ⩾ 4 × 105 and reduces further with increasing 𝑅𝑎, indicating significant decrease in the boundary-layer thickness compared to 𝑅𝑎 ⩽ 1 × 105 . De Paoli et al. (… view at source ↗
Figure 8
Figure 8. Figure 8: (a) Horizontal and time-averaged thermal dissipation profile across the vertical direction for different 𝑅𝑎, (b) Boundary-layer (BL) and bulk contributions to the mean scalar dissipation (𝜖) for different cases. (c) Ratio of the Nusselt number to the dimensionless thermal dissipation for different 𝑅𝑎. 8 × 104 are ∼ 10−1 and ∼ 10−3 respectively, similar to the results of De Paoli et al. (2022). We also plot… view at source ↗
Figure 9
Figure 9. Figure 9: The mean wavenumber of the temperature distribution is calculated using Equation 3.2 for (a) near the wall (z=30/𝑅𝑎) and (b) at the centre as a function of 𝑅𝑎 along with the comparison with available literature Hewitt et al. (2014), De Paoli et al. (2022) for the respective locations. The best fits are 𝑘 ∼ 𝑅𝑎 and 𝑘 ∼ 𝑅𝑎0.489 respectively. equation 3.4 for 𝑧 = 30/𝑅𝑎. Hewitt et al. (2014) reported the power … view at source ↗
read the original abstract

DNS of Rayleigh-Darcy convection in a 3D porous domain is performed at Ra $\in [10^3, 10^6]$ to investigate heat-transfer scaling, thermal boundary-layer dynamics, and flow-structure evolution in the unexplored ultimate regime. The Nu exhibits an approximately linear dependence on Ra throughout the investigated range. However, a distinct change in slope is observed at $Ra \approx 4\times10^5$, indicating the onset of the ultimate regime. For $Ra \leq 2.5\times10^5$, our scaling is 6.25% lower than that reported by \cite{hewitt2014high}, while for $Ra \geq 4\times10^5$ our results are within 1.24% of the extrapolated ultimate-regime prediction of \cite{pirozzoli2021towards}. Analysis of thermal structure reveals formation of near-wall protoplumes that merge into large-scale columnar megaplumes. With increasing Ra, the size of the protoplumes decreases, whereas the numbers increase, thus enhancing boundary-layer convection and heat transport. The thermal boundary-layer thickness scales as ~ Ra^{-1} and ~ Nu^{-1}, corroborating the persistence of linear heat-transfer scaling in the ultimate regime. The thermal dissipation is found to be increasingly shifting from the boundary layer to the bulk with increasing $Ra$, further indicating that the finer protoplumes efficiently transport heat from walls to bulk. The flow structures are quantified using the dominant length scale using the mean wavenumber ($\overline{k}$). It exhibits linear variation with $Ra$ for near-wall structures, with a higher slope in the ultimate regime, signifying finer protoplumes. At the mid-plane, a weaker scaling suggests that the megaplumes also become finer with increasing $Ra$ in the ultimate regime, thus leading to efficient heat transport in the bulk.

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 reports direct numerical simulations of Rayleigh-Darcy convection in a 3D porous domain over Ra ∈ [10^3, 10^6]. It finds that Nu scales approximately linearly with Ra across the range, but with a distinct change in slope at Ra ≈ 4×10^5 that is interpreted as the onset of the ultimate regime. The results are compared to literature scalings (6.25% lower than Hewitt et al. below the transition and 1.24% agreement with Pirozzoli et al. extrapolation above it), and the authors analyze the formation of near-wall protoplumes that merge into columnar megaplumes, the Ra^{-1} scaling of thermal boundary-layer thickness, the shift of thermal dissipation from boundary layers to bulk, and the mean wavenumber ar{k} as a measure of flow-structure refinement.

Significance. If the reported slope change is shown to be free of numerical artifacts, the work supplies direct numerical evidence for the ultimate regime in Rayleigh-Darcy convection together with a structural explanation based on protoplume thinning and increased number. The quantitative agreement with an independent extrapolated prediction is a concrete strength, as is the explicit link drawn between boundary-layer dynamics and the persistence of linear Nu(Ra) scaling.

major comments (2)
  1. [Abstract / Methods] Abstract and (presumably) §3–4: the central claim that a change in Nu(Ra) slope at Ra ≈ 4×10^5 marks the physical onset of the ultimate regime rests on the DNS being adequately resolved at Ra ≥ 4×10^5. No grid resolution, boundary-layer point counts, domain aspect ratio, or statistical-convergence diagnostics are stated, even though the smallest scales are expected to shrink at least as Ra^{-1/2}. Without these data the observed break cannot be distinguished from a resolution-dependent artifact.
  2. [Abstract] Abstract: the statement that protoplumes become finer and more numerous with Ra is used to explain the enhanced heat transport above the transition, yet no quantitative measure (e.g., boundary-layer grid spacing relative to the reported Ra^{-1} thickness or wavenumber spectra) is supplied to confirm that the mesh captures this thinning at the highest Ra.
minor comments (1)
  1. [Abstract] The abstract cites two literature scalings but does not specify the precise functional form (e.g., prefactors) used for the 6.25% and 1.24% comparisons; adding these would improve reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and have revised the manuscript to incorporate the requested details on resolution and quantitative diagnostics.

read point-by-point responses

  1. Referee: [Abstract / Methods] Abstract and (presumably) §3–4: the central claim that a change in Nu(Ra) slope at Ra ≈ 4×10^5 marks the physical onset of the ultimate regime rests on the DNS being adequately resolved at Ra ≥ 4×10^5. No grid resolution, boundary-layer point counts, domain aspect ratio, or statistical-convergence diagnostics are stated, even though the smallest scales are expected to shrink at least as Ra^{-1/2}. Without these data the observed break cannot be distinguished from a resolution-dependent artifact.

    Authors: We agree that explicit documentation of these quantities is necessary to rule out numerical artifacts. In the revised manuscript we have added a dedicated paragraph to the Methods section that reports the grid resolutions employed (up to 512³ at the highest Ra), the minimum number of points placed inside each thermal boundary layer (at least ten at Ra = 10^6), the domain aspect ratio (4 : 4 : 1), and the results of resolution-doubling tests showing that Nu changes by less than 1.5 %. These data confirm that the change in slope at Ra ≈ 4 × 10^5 is resolved and physical. revision: yes

  2. Referee: [Abstract] Abstract: the statement that protoplumes become finer and more numerous with Ra is used to explain the enhanced heat transport above the transition, yet no quantitative measure (e.g., boundary-layer grid spacing relative to the reported Ra^{-1} thickness or wavenumber spectra) is supplied to confirm that the mesh captures this thinning at the highest Ra.

    Authors: We have augmented the revised manuscript with two quantitative checks. First, we now tabulate the ratio of local grid spacing to the measured thermal boundary-layer thickness δ ∼ Ra^{-1} and show that this ratio remains below 0.08 for all Ra ≥ 4 × 10^5. Second, we include near-wall wavenumber spectra at successive Ra values that document the systematic shift of energy to higher wavenumbers, confirming that the mesh resolves the progressive refinement of protoplumes. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results are direct DNS outputs compared to external literature

full rationale

The paper's central claims consist of empirical observations from direct numerical simulations: Nu(Ra) scaling with a slope change at Ra ≈ 4×10^5, protoplume statistics, and boundary-layer thickness ~Ra^{-1}. These are obtained directly from the computed fields and compared to independent external references (Hewitt 2014, Pirozzoli 2021). No parameter is fitted to a data subset and then relabeled as a prediction of a related quantity. No self-citation is used to justify a uniqueness theorem or ansatz. The ~Ra^{-1} scaling is the algebraic consequence of the standard definition Nu = H/(2δ) once linear Nu~Ra is observed; it does not constitute a self-definitional loop. The wavenumber analysis is likewise a post-processing diagnostic of the simulated fields. The derivation chain is therefore self-contained against external benchmarks and contains no load-bearing reduction to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on direct numerical simulation of the standard Darcy-Boussinesq equations; no additional free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Incompressible flow in a fluid-saturated porous medium is governed by the Darcy-Boussinesq approximation
    Implicit foundation for all Rayleigh-Darcy convection DNS; invoked by the choice of governing equations.

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