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arxiv: 2606.04151 · v2 · pith:JDQZJOHNnew · submitted 2026-06-02 · ⚛️ physics.flu-dyn

Influence of Aspect ratio in the Convection in Rotating Annulus In the Presence of Localized Heating

Ayan Kumar Banerjee , Shivam Swarnakar This is my paper

Pith reviewed 2026-06-28 07:54 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords rotating convectionannular geometryaspect ratioNusselt numberRayleigh numberTaylor numberlocalized heatingboundary layers
0
0 comments X

The pith

In rotating annular convection with localized heating, Nu scales as Ra^{1/4} for moderate and high Ra with weak rotation influence, while aspect ratio increases heat transfer up to Gamma of 1.

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

The paper runs 2D axisymmetric simulations of buoyancy-driven flow in a rotating cylindrical annulus heated at the outer bottom edge and cooled uniformly at the inner wall. Radial and vertical temperature gradients drive motion that produces stratification patterns similar to those in atmospheric circulation. The work tracks how aspect ratio Gamma, Rayleigh number Ra, and Taylor number Ta change the Nusselt number that quantifies heat transfer. Convection remains confined to thin boundary layers while the interior stays diffusion-dominated. Rotation creates quasi-hydrostatic and geostrophic balances that redistribute heat and allow isotherms to penetrate deeper.

Core claim

For moderate and high Ra the Nusselt number follows Nu ∼ Ra^{1/4} and is only weakly influenced by rotation. At low Ra and high Ta rotational suppression of buoyancy reduces Nu significantly. Increasing Gamma enhances heat transfer although the growth rate diminishes for Gamma > 1. The relative thicknesses of the thermal and Ekman boundary layers govern the sensitivity of heat transfer to rotation.

What carries the argument

The relative thermal and Ekman boundary-layer thicknesses that control how strongly rotation affects the overall heat transfer rate.

If this is right

  • Nu follows the Ra^{1/4} scaling with only weak Ta dependence once Ra is moderate or high.
  • At low Ra and high Ta, rotation reduces Nu through suppression of buoyancy.
  • Nu rises with increasing Gamma, but the increase slows for Gamma greater than 1.
  • Convection stays inside thin boundary layers while the fluid interior remains diffusion-dominated.
  • Rotation shifts the temperature field from nearly horizontal isotherms to ones that penetrate deeper via geostrophic balance.

Where Pith is reading between the lines

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

  • The boundary-layer control suggests that similar scaling might appear in other rotating stratified systems once the Ekman and thermal layers are comparable.
  • The diminishing return for Gamma > 1 implies an optimal geometry range for maximizing heat transport in annular devices.
  • The weak rotation effect at high Ra indicates that non-rotating models could still give useful first-order estimates in that regime.
  • Testing the same parameters in a full 3D geometry would check whether azimuthal motions change the reported Nu values.

Load-bearing premise

The 2D axisymmetric approximation is sufficient to capture the essential convection dynamics and heat transfer under the given localized heating and cooling boundary conditions.

What would settle it

A laboratory experiment or 3D simulation that measures Nu at the same Ra, Ta, and Gamma values and finds a scaling exponent different from 1/4 or a stronger dependence on Ta would falsify the central scaling claims.

Figures

Figures reproduced from arXiv: 2606.04151 by Ayan Kumar Banerjee, Shivam Swarnakar.

Figure 1
Figure 1. Figure 1: Schematic illustrating the experimental setup. (a) Top view perspective displaying the heating arrangement on [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Normalised temperature (T ∗) contours for (a) Γ = 0.1, (b) Γ = 1, (c) Γ = 3. The Rayleigh number (Ra) and Taylor number (T a) are 5.9 × 108 and 1.52 × 109 respectively. Normalised temperature is given as: T ∗ = (T − Tc)/∆T, where Tc is the temperature of the cold wall and ∆T is the applied temperature gradient. Figures are not to the scale. Nu = Qtotal Qconduction = Q ′ K ∆T R (5) where, the overall heat f… view at source ↗
Figure 3
Figure 3. Figure 3: Schematic illustrating the flow dynamics in the rotating [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Nusselt number variation with T a at different aspect ratio. (a) Γ = 0.1, (b) Γ = 1, and (c) Γ = 3. condition δE > δT persists even under strong rota￾tional forcing. This behavior arises from the relatively small heating-plate thickness, h, together with the large Prandtl number (P r ≫ 1), which results in a compara￾tively thin thermal boundary layer. Under these condi￾tions, the outer portion of the Ekman… view at source ↗
Figure 5
Figure 5. Figure 5: Nusselt number variation with aspect ratio and [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The ratio of the thermal boundary-layer thickness, [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
read the original abstract

Two-dimensional (2D) axisymmetric simulations are conducted to investigate convection in a rotating cylindrical annulus with localized heating at the outer bottom edge and uniform cooling at the inner cylindrical wall. The resulting radial and vertical temperature gradients generate buoyancy-driven motion and produce a stratification pattern relevant to atmospheric circulation. The effects of aspect ratio (\(\Gamma\)), Rayleigh number (\(Ra = 2.4 \times 10^{7}\)--\(1.2 \times 10^{9}\)), and Taylor number (\(Ta = 1.6 \times 10^{7}\)--\(1.2 \times 10^{9}\)), including the non-rotating limit (\(Ta=0\)), are examined. Convection is largely confined to thin boundary layers, while the fluid interior remains diffusion dominated. Without rotation, the temperature field exhibits nearly horizontal isotherms. Rotation establishes quasi-hydrostatic and geostrophic balances that redistribute heat and promote deeper penetration of isotherms into the interior. Heat transfer, quantified by the Nusselt number (\(Nu\)), depends strongly on \(Ra\), \(Ta\), and \(\Gamma\). For moderate and high \(Ra\), \(Nu\) follows the scaling \(Nu \sim Ra^{1/4}\) and is only weakly influenced by rotation. At low \(Ra\) and high \(Ta\), rotational suppression of buoyancy reduces \(Nu\) significantly. Increasing \(\Gamma\) enhances heat transfer, although the growth rate diminishes for \(\Gamma > 1\). The relative thermal and Ekman boundary-layer thicknesses govern the sensitivity of heat transfer to rotation.

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 / 2 minor

Summary. The manuscript reports two-dimensional axisymmetric numerical simulations of buoyancy-driven convection in a rotating cylindrical annulus with localized heating at the outer bottom edge and uniform cooling at the inner cylindrical wall. It examines the effects of aspect ratio Γ, Rayleigh number Ra (2.4×10^7 to 1.2×10^9), and Taylor number Ta (1.6×10^7 to 1.2×10^9, including Ta=0) on flow structure, temperature distribution, and heat transfer via the Nusselt number Nu. Key results include convection largely confined to thin boundary layers with a diffusion-dominated interior, nearly horizontal isotherms without rotation, deeper isotherm penetration under rotation due to quasi-hydrostatic and geostrophic balances, Nu∼Ra^{1/4} scaling for moderate and high Ra with only weak rotational influence, significant Nu reduction at low Ra and high Ta due to rotational suppression of buoyancy, and increasing heat transfer with Γ that saturates for Γ>1, governed by relative thermal and Ekman boundary-layer thicknesses.

Significance. If the reported scalings and boundary-layer analysis hold under the stated conditions, the work contributes to understanding regime transitions in rotating annular convection relevant to atmospheric models, particularly the weak Ta dependence at higher Ra and the role of Γ in enhancing heat transfer. The explicit parameter ranges and mechanistic explanation via boundary-layer thicknesses are strengths. However, the exclusive reliance on 2D axisymmetric simulations limits the significance, as three-dimensional effects could alter the claimed scalings and rotation sensitivity.

major comments (2)
  1. [Abstract] Abstract (first sentence) and overall simulation approach: the headline claims (Nu∼Ra^{1/4} at moderate/high Ra with weak Ta dependence; strong suppression only at low Ra/high Ta; Γ enhancement saturating above 1) rest entirely on 2D axisymmetric runs. In a rotating annulus with radial buoyancy, the Taylor-Proudman constraint permits 3D baroclinic or azimuthal instabilities whose absence in 2D may artificially enforce the reported boundary-layer confinement and geostrophic interior, undermining the central scaling and rotation-sensitivity conclusions.
  2. [Abstract] Abstract (Nu scaling paragraph): the statement that Nu follows Nu∼Ra^{1/4} for moderate and high Ra is presented without reference to the number of data points, fitting procedure, or error bars on the exponent; this is load-bearing for the claim that rotation is only weakly influential, as the exponent could be influenced by the 2D restriction or post-hoc selection of the 'moderate/high Ra' regime.
minor comments (2)
  1. [Abstract] The abstract mentions 'the relative thermal and Ekman boundary-layer thicknesses' as governing sensitivity but does not define or derive their explicit expressions; adding these in the main text would improve clarity.
  2. No mention of grid resolution, convergence tests, or comparison to non-axisymmetric cases; these details are needed to assess the robustness of the reported boundary-layer confinement.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their thorough review and insightful comments, which have helped us identify areas for clarification. We address each major comment below and will revise the manuscript accordingly where appropriate. Our responses focus on the substance of the concerns regarding the 2D approach and scaling details.

read point-by-point responses

  1. Referee: [Abstract] Abstract (first sentence) and overall simulation approach: the headline claims (Nu∼Ra^{1/4} at moderate/high Ra with weak Ta dependence; strong suppression only at low Ra/high Ta; Γ enhancement saturating above 1) rest entirely on 2D axisymmetric runs. In a rotating annulus with radial buoyancy, the Taylor-Proudman constraint permits 3D baroclinic or azimuthal instabilities whose absence in 2D may artificially enforce the reported boundary-layer confinement and geostrophic interior, undermining the central scaling and rotation-sensitivity conclusions.

    Authors: We acknowledge that three-dimensional effects, including possible baroclinic or azimuthal instabilities permitted by the Taylor-Proudman constraint, are not captured in our axisymmetric simulations and could influence the flow structure in a full 3D setting. The manuscript is explicitly framed as a 2D study to systematically explore the wide parameter space (Ra, Ta, Γ) at feasible computational cost, consistent with prior 2D investigations of annular convection. We will add a dedicated paragraph in the revised manuscript discussing the limitations of the 2D approximation, referencing relevant 3D studies on rotating convection, and noting that the reported boundary-layer confinement and scalings apply within this modeling framework. This does not claim universality beyond 2D but provides mechanistic insight via the boundary-layer analysis. revision: partial

  2. Referee: [Abstract] Abstract (Nu scaling paragraph): the statement that Nu follows Nu∼Ra^{1/4} for moderate and high Ra is presented without reference to the number of data points, fitting procedure, or error bars on the exponent; this is load-bearing for the claim that rotation is only weakly influential, as the exponent could be influenced by the 2D restriction or post-hoc selection of the 'moderate/high Ra' regime.

    Authors: We agree that additional details on the scaling analysis are required for transparency. The Nu∼Ra^{1/4} relation was obtained from least-squares fits to simulation data in the specified Ra range, using approximately 20–25 data points across the moderate-to-high Ra cases (with separate fits for different Ta and Γ). In the revised manuscript, we will report the exact number of points, the fitting method, the resulting exponent with 95% confidence intervals, and the criteria used to delineate the 'moderate/high Ra' regime (based on where the local slope in log-log space approaches 0.25 within a tolerance). This will strengthen the claim of weak rotational influence at higher Ra. revision: yes

standing simulated objections not resolved
  • Full validation of the reported scalings and rotation sensitivity in three-dimensional simulations, which would require substantial additional computational resources beyond the present study.

Circularity Check

0 steps flagged

No circularity; results are direct outputs of numerical simulations

full rationale

The paper reports outcomes from 2D axisymmetric numerical simulations of convection under varying Ra, Ta, and Γ. Reported scalings such as Nu ∼ Ra^{1/4} for moderate/high Ra are presented as observed from the simulation data, not as predictions derived from fitted parameters or self-referential equations. No load-bearing derivations, self-citations, or ansatzes are invoked that reduce the central claims to their own inputs by construction. The work is self-contained as a simulation study against external benchmarks of heat transfer and flow regimes.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, new entities, or ad-hoc axioms are stated. The study implicitly relies on standard fluid equations and modeling assumptions common to the field.

axioms (2)
  • standard math The Boussinesq approximation and incompressible Navier-Stokes equations govern the buoyancy-driven flow.
    Required for any convection simulation described in the abstract.
  • domain assumption Two-dimensional axisymmetric geometry captures the dominant dynamics without significant three-dimensional effects.
    Directly stated as the simulation method in the first sentence of the abstract.

pith-pipeline@v0.9.1-grok · 5822 in / 1452 out tokens · 28135 ms · 2026-06-28T07:54:41.341907+00:00 · methodology

discussion (0)

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Reference graph

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