pith. sign in

arxiv: 2606.06331 · v1 · pith:I4LSQOYEnew · submitted 2026-06-04 · ⚛️ physics.flu-dyn · physics.geo-ph

An experimental study on the heat transport in porous media convection

Jing Dong , Lu Zhang , Ke-Qing Xia This is my paper

Pith reviewed 2026-06-27 23:25 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn physics.geo-ph
keywords porous media convectionheat transportRayleigh-Darcy numberNusselt numberflow regimesDarcy numberRayleigh-Bénard convection3D-printed lattice
0
0 comments X

The pith

Porous media convection transitions through five regimes as the Rayleigh-Darcy number rises, shifting from Darcy-like to classical Rayleigh-Bénard behavior.

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

The study measures heat transport in water-filled 3D-printed lattice structures over a wide range of Rayleigh-Darcy numbers and Darcy numbers. Analyses of the porous-medium Nusselt number and local temperature fluctuations identify five successive regimes: conduction, convection, oscillation, transition, and classical Rayleigh-Bénard convection. This sequence connects the two limiting behaviors previously studied separately in porous-media flows. Changing the matrix permeability shows that the Darcy number sets the locations of the transitions between regimes. Flow visualizations further indicate that large-scale rolls reorganize and pore-scale Reynolds numbers exceed one once the system enters the later regimes.

Core claim

The present system undergoes a transition through five distinct regimes: I. Conduction, II. Convection, III. Oscillation, IV. Transition, V. Classical Rayleigh–Bénard convection. This transitional process bridges the gap between Rayleigh–Darcy-like behaviour and Rayleigh–Bénard-like behaviour in porous media convection. By varying the permeability of the matrix, the Darcy number is shown to have a profound impact on the transitional processes across different regimes. Flow field measurements reveal that the flow structures within Regime IV and Regime V evolve from several horizontally stacked convection rolls to a single-roll structure, and the pore-scale Reynolds number exceeds unity in the

What carries the argument

Porous-medium Nusselt number Nu_m and local temperature statistics used to delineate the five regimes while the 3D-printed lattices allow controlled variation of Darcy number.

If this is right

  • The Darcy number controls the Rayleigh-Darcy number values at which each regime boundary occurs.
  • In the transition and classical regimes the large-scale flow changes from multiple stacked rolls to a single roll.
  • Pore-scale Reynolds number exceeds one precisely when the system leaves the oscillation regime.
  • The phase diagram in Ra-Da space locates all five regimes for the range of permeabilities tested.

Where Pith is reading between the lines

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

  • The reported sequence supplies a concrete map that could be used to predict regime boundaries in geothermal or subsurface flow models once permeability is known.
  • If the oscillation regime is tied to a specific instability, its boundaries might be predictable from linear stability analysis of the lattice geometry.
  • Extending the measurements to higher Rayleigh-Darcy numbers could reveal whether additional regimes appear beyond classical Rayleigh-Bénard convection.

Load-bearing premise

The 3D-printed lattice structures produce flow and heat transport statistics representative of natural porous media without introducing dominant artifacts from the printing process or lattice geometry.

What would settle it

Repeating the experiment with a natural sand or bead pack instead of the 3D-printed lattice and finding a different number of regimes or no clear bridging between Darcy-like and Rayleigh-Bénard-like behavior would falsify the reported sequence.

Figures

Figures reproduced from arXiv: 2606.06331 by Jing Dong, Ke-Qing Xia, Lu Zhang.

Figure 1
Figure 1. Figure 1: Experimental setups. (a) Schematic drawing of the convection cell. (b) A magnified portion of the porous structure. The elemental unit is outlined by the red lines. (c) A photo of the 3D printed porous samples used in the experiment. Samples labeled with 1, 2, and 3 correspond to the same 𝐷𝑎 number. The remainder of this paper is organized as follows: In §2, we describe the experimental setup; then we pres… view at source ↗
Figure 2
Figure 2. Figure 2: Results of the global heat transport. (a) Fluid Nusselt number 𝑁𝑢 𝑓 versus fluid Rayleigh number 𝑅𝑎 𝑓 . (b) Compensated plots of 𝑁𝑢 𝑓 /𝑅𝑎0.3 𝑓 versus 𝑅𝑎 𝑓 . (c) Medium Nusselt number 𝑁𝑢𝑚 versus Rayleigh–Darcy number 𝑅𝑎. (d) Compensated plot of 𝑁𝑢𝑚/𝑅𝑎 versus 𝑅𝑎. 0 X0-7 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (a) The porous medium Nusselt number 𝑁𝑢𝑚 as a function of Rayleigh–Darcy number 𝑅𝑎. The solid lines show fittings with equation 3.1. The dashed black line and blue line show the fitting functions adopted from references Pirozzoli et al. (2021) and Keene & Goldstein (2015), respectively. The blue solid diamonds represent values of 𝑅𝑎 at which the centre temperature measurements are conducted. (b) The local … view at source ↗
Figure 4
Figure 4. Figure 4: (a-f) The time series of the experimentally measured temperature fluctuation 𝑇𝑐 − ⟨𝑇𝑐⟩ at the cell centre for different values of 𝑅𝑎. The horizontal axis are normalised by the free-fall time 𝑡 𝑓 𝑓 = √︁ 𝐻/(𝛼𝑔𝛥𝑇). (g-l) The PSDs of the corresponding time series in (a-f). The black line denotes the 𝑓 −5/3 power law. turbulence. At 𝑅𝑎 = 3.27 × 104 , the time series of 𝑇𝑐 displays more intense fluctuations (fig… view at source ↗
Figure 5
Figure 5. Figure 5: (a) Compensated plot of the experimentally measured media Nusselt number 𝑁𝑢𝑚/𝑅𝑎 versus 𝑅𝑎 for different values of 𝐷𝑎. The solid lines represent fitting curves of equation 3.1 with 𝑛 = 3. (b) The phase diagram of different heat transport regimes in the 𝑅𝑎-𝐷𝑎 space. The solid symbols in both panel (a) and panel (b) present the critical Rayleigh–Darcy numbers. 𝐷𝑎 𝐴 𝑅𝑎𝑡,1 𝑅𝑎𝑡,2 𝛽1 𝛽2 𝛽3 𝑅𝑎III 𝑅𝑎IV 1.21 × 10−5 … view at source ↗
Figure 6
Figure 6. Figure 6: (a-d) Experimentally measured velocity fields at the centre 𝑥𝑦-plane (𝑧 = 1/2𝑊) for (a) 𝑅𝑎 = 2.12 × 103 , (b) 𝑅𝑎 = 1.06 × 104 and (c) 𝑅𝑎 = 3.35 × 104 in a 𝛤𝑥 𝑦 = 1 cell; (d) 𝑅𝑎 = 1.34 × 105 in a 𝛤𝑥 𝑦 = 1/2 cell. (e) Pore-scale Reynolds number 𝑅𝑒 𝑝 based on maximum or volume-averaged root-mean-square velocity as a function of Rayleigh–Darcy number 𝑅𝑎. The solid lines represent the power-law fit to the exper… view at source ↗
read the original abstract

We investigate the heat transport in porous media convection over a wide Rayleigh--Darcy number range of $26.8\leq Ra\leq 2.62\times 10^5$, and a Darcy number range of $6.18\times10^{-7}\leq Da\leq 1.21\times 10^{-5}$. In the experiments, we employ 3D-printed lattice structures as the solid porous matrix and water as the working fluid. Quantitative analyses of the porous medium Nusselt number $Nu_m$ and local temperature statistics reveal that the present system undergoes a transition through five distinct regimes: I. Conduction, II. Convection, III. Oscillation, IV. Transition, V. Classical Rayleigh--B\'enard convection. This transitional process bridges the gap between Rayleigh--Darcy-like behaviour and Rayleigh--B\'enard-like behaviour in porous media convection. By varying the permeability of the matrix, we further examine the role of the Darcy number $Da$, which turns out to have a profound impact on the transitional processes across different regimes. Flow field measurements reveal that the flow structures within Regime IV and Regime V evolve from several horizontally stacked convection rolls to a single-roll structure, and the pore-scale Reynolds number both exceeds unity in these two regimes. Finally, we report the corresponding phase diagram in the $Ra$-$Da$ space.

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

0 major / 3 minor

Summary. The manuscript reports experiments on heat transport in porous media convection using 3D-printed lattice structures with water as the fluid. Over 26.8 ≤ Ra ≤ 2.62×10^5 and 6.18×10^{-7} ≤ Da ≤ 1.21×10^{-5}, quantitative Nu_m(Ra,Da) and temperature statistics identify five regimes (I: conduction, II: convection, III: oscillation, IV: transition, V: classical Rayleigh-Bénard convection). Da strongly affects the transitions; flow visualizations show stacked rolls evolving to a single-roll structure with pore-scale Re > 1 in regimes IV and V. A phase diagram in Ra-Da space is presented.

Significance. If the regime boundaries and lattice representativeness hold, the work supplies a useful experimental bridge between Rayleigh-Darcy and Rayleigh-Bénard limits in porous media, including the quantified role of permeability, flow structure evolution, and a phase diagram. The controlled variation of Da via 3D-printed matrices and the inclusion of permeability calibration and visualizations are strengths.

minor comments (3)
  1. [Abstract] Abstract: The regime list is given without any numerical boundaries in Ra or Da; the full text supplies these but the abstract would benefit from at least one example boundary value per transition for immediate context.
  2. [Introduction/Notation] Notation: The symbol Ra is used for the Rayleigh-Darcy number throughout; an explicit definition (including the permeability scaling) in the introduction or methods would prevent confusion with the standard Rayleigh number.
  3. [Results/Figures] Figure clarity: Ensure all regime-transition figures include error bars on Nu_m and temperature statistics, and label the specific Ra-Da values at which regime boundaries occur.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript, including recognition of the controlled variation of Da, permeability calibration, flow visualizations, and the phase diagram in Ra-Da space. The referee's summary accurately reflects our experimental findings on the five regimes bridging Rayleigh-Darcy and Rayleigh-Bénard behaviors. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

This paper is a purely experimental report on heat transport in 3D-printed porous media. It identifies five regimes via direct measurements of Nu_m(Ra,Da) and temperature statistics without any derivations, fitted models, predictions, or load-bearing self-citations. Regime boundaries and flow visualizations are presented as empirical observations, with no equations or ansatzes that reduce to inputs by construction. The central claims rest on experimental data reduction rather than any self-referential chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard dimensionless groups and experimental assumptions of porous-media convection; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • domain assumption Rayleigh-Darcy and Darcy numbers are the controlling dimensionless parameters for the observed transitions.
    Standard choice in the field of porous media convection.

pith-pipeline@v0.9.1-grok · 5774 in / 1160 out tokens · 25378 ms · 2026-06-27T23:25:36.603698+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

77 extracted references · 71 canonical work pages

  1. [1]

    The training process of many deep networks explores the same low-dimensional manifold

    Noto, Daisuke and Letelier, Juvenal A. and Ulloaet, Hugo N. , title =. Proceedings of the National Academy of Sciences , ISSN =. doi:10.1073/pnas , year =

    work page doi:10.1073/pnas

  2. [2]

    and DiCarlo, David , title =

    Liang, Yu and Wen, Baole and Hesse, Marc A. and DiCarlo, David , title =. Geophys. Res. Lett. , volume =. doi:10.1029/2018gl079849 , year =

    work page doi:10.1029/2018gl079849

  3. [3]

    Huang, S. D. and Kaczorowski, M. and Ni, R. and Xia, K. Q. , title =. Phys. Rev. Lett. , volume =. 2013 , type =. doi:10.1103/PhysRevLett.111.104501 , url =

    work page doi:10.1103/physrevlett.111.104501 2013

  4. [4]

    Xia, Ke-Qing and Huang, Shi-Di and Xie, Yi-Chao and Zhang, Lu , title =. Nat. Sci. Rev. , pages =. 2023 , type =. doi:10.1093/nsr/nwad012 , url =

    work page doi:10.1093/nsr/nwad012 2023

  5. [5]

    De Paoli, Marco and Yerragolam, Guru Sreevanshu and Lohse, Detlef and Verzicco, Roberto , title =. Comput. Phys. Commun. , volume =. doi:10.1016/j.cpc.2025.109579 , year =

    work page doi:10.1016/j.cpc.2025.109579 2025

  6. [6]

    Joseph, D. D. and Nield, D. A. and Papanicolaou, G. , title =. Water Resour. Res. , volume =. doi:https://doi.org/10.1029/WR018i004p01049 , url =

    work page doi:10.1029/wr018i004p01049

  7. [7]

    and Bories, S.A

    Combarnous, M.A. and Bories, S.A. , title =. Adv. Hydrosci. , volume =. doi:10.1016/B978-0-12-021810-3.50008-4 , year =

    work page doi:10.1016/b978-0-12-021810-3.50008-4

  8. [8]

    2013 , issn =

    Current trends and future directions in turbulent thermal convection , journal =. 2013 , issn =. doi:https://doi.org/10.1063/2.1305201 , url =

    work page doi:10.1063/2.1305201 2013

  9. [9]

    and Ennis-King, Jonathan , title =

    Emami-Meybodi, Hamid and Hassanzadeh, Hassan and Green, Christopher P. and Ennis-King, Jonathan , title =. Int. J. Greenh. Gas Control , volume =. doi:10.1016/j.ijggc.2015.04.003 , year =

    work page doi:10.1016/j.ijggc.2015.04.003 2015

  10. [10]

    and Bejan, Adrian , title =

    Nield, Donald A. and Bejan, Adrian , title =. doi:10.1007/978-3-319-49562-0 , year =

    work page doi:10.1007/978-3-319-49562-0

  11. [11]

    and Pedretti, A

    Zanganeh, G. and Pedretti, A. and Zavattoni, S. and Barbato, M. and Steinfeld, A. , title =. Sol. Energy , volume =. doi:10.1016/j.solener.2012.07.019 , year =

    work page doi:10.1016/j.solener.2012.07.019 2012

  12. [12]

    , title =

    De Paoli, M. , title =. Eur. Phys. J. E , volume =. doi:10.1140/epje/s10189-023-00390-8 , year =

    work page doi:10.1140/epje/s10189-023-00390-8

  13. [13]

    Malkus, W. V. R. , title =. Proc. R. Soc. Lond. A , volume =. doi:10.1098/rspa.1954.0197 , year =

    work page doi:10.1098/rspa.1954.0197 1954

  14. [14]

    Howard, L. N. , title =. Applied Mechanics , editor =. 1966 , publisher =

    1966

  15. [15]

    and Constantin, Peter , title =

    Doering, Charles R. and Constantin, Peter , title =. J. Fluid Mech. , volume =. doi:10.1017/S002211209800281X , year =

    work page doi:10.1017/s002211209800281x

  16. [16]

    and Johnston, Hans and Worthing, Rodney A

    Otero, Jesse and Dontcheva, Lubomira A. and Johnston, Hans and Worthing, Rodney A. and Kurganov, Alexander and Petrova, Guergana and Doering, Charles R. , title =. J. Fluid Mech. , volume =. doi:10.1017/S0022112003007298 , year =

    work page doi:10.1017/s0022112003007298

  17. [17]

    Ahlers, Guenter and Grossmann, Siegfried and Lohse, Detlef , title =. Rev. Mod. Phys. , volume =. doi:10.1103/RevModPhys.81.503 , year =

    work page doi:10.1103/revmodphys.81.503

  18. [18]

    and Xia, K

    Lohse, D. and Xia, K. Q. , title =. Annu. Rev. Fluid Mech. , volume =. doi:10.1146/annurev.fluid.010908.165152 , year =

    work page doi:10.1146/annurev.fluid.010908.165152

  19. [19]

    Acta Mech

    Xia, Ke-Qing and Chong, Kai Leong and Ding, Guang-Yu and Zhang, Lu , title =. Acta Mech. Sin. , volume =. doi:10.1007/s10409-024-24287-x , year =

    work page doi:10.1007/s10409-024-24287-x

  20. [20]

    and Neufeld, Jerome A

    Huppert, Herbert E. and Neufeld, Jerome A. , title =. Annu. Rev. Fluid Mech. , volume =. doi:10.1146/annurev-fluid-011212-140627 , year =

    work page doi:10.1146/annurev-fluid-011212-140627

  21. [21]

    Hewitt, D. R. and Neufeld, J. A. and Lister, J. R. , title =. Phys. Rev. Lett. , volume =. doi:10.1103/PhysRevLett.108.224503 , year =

    work page doi:10.1103/physrevlett.108.224503

  22. [22]

    Pirozzoli, Sergio and De Paoli, Marco and Zonta, Francesco and Soldati, Alfredo , title =. J. Fluid Mech. , volume =. doi:10.1017/jfm.2020.1178 , year =

    work page doi:10.1017/jfm.2020.1178 2020

  23. [23]

    Liu, Shuang and Jiang, Linfeng and Chong, Kai Leong and Zhu, Xiaojue and Wan, Zhen-Hua and Verzicco, Roberto and Stevens, Richard J. A. M. and Lohse, Detlef and Sun, Chao , title =. J. Fluid Mech. , volume =. doi:10.1017/jfm.2020.309 , year =

    work page doi:10.1017/jfm.2020.309 2020

  24. [24]

    and Avila, Marc and Jin, Yan , title =

    Gasow, Stefan and Lin, Zhe and Zhang, Hao Chun and Kuznetsov, Andrey V. and Avila, Marc and Jin, Yan , title =. J. Fluid Mech. , volume =. doi:10.1017/jfm.2020.164 , year =

    work page doi:10.1017/jfm.2020.164 2020

  25. [25]

    Korba, David and Li, Like , title =. J. Fluid Mech. , volume =. doi:10.1017/jfm.2022.491 , year =

    work page doi:10.1017/jfm.2022.491 2022

  26. [26]

    Xu, Ao and Xu, Ben-Rui and Xi, Heng-Dong , title =. Phys. Rev. Fluids , volume =. doi:10.1103/PhysRevFluids.8.093504 , year =

    work page doi:10.1103/physrevfluids.8.093504

  27. [27]

    Li, Junyi and Yang, Yantao and Sun, Chao , title =. J. Fluid Mech. , volume =. doi:10.1017/jfm.2025.213 , year =

    work page doi:10.1017/jfm.2025.213 2025

  28. [28]

    Graham, M. D. and Steen, P. H. , title =. J. Fluid Mech. , volume =. doi:10.1017/S0022112094004386 , year =

    work page doi:10.1017/s0022112094004386

  29. [29]

    De Paoli, Marco and Zonta, Francesco and Soldati, Alfredo , title =. Phys. Fluids , volume =. doi:10.1063/1.4947425 , year =

    work page doi:10.1063/1.4947425

  30. [30]

    Lister, C. R. B. , title =. J. Fluid Mech. , volume =. doi:10.1017/S0022112090000143 , year =

    work page doi:10.1017/s0022112090000143

  31. [31]

    and Prasad, V

    Kladias, N. and Prasad, V. , title =. J. Thermophys. Heat Transf. , volume =. doi:10.2514/3.301 , year =

    work page doi:10.2514/3.301

  32. [32]

    and Goldstein, R

    Keene, Daniel J. and Goldstein, R. J. , title =. J. Heat Transfer , volume =. doi:10.1115/1.4029087 , year =

    work page doi:10.1115/1.4029087

  33. [33]

    and Tummers, Mark J

    Ataei-Dadavi, Iman and Chakkingal, Manu and Kenjeres, Sasa and Kleijn, Chris R. and Tummers, Mark J. , title =. Int. J. Heat Mass Transf. , volume =. doi:10.1016/j.ijheatmasstransfer.2018.10.118 , year =

    work page doi:10.1016/j.ijheatmasstransfer.2018.10.118 2018

  34. [34]

    , title =

    Schneider, K.-J. , title =. Progress in Refrigeration Science and Technology , publisher =. 1965 , doi =

    1965

  35. [35]

    Zhang, Lu and Xia, Ke-Qing , title =. J. Fluid Mech. , volume =. doi:10.1017/jfm.2023.600 , year =

    work page doi:10.1017/jfm.2023.600 2023

  36. [36]

    Hewitt, Duncan R and Neufeld, Jerome A and Lister, John R , title =. J. Fluid Mech. , volume =. doi:10.1017/jfm.2014.216 , year =

    work page doi:10.1017/jfm.2014.216 2014

  37. [37]

    Zhang, Lu and Xia, Ke-Qing , title =. Phys. Rev. Fluids , volume =. doi:10.1103/PhysRevFluids.8.023501 , year =

    work page doi:10.1103/physrevfluids.8.023501

  38. [38]

    Horton, C. W. and Rogers, F. T., Jr. , title =. J. Appl. Phys. , volume =. doi:10.1063/1.1707601 , year =

    work page doi:10.1063/1.1707601

  39. [39]

    Lapwood, E. R. , title =. Math. Proc. Camb. Philos. Soc. , volume =. doi:10.1017/S030500410002452X , year =

    work page doi:10.1017/s030500410002452x

  40. [40]

    Xi, H. D. and Xia, K. Q. , title =. Phys. Fluids , volume =. doi:10.1063/1.2920444 , year =

    work page doi:10.1063/1.2920444

  41. [41]

    Huang, S. D. and Xia, K. Q. , title =. J. Fluid Mech. , volume =. doi:10.1017/jfm.2016.181 , year =

    work page doi:10.1017/jfm.2016.181 2016

  42. [42]

    Chen, Xin and Huang, Shi-Di and Xia, Ke-Qing and Xi, Heng-Dong , title =. J. Fluid Mech. , volume =. doi:10.1017/jfm.2019.624 , year =

    work page doi:10.1017/jfm.2019.624 2019

  43. [43]

    Zhang, Lu and Dong, Jing and Xia, Ke-Qing , title =. J. Fluid Mech. , volume =. doi:10.1017/jfm.2022.187 , year =

    work page doi:10.1017/jfm.2022.187 2022

  44. [44]

    and Ni, R

    Sugiyama, K. and Ni, R. and Stevens, R. J. A. M. and Chan, T. S. and Zhou, S. Q. and Xi, H. D. and Sun, C. and Grossmann, S. and Xia, K. Q. and Lohse, D. , title =. Phys. Rev. Lett. , volume =. doi:10.1103/PhysRevLett.105.034503 , year =

    work page doi:10.1103/physrevlett.105.034503

  45. [45]

    and Huang, S

    Ni, R. and Huang, S. D. and Xia, K. Q. , title =. J. Fluid Mech. , volume =. doi:10.1017/jfm.2015.433 , year =

    work page doi:10.1017/jfm.2015.433 2015

  46. [46]

    Xi, H. D. and Xia, K. Q. , title =. Phys. Rev. E , volume =. doi:10.1103/PhysRevE.75.066307 , year =

    work page doi:10.1103/physreve.75.066307

  47. [47]

    and Lister, C

    Hartline, Beverly K. and Lister, C. R. B. , title =. J. Fluid Mech. , volume =. doi:10.1017/S0022112077000202 , year =

    work page doi:10.1017/s0022112077000202

  48. [48]

    Elder, J. W. , title =. J. Fluid Mech. , volume =. doi:10.1017/S0022112067000023 , year =

    work page doi:10.1017/s0022112067000023

  49. [49]

    and LeFur, B

    Combarnous, M. and LeFur, B. , title =. C. R. Acad. Sci. Paris B , volume =

  50. [50]

    Yen, Yin-Chao , title =. Int. J. Heat Mass Transf. , volume =. doi:10.1016/0017-9310(74)90136-7 , year =

    work page doi:10.1016/0017-9310(74)90136-7

  51. [51]

    Buretta, R. J. and Berman, A. S. , title =. J. Appl. Mech. , volume =. doi:10.1115/1.3423818 , year =

    work page doi:10.1115/1.3423818

  52. [52]

    Caltagirone, J. P. and Meyer, G. and Mojtabi, A. , title =. J. Mécanique , volume =

  53. [53]

    Holst, P. H. and Aziz, K. , title =. Int. J. Heat Mass Transf. , volume =. doi:10.1016/0017-9310(72)90167-6 , year =

    work page doi:10.1016/0017-9310(72)90167-6

  54. [54]

    Caltagirone, J. P. and Fabrie, P. , title =. Eur. J. Mech. B Fluids , volume =

  55. [55]

    Caltagirone, J. P. and Cloupeau, M. and Combarnous, M. , title =. C. R. Acad. Sci. Paris B , volume =

  56. [56]

    and Homsy, G

    Walker, K. and Homsy, G. M. , title =. J. Heat Transfer , volume =. doi:10.1115/1.3450692 , year =

    work page doi:10.1115/1.3450692

  57. [57]

    Ren, Lei and Tao, Xin and Zhang, Lu and Ni, Ming-Jiu and Xia, Ke-Qing and Xie, Yi-Chao , title =. J. Fluid Mech. , volume =. doi:10.1017/jfm.2022.866 , year =

    work page doi:10.1017/jfm.2022.866 2022

  58. [58]

    Niemela, J. J. and Skrbek, L. and Sreenivasan, K. R. and Donnelly, R. J. , title =. Nature , volume =. 2000 , type =. doi:10.1038/35009036 , url =

    work page doi:10.1038/35009036 2000

  59. [59]

    and Wu, X

    Sano, M. and Wu, X. Z. and Libchaber, A. , title =. Phys. Rev. A Gen. Phys. , volume =. 1989 , type =. doi:10.1103/physreva.40.6421 , url =

    work page doi:10.1103/physreva.40.6421 1989

  60. [60]

    Zhou, S. Q. and Xia, K. Q. , title =. Phys. Rev. Lett. , volume =. 2001 , type =. doi:10.1103/PhysRevLett.87.064501 , url =

    work page doi:10.1103/physrevlett.87.064501 2001

  61. [61]

    and Verzicco, R

    Wang, Q. and Verzicco, R. and Lohse, D. and Shishkina, O. , title =. Phys. Rev. Lett. , volume =. 2020 , type =. doi:10.1103/PhysRevLett.125.074501 , url =

    work page doi:10.1103/physrevlett.125.074501 2020

  62. [62]

    and Stevens, Richard J

    van der Poel, Erwin P. and Stevens, Richard J. A. M. and Sugiyama, Kazuyasu and Lohse, Detlef , title =. Phys. Fluids , volume =. doi:10.1063/1.4744988 , year =

    work page doi:10.1063/1.4744988

  63. [63]

    De Paoli, Marco and Pirozzoli, Sergio and Zonta, Francesco and Soldati, Alfredo , title =. J. Fluid Mech. , volume =. doi:10.1017/jfm.2022.461 , year =

    work page doi:10.1017/jfm.2022.461 2022

  64. [64]

    Zhu, Xiaojue and Fu, Yifeng and De Paoli, Marco , title =. J. Fluid Mech. , volume =. doi:10.1017/jfm.2024.528 , year =

    work page doi:10.1017/jfm.2024.528 2024

  65. [65]

    and Chini, Gregory P

    Wen, Baole and Corson, Lindsey T. and Chini, Gregory P. , title =. J. Fluid Mech. , volume =. doi:10.1017/jfm.2015.205 , year =

    work page doi:10.1017/jfm.2015.205 2015

  66. [66]

    Xia, K. Q. and Sun, C. and Zhou, S. Q. , title =. Phys. Rev. E , volume =. 2003 , type =. doi:10.1103/PhysRevE.68.066303 , url =

    work page doi:10.1103/physreve.68.066303 2003

  67. [67]

    Xi, H. D. and Xia, K. Q. , title =. Physical Review E , volume =. 2008 , type =. doi:10.1103/PhysRevE.78.036326 , url =

    work page doi:10.1103/physreve.78.036326 2008

  68. [68]

    and Verzicco, R

    Stringano, G. and Verzicco, R. , title =. J. Fluid Mech. , volume =. doi:10.1017/s0022112005007378 , year =

    work page doi:10.1017/s0022112005007378

  69. [69]

    Weiss, Stephan and Ahlers, Guenter , title =. J. Fluid Mech. , volume =. doi:10.1017/s0022112010005963 , year =

    work page doi:10.1017/s0022112010005963

  70. [70]

    Elder, J. W. , title =. J. Fluid Mech. , volume =. doi:10.1017/s0022112065001258 , year =

    work page doi:10.1017/s0022112065001258

  71. [71]

    Xu, Fang and Zhang, Lu and Xia, Ke-Qing , title =. J. Fluid Mech. , volume =. doi:10.1017/jfm.2024.164 , year =

    work page doi:10.1017/jfm.2024.164 2024

  72. [72]

    Alam, Parvez and Madanan, Umesh , title =. Int. J. Therm. Sci. , volume =. doi:10.1016/j.ijthermalsci.2025.110409 , year =

    work page doi:10.1016/j.ijthermalsci.2025.110409 2025

  73. [73]

    Bavandla, K. C. and Srinivasan, V. , title =. J. Heat Transfer , volume =. doi:10.1115/1.4064327 , year =

    work page doi:10.1115/1.4064327

  74. [74]

    Bavandla, K. C. and Srinivasan, V. , title =. J. Heat Transfer , volume =. doi:10.1115/1.4067338 , year =

    work page doi:10.1115/1.4067338

  75. [75]

    Schwendener, Dario and Noir, Jerome and Latt, Jonas and Coreixas, Christophe and Kong, Xiang-Zhao , title =. J. Fluid Mech. , volume =. doi:10.1017/jfm.2025.11014 , year =

    work page doi:10.1017/jfm.2025.11014 2025

  76. [76]

    and Huber, C

    Karani, H. and Huber, C. , title =. Phys. Rev. E , volume =. 2017 , type =. doi:10.1103/PhysRevE.95.033123 , url =

    work page doi:10.1103/physreve.95.033123 2017

  77. [77]

    Wang, M. Y. and Bejan, A. , title =. Int. Commun. Heat Mass Transf. , volume =. 1987 , type =. doi:10.1016/0735-1933(87)90041-8 , url =

    work page doi:10.1016/0735-1933(87)90041-8 1987