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On the role of the Prandtl number in convection driven by heat sources and sinks

Published online by Cambridge University Press:  04 August 2020

Benjamin Miquel Open the ORCID record for Benjamin Miquel [Opens in a new window] ,
Vincent Bouillaut ,
Sébastien Aumaître  and
Basile Gallet Open the ORCID record for Basile Gallet [Opens in a new window]
Benjamin Miquel*
Affiliation:
Université Paris-Saclay, CEA, CNRS, Service de Physique de l'Etat Condensé, 91191Gif-sur-Yvette, France
Vincent Bouillaut
Affiliation:
Université Paris-Saclay, CEA, CNRS, Service de Physique de l'Etat Condensé, 91191Gif-sur-Yvette, France
Sébastien Aumaître
Affiliation:
Université Paris-Saclay, CEA, CNRS, Service de Physique de l'Etat Condensé, 91191Gif-sur-Yvette, France
Basile Gallet
Affiliation:
Université Paris-Saclay, CEA, CNRS, Service de Physique de l'Etat Condensé, 91191Gif-sur-Yvette, France
*
†Email address for correspondence: benjamin.miquel@cea.fr
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Abstract

We report on a numerical study of turbulent convection driven by a combination of internal heat sources and sinks. Motivated by a recent experimental realisation (Lepot etal., Proc. Natl Acad. Sci. USA, vol. 115 (36), 2018, pp. 8937–8941), we focus on the situation where the cooling is uniform, while the internal heating is localised near the bottom boundary, over approximately one tenth of the domain height. We obtain scaling laws ${Nu} \sim {Ra} ^{\gamma } {Pr}^{\chi }$ for the heat transfer as measured by the Nusselt number ${Nu}$ expressed as a function of the Rayleigh number ${Ra}$ and the Prandtl number ${Pr}$. After confirming the experimental value $\gamma \approx 1/2$ for the dependence on ${Ra}$, we identify several regimes of dependence on ${Pr}$. For a stress-free bottom surface and within a range as broad as ${Pr} \in [0.003, 10]$, we observe the exponent $\chi \approx 1/2$, in agreement with Spiegel's mixing-length theory. For a no-slip bottom surface we observe a transition from $\chi \approx 1/2$ for ${Pr} \leq 0.04$ to $\chi \approx 1/6$ for ${Pr} \geq 0.04$, in agreement with scaling predictions by Bouillaut etal. (J. Fluid Mech. vol. 861, 2019, R5). The latter scaling regime stems from heat accumulation in the stagnant layer adjacent to a no-slip bottom boundary, which we characterise by comparing the local contributions of diffusive and convective thermal fluxes.

JFM classification

Turbulent Flows: Turbulent convection
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 900 , 10 October 2020 , R1
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

REFERENCES

Ahlers, G., Grossmann, S. & Lohse, D. 2009 Heat transfer and large scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81, 503–537.CrossRefGoogle Scholar
Aurnou, J. M., Bertin, V., Grannan, A. M., Horn, S. & Vogt, T. 2018 Rotating thermal convection in liquid gallium: multi-modal flow, absent steady columns. J. Fluid Mech. 846, 846–876.CrossRefGoogle Scholar
Aurnou, J. M., Calkins, M. A., Cheng, J. S., Julien, K., King, E. M., Nieves, D., Soderlund, K. M. & Stellmach, S. 2015 Rotating convective turbulence in earth and planetary cores. Phys. Earth Planet. Inter. 246, 52–71.CrossRefGoogle Scholar
Barker, A. J., Dempsey, A. M. & Lithwick, Y. 2014 Theory and simulations of rotating convection. Astrophys. J. 791 (1), 13.CrossRefGoogle Scholar
Bouillaut, V., Lepot, S., Aumaître, S. & Gallet, B. 2019 Transition to the ultimate regime in a radiatively driven convection experiment. J. Fluid Mech. 861, R5.CrossRefGoogle Scholar
Browning, M. K 2008 Simulations of dynamo action in fully convective stars. Astrophys. J. 676 (2), 1262.CrossRefGoogle Scholar
Burns, K. J., Vasil, G. M., Oishi, J. S., Lecoanet, D. & Brown, B. P. 2020 Dedalus: a flexible framework for numerical simulations with spectral methods. Phys. Rev. Res. 2, 023068.CrossRefGoogle Scholar
Calkins, M. A., Julien, K., Tobias, S. M. & Aurnou, J. M. 2015 A multiscale dynamo model driven by quasi-geostrophic convection. J. Fluid Mech. 780, 143–166.CrossRefGoogle Scholar
Castaing, B., Gunaratne, G., Heslot, F., Kadanoff, L., Libchaber, A., Thomae, S., Wu, X.-Z., Zaleski, S. & Zanetti, G. 1989 Scaling of hard thermal turbulence in Rayleigh–Bénard convection. J. Fluid Mech. 204, 1–30.CrossRefGoogle Scholar
Chavanne, X., Chillà, F., Castaing, B., Hébral, B., Chabaud, B. & Chaussy, J. 1997 Observation of the ultimate regime in Rayleigh–Bénard convection. Phys. Rev. Lett. 79, 3648–3651.CrossRefGoogle Scholar
Chillà, F. & Schumacher, J. 2012 New perspectives in turbulent Rayleigh–Bénard convection. Eur. Phys. J. E 35 (7), 58.CrossRefGoogle ScholarPubMed
Cioni, S., Ciliberto, S. & Sommeria, J. 1997 Strongly turbulent Rayleigh–Bénard convection in mercury: comparison with results at moderate Prandtl number. J. Fluid Mech. 335, 111–140.CrossRefGoogle Scholar
Deardorff, J. W. 1974 Three-dimensional numerical study of the height and mean structure of a heated planetary boundary layer. Boundary-Layer Meteorol. 7 (1), 81–106.CrossRefGoogle Scholar
Doering, C. R., Toppaladoddi, S. & Wettlaufer, J. S. 2019 Absence of evidence for the ultimate regime in two-dimensional Rayleigh–Bénard convection. Phys. Rev. Lett. 123, 259401.CrossRefGoogle ScholarPubMed
Fauve, S, Laroche, C & Libchaber, A 1981 Effect of a horizontal magnetic field on convective instabilities in mercury. J. Phys. Lett. 42 (21), 455–457.CrossRefGoogle Scholar
Goluskin, D. 2015 Internally heated convection beneath a poor conductor. J. Fluid Mech. 771, 36–56.CrossRefGoogle Scholar
Goluskin, D. & van der Poel, E. P. 2016 Penetrative internally heated convection in two and three dimensions. J. Fluid Mech. 791, R6.CrossRefGoogle Scholar
Guervilly, C., Cardin, P. & Schaeffer, N. 2019 Turbulent convective length scale in planetary cores. Nature 570 (7761), 368–371.CrossRefGoogle Scholar
He, X., Funfschilling, D., Nobach, H., Bodenschatz, E. & Ahlers, G. 2012 Transition to the ultimate state of turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 108, 024502.CrossRefGoogle ScholarPubMed
Lepot, S., Aumaître, S. & Gallet, B. 2018 Radiative heating achieves the ultimate regime of thermal convection. Proc. Natl Acad. Sci. USA 115 (36), 8937–8941.CrossRefGoogle ScholarPubMed
Miquel, B., Lepot, S., Bouillaut, V. & Gallet, B. 2019 Convection driven by internal heat sources and sinks: heat transport beyond the mixing-length or ‘ultimate’ scaling regime. Phys. Rev. Fluids 4, 121501.CrossRefGoogle Scholar
Miquel, B., Xie, J.-H., Featherstone, N., Julien, K. & Knobloch, E. 2018 Equatorially trapped convection in a rapidly rotating shallow shell. Phys. Rev. Fluids 3, 053801.CrossRefGoogle Scholar
Niemela, J. J., Skrbek, L., Sreenivasan, K. R. & Donnelly, R. J. 2000 Turbulent convection at very high Rayleigh numbers. Nature 404 (6780), 837–840.CrossRefGoogle ScholarPubMed
Plant, R. S. & Yano, J.-I. 2015 Parameterization of Atmospheric Convection. Imperial College Press.CrossRefGoogle Scholar
Rocha, C. B., Bossy, T., Llewellyn Smith, S. G. & Young, W. R. 2020 Improved bounds on horizontal convection. J. Fluid Mech. 883, A41.CrossRefGoogle Scholar
Shishkina, O., Emran, M. S., Grossmann, S. & Lohse, D. 2017 Scaling relations in large-Prandtl-number natural thermal convection. Phys. Rev. Fluids 2, 103502.CrossRefGoogle Scholar
Shishkina, O. & Wagner, S. 2016 Prandtl-number dependence of heat transport in laminar horizontal convection. Phys. Rev. Lett. 116, 024302.CrossRefGoogle ScholarPubMed
Soderlund, K. M. 2019 Ocean dynamics of outer solar system satellites. Geophys. Res. Lett. 46 (15), 8700–8710.CrossRefGoogle Scholar
Soderlund, K. M., King, E. M. & Aurnou, J. M. 2012 The influence of magnetic fields in planetary dynamo models. Earth Planet. Sci. Lett. 333–334, 9–20.CrossRefGoogle Scholar
Sutherland, B. R., Achatz, U., Caulfield, C. P. & Klymak, J. M. 2019 Recent progress in modeling imbalance in the atmosphere and ocean. Phys. Rev. Fluids 4, 010501.CrossRefGoogle Scholar
Vreugdenhil, C. A., Griffiths, R. W. & Gayen, B. 2017 Geostrophic and chimney regimes in rotating horizontal convection with imposed heat flux. J. Fluid Mech. 823, 57–99.CrossRefGoogle Scholar
Yano, J.-I., Talagrand, O. & Drossard, P. 2003 Origins of atmospheric zonal winds. Nature 421, 36.CrossRefGoogle ScholarPubMed