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Asymptotic ultimate regime of homogeneous Rayleigh–Bénard convection on logarithmic lattices

Published online by Cambridge University Press:  26 April 2023

Amaury Barral Open the ORCID record for Amaury Barral [Opens in a new window]  and
Berengere Dubrulle Open the ORCID record for Berengere Dubrulle [Opens in a new window]
Amaury Barral
Affiliation:
Université Paris-Saclay, CEA, CNRS, SPEC, 91191 Gif-sur-Yvette, France
Berengere Dubrulle*
Affiliation:
Université Paris-Saclay, CEA, CNRS, SPEC, 91191 Gif-sur-Yvette, France
*
†Email address for correspondence: berengere.dubrulle@cea.fr
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Abstract

We investigate how the heat flux $Nu$ scales with the imposed temperature gradient $Ra$ in homogeneous Rayleigh–Bénard convection using one-, two- and three-dimensional simulations on logarithmic lattices. Logarithmic lattices are a new spectral decimation framework which enables us to span an unprecedented range of parameters ($Ra$, $Re$, $\Pr$) and test existing theories using little computational power. We first show that known diverging solutions can be suppressed with a large-scale friction. In the turbulent regime, for $\Pr \approx 1$, the heat flux becomes independent of viscous processes (‘asymptotic ultimate regime’, $Nu\sim Ra ^{1/2}$ with no logarithmic correction). We recover scalings coherent with the theory developed by Grossmann and Lohse, for all situations where the large-scale frictions keep a constant magnitude with respect to viscous and diffusive dissipation. We also identify another turbulent friction-dominated regime at $\Pr \ll 1$, where deviations from the Grossmann and Lohse prediction are observed. These two friction-dominated regimes may be relevant in some geophysical or astrophysical situations, where large-scale friction arises due to rotation, stratification or magnetic field.

JFM classification

Nonlinear Dynamical Systems: Low-dimensional models Turbulent Flows: Turbulence simulation
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 962 , 10 May 2023 , A2
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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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
Biferale, L. 2003 Shell models of energy cascade in turbulence. Annu. Rev. Fluid Mech. 35, 441–468.CrossRefGoogle Scholar
Borue, V. & Orszag, S.A. 1997 Turbulent convection driven by a constant temperature gradient. J. Sci. Comput. 12 (3), 305–351.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
Brandenburg, A. 1992 Energy spectra in a model for convective turbulence. Phys. Rev. Lett. 69, 605–608.CrossRefGoogle Scholar
Calzavarini, E., Doering, C.R., Gibbon, J.D., Lohse, D., Tanabe, A. & Toschi, F. 2006 Exponentially growing solutions in homogeneous Rayleigh–Bénard convection. Phys. Rev. E 73, 035301.CrossRefGoogle ScholarPubMed
Calzavarini, E., Lohse, D. & Toschi, F. 2007 Homogeneous Rayleigh–Bénard convection. In Progress in Turbulence II (ed. M. Oberlack, G. Khujadze, S. Günther, T. Weller, M. Frewer, J. Peinke & S. Barth), pp. 181–184. Springer.CrossRefGoogle Scholar
Calzavarini, E., Lohse, D., Toschi, F. & Tripiccione, R. 2005 Rayleigh and Prandtl number scaling in the bulk of Rayleigh–Bénard turbulence. Phys. Fluids 17 (5), 055107.CrossRefGoogle Scholar
Campolina, C.S. & Mailybaev, A.A. 2018 Chaotic blowup in the 3D incompressible euler equations on a logarithmic lattice. Phys. Rev. Lett. 121, 064501.CrossRefGoogle ScholarPubMed
Campolina, C.S. & Mailybaev, A.A. 2021 Fluid dynamics on logarithmic lattices. Nonlinearity 34 (7), 4684–4715.CrossRefGoogle Scholar
Castaing, B., Rusaouen, E., Salort, J. & Chilla, F. 2017 Turbulent heat transport regimes in a channel. Phys. Rev. Fluids 2, 062801.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
Ching, E.S.C. & Ko, T.C. 2008 Ultimate-state scaling in a shell model for homogeneous turbulent convection. Phys. Rev. E 78, 036309.CrossRefGoogle Scholar
Ciliberto, S. & Laroche, C. 1999 Random roughness of boundary increases the turbulent convection scaling exponent. Phys. Rev. Lett. 82, 3998–4001.CrossRefGoogle Scholar
Doering, C.R. & Constantin, P. 1996 Variational bounds on energy dissipation in incompressible flows. III. Convection. Phys. Rev. E 53, 5957–5981.CrossRefGoogle ScholarPubMed
Frisch, U., Pomyalov, A., Procaccia, I. & Ray, S.S. 2012 Turbulence in noninteger dimensions by fractal fourier decimation. Phys. Rev. Lett. 108, 074501.CrossRefGoogle ScholarPubMed
Gloaguen, C., Léorat, J., Pouquet, A. & Grappin, R. 1985 A scalar model for MHD turbulence. Physica D 17 (2), 154–182.CrossRefGoogle Scholar
Grossmann, S. & Lohse, D. 2000 Scaling in thermal convection: a unifying theory. J. Fluid Mech. 407, 27–56.CrossRefGoogle Scholar
Grossmann, S. & Lohse, D. 2011 Multiple scaling in the ultimate regime of thermal convection. Phys. Fluids 23 (4), 045108.CrossRefGoogle Scholar
Grossmann, S., Lohse, D., L'vov, V. & Procaccia, I. 1994 Finite size corrections to scaling in high Reynolds number turbulence. Phys. Rev. Lett. 73, 432–435.CrossRefGoogle ScholarPubMed
Grossmann, S., Lohse, D. & Reeh, A. 1996 Developed turbulence: from full simulations to full mode reductions. Phys. Rev. Lett. 77, 5369–5372.CrossRefGoogle ScholarPubMed
Jiang, H., Wang, D., Liu, S. & Sun, C. 2022 Experimental evidence for the existence of the ultimate regime in rapidly rotating turbulent thermal convection. Phys. Rev. Lett. 129, 204502.CrossRefGoogle ScholarPubMed
Kawano, K., Motoki, S., Shimizu, M. & Kawahara, G. 2021 Ultimate heat transfer in ‘wall-bounded’ convective turbulence. J. Fluid Mech. 914, A13.CrossRefGoogle Scholar
Kraichnan, R.H. 1962 Turbulent thermal convection at arbitrary Prandtl number. Phys. Fluids 5 (11), 1374–1389.CrossRefGoogle Scholar
Lanotte, A.S., Benzi, R., Malapaka, S.K., Toschi, F. & Biferale, L. 2015 Turbulence on a fractal fourier set. Phys. Rev. Lett. 115, 264502.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
Lohse, D. & Toschi, F. 2003 Ultimate state of thermal convection. Phys. Rev. Lett. 90, 034502.CrossRefGoogle ScholarPubMed
Malkus, W.V.R. & Chandrasekhar, S. 1954 The heat transport and spectrum of thermal turbulence. Proc. R. Soc. Lond. A 225 (1161), 196–212.Google Scholar
Motoki, S., Kawahara, G. & Shimizu, M. 2022 Steady thermal convection representing the ultimate scaling. Phil. Trans. R. Soc. A 380 (2225), 20210037.CrossRefGoogle ScholarPubMed
Pawar, S.S. & Arakeri, J.H. 2016 Two regimes of flux scaling in axially homogeneous turbulent convection in vertical tube. Phys. Rev. Fluids 1, 042401.CrossRefGoogle Scholar
Pumir, A. & Shraiman, B.I. 1995 Persistent small scale anisotropy in homogeneous shear flows. Phys. Rev. Lett. 75, 3114–3117.CrossRefGoogle ScholarPubMed
Roche, P.-E. 2020 The ultimate state of convection: a unifying picture of very high Rayleigh numbers experiments. New J. Phys. 22 (7), 073056.CrossRefGoogle Scholar
Rusaouën, E., Liot, O., Castaing, B., Salort, J. & Chilla, F. 2018 Thermal transfer in Rayleigh–Bénard cell with smooth or rough boundaries. J. Fluid Mech. 837, 443–460.CrossRefGoogle Scholar
Schmidt, L.E., Calzavarini, E., Lohse, D., Toschi, F. & Verzicco, R. 2012 Axially homogeneous Rayleigh–Bénard convection in a cylindrical cell. J. Fluid Mech. 691, 52–68.CrossRefGoogle Scholar
Spiegel, E.A. 1963 A generalization of the mixing-length theory of turbulent convection. Astrophys. J. 138, 216.CrossRefGoogle Scholar
Stevens, B., Duan, J., McWilliams, J.C., Münnich, M. & Neelin, J.D. 2002 Entrainment, Rayleigh friction, and boundary layer winds over the tropical pacific. J. Clim. 15 (1), 30–44.2.0.CO;2>CrossRefGoogle Scholar
Sukoriansky, S., Galperin, B. & Chekhlov, A. 1999 Large scale drag representation in simulations of two-dimensional turbulence. Phys. Fluids 11 (10), 3043–3053.CrossRefGoogle Scholar
Urban, P., Hanzelka, P., Králík, T., Macek, M., Musilová, V. & Skrbek, L. 2019 Elusive transition to the ultimate regime of turbulent Rayleigh–Bénard convection. Phys. Rev. E 99, 011101.CrossRefGoogle Scholar
Whalen, P., Brio, M. & Moloney, J.V. 2015 Exponential time-differencing with embedded Runge–Kutta adaptive step control. J. Comput. Phys. 280, 579–601.CrossRefGoogle Scholar
Yeung, P.K., Sreenivasan, K.R. & Pope, S.B. 2018 Effects of finite spatial and temporal resolution in direct numerical simulations of incompressible isotropic turbulence. Phys. Rev. Fluids 3, 064603.CrossRefGoogle Scholar
Zhu, X., Mathai, V., Stevens, R.J.A.M., Verzicco, R. & Lohse, D. 2018 Transition to the ultimate regime in two-dimensional Rayleigh–Bénard convection. Phys. Rev. Lett. 120, 144502.CrossRefGoogle Scholar
Zhu, X., Mathai, V., Stevens, R.J.A.M., Verzicco, R. & Lohse, D. 2019 a Reply to “absence of evidence for the ultimate regime in two-dimensional Rayleigh–Bénard convection”. Phys. Rev. Lett. 123 (25), 259402.CrossRefGoogle Scholar
Zhu, X., Stevens, R.J.A.M., Shishkina, O., Verzicco, R. & Lohse, D. 2019 b $Nu\sim Ra^{1/2}$ scaling enabled by multiscale wall roughness in Rayleigh–Bénard turbulence. J. Fluid Mech. 869, R4.CrossRefGoogle Scholar
Zou, S. & Yang, Y. 2021 Realizing the ultimate scaling in convection turbulence by spatially decoupling the thermal and viscous boundary layers. J. Fluid Mech. 919, R3.CrossRefGoogle Scholar
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