Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-28T03:12:22.210Z Has data issue: false hasContentIssue false
  • English
  • Français

Heat transport and temperature boundary-layer profiles in closed turbulent Rayleigh–Bénard convection with slippery conducting surfaces

Published online by Cambridge University Press:  06 June 2022

Maojing Huang ,
Yin Wang Open the ORCID record for Yin Wang [Opens in a new window] ,
Yun Bao  and
Xiaozhou He Open the ORCID record for Xiaozhou He [Opens in a new window]
Maojing Huang
Affiliation:
School of Mechanical Engineering and Automation, Harbin Institute of Technology, Shenzhen 518055, China
Yin Wang
Affiliation:
Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China Princeton Plasma Physics Laboratory, Princeton University, Princeton, NJ 08543, USA
Yun Bao
Affiliation:
Department of Mechanics, Sun Yat-Sen University, Guangzhou 510275, China
Xiaozhou He*
Affiliation:
School of Mechanical Engineering and Automation, Harbin Institute of Technology, Shenzhen 518055, China Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, 37077 Göttingen, Germany
*
†Email address for correspondence: hexiaozhou@hit.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

We report direct numerical simulations (DNS) of the Nusselt number $Nu$, the vertical profiles of mean temperature $\varTheta (z)$ and temperature variance $\varOmega (z)$ across the thermal boundary layer (BL) in closed turbulent Rayleigh–Bénard convection (RBC) with slippery conducting surfaces ($z$ is the vertical distance from the bottom surface). The DNS study was conducted in three RBC samples: a three-dimensional cuboid with length $L = H$ and width $W = H/4$ ($H$ is the sample height), and two-dimensional rectangles with aspect ratios $\varGamma \equiv L/H = 1$ and $10$. The slip length $b$ for top and bottom plates varied from $0$ to $\infty$. The Rayleigh numbers $Ra$ were in the range $10^{6} \leqslant Ra \leqslant 10^{10}$ and the Prandtl number $Pr$ was fixed at $4.3$. As $b$ increases, the normalised $Nu/Nu_0$ ($Nu_0$ is the global heat transport for $b = 0$) from the three samples for different $Ra$ and $\varGamma$ can be well described by the same function $Nu/Nu_0 = N_0 \tanh (b/\lambda _0) + 1$, with $N_0 = 0.8 \pm 0.03$. Here $\lambda _0 \equiv L/(2Nu_0)$ is the thermal boundary layer thickness for $b = 0$. Considering the BL fluctuations for $Pr>1$, one can derive solutions of temperature profiles $\varTheta (z)$ and $\varOmega (z)$ near the thermal BL for $b \geqslant 0$. When $b=0$, the solutions are equivalent to those reported by Shishkina et al. (Phys. Rev. Lett., vol. 114, 2015, 114302) and Wang et al. (Phys. Rev. Fluids, vol. 1, 2016, 082301(R)), respectively, for no-slip plates. For $b > 0$, the derived solutions are in excellent agreement with our DNS data for slippery plates.

JFM classification

Boundary Layers: Boundary layer structure
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 943 , 25 July 2022 , A2
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Ahlers, G. 2009 Turbulent convection. Physics 2, 74.CrossRefGoogle Scholar
Ahlers, G., Bodenschatz, E., Funfschilling, D., Grossmann, S., He, X., Lohse, D., Stevens, R.J.A.M. & Verzicco, R. 2012 a Logarithmic temperature profiles in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 109, 114501.CrossRefGoogle ScholarPubMed
Ahlers, G., Bodenschatz, E., Hartmann, R., He, X., Lohse, D., Reiter, P., Stevens, R.J.A.M., Verzicco, R., Wedi, M. & Weiss, S. 2022 Aspect ratio dependence of heat transfer in a cylindrical Rayleigh–Bénard cell. Phys. Rev. Lett. 128, 084501.CrossRefGoogle Scholar
Ahlers, G., Bodenschatz, E. & He, X. 2014 Logarithmic temperature profiles of turbulent Rayleigh–Bénard convection in the classical and ultimate state for a Prandtl number of 0.8. J. Fluid Mech. 758, 436–467.CrossRefGoogle Scholar
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
Ahlers, G., He, X., Funfschilling, D. & Bodenschatz, E. 2012 b Heat transport by turbulent Rayleigh–Bénard convection for $Pr \simeq 0.8$ and $3\times 10^{12} \lesssim Ra \lesssim 10^{15}$: aspect ratio ${\varGamma } = 0.50$. New J. Phys. 14, 103012.CrossRefGoogle Scholar
Bao, Y., Chen, J., Liu, B.F., She, Z.S., Zhang, J. & Zhou, Q. 2015 Enhanced heat transport in partitioned thermal convection. J. Fluid Mech. 784, R5.CrossRefGoogle Scholar
Ceccio, S.L. 2010 Friction drag reduction of external flows with bubble and gas injection. Annu. Rev. Fluid Mech. 42, 183–203.CrossRefGoogle Scholar
Chen, J., Bao, Y., Yin, Z.X. & She, Z.S. 2017 Theoretical and numerical study of enhanced heat transfer in partitioned thermal convection. Intl J. Heat Mass Transfer 115, 556–569.CrossRefGoogle Scholar
Ching, E.S.C., Dung, O.Y. & Shishkina, O. 2017 Fluctuating thermal boundary layers and heat transfer in turbulent Rayleigh–Bénard convection. J. Stat. Phys. 167, 626–635.CrossRefGoogle Scholar
Ching, E.S.C., Leung, H.S., Zwirner, L. & Shishkina, O. 2019 Velocity and thermal boundary layer equations for turbulent Rayleigh–Bénard convection. Phys. Rev. Res. 1, 033037.CrossRefGoogle Scholar
Choi, C.H. & Kim, C.J. 2006 Large slip of aqueous liquid flow over a nanoengineered superhydrophobic surface. Phys. Rev. Lett. 96, 066001.CrossRefGoogle Scholar
Davis, R.E. 1991 Lagrangian ocean studies. Annu. Rev. Fluid Mech. 23, 43–64.CrossRefGoogle Scholar
Goluskin, D. 2015 Internally heated convection beneath a poor conductor. J. Fluid Mech. 771, 36–56.CrossRefGoogle Scholar
Goluskin, D., Johnston, H., Flierl, G.R. & Spiegel, E.A. 2014 Convectively driven shear and decreased heat flux. J. Fluid Mech. 759, 360–385.CrossRefGoogle Scholar
Grossmann, S. & Lohse, D. 2000 Scaling in thermal convection: a unifying view. J. Fluid Mech. 407, 27–56.CrossRefGoogle Scholar
Grossmann, S. & Lohse, D. 2001 Thermal convection for large Prandtl number. Phys. Rev. Lett. 86, 3316–3319.CrossRefGoogle Scholar
Grossmann, S. & Lohse, D. 2011 Multiple scaling in the ultimate regime of thermal convection. Phys. Fluids 23, 045108.CrossRefGoogle Scholar
He, X., Bodenschatz, E. & Ahlers, G. 2020 a Absence of evidence for the ultimate state of turbulent Rayleigh–Bénard convection reply. Phys. Rev. Lett. 124, 229402.CrossRefGoogle Scholar
He, X., Bodenschatz, E. & Ahlers, G. 2020 b Aspect ratio dependence of the ultimate-state transition in turbulent thermal convection. Proc. Natl Acad. Sci. USA 117, 30022.CrossRefGoogle ScholarPubMed
He, X., Bodenschatz, E. & Ahlers, G. 2021 a A model for universal spatial variations of temperature fluctuations in turbulent Rayleigh–Bénard convection. Theor. Appl. Mech. Lett. 11, 1.CrossRefGoogle Scholar
He, X., Bodenschatz, E. & Ahlers, G. 2021 b Universal scaling of temperature variance in Rayleigh–Bénard convection near the transition to the ultimate state. J. Fluid Mech. 931, A7.CrossRefGoogle Scholar
He, X., Funfschilling, D., Bodenschatz, E. & Ahlers, G. 2012 a Heat transport by turbulent Rayleigh–Bénard convection for $Pr \simeq 0.8$ and $4\times 10^{11} \lesssim Ra \lesssim 2\times 10^{14}$: ultimate-state transition for aspect ratio $\gamma = 1.00$. New J. Phys. 14, 063030.CrossRefGoogle Scholar
He, X., Funfschilling, D., Nobach, H., Bodenschatz, E. & Ahlers, G. 2012 b Transition to the ultimate state of turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 108, 024502.CrossRefGoogle Scholar
Huang, M. & He, X. 2022 Heat transport in horizontally periodic and confined Rayleigh–Bénard convection with no-slip and free-slip plates. Theor. Appl. Mech. Lett. 12, 100330.CrossRefGoogle Scholar
Iyer, K.P., Scheel, J.D., Schumacher, J. & Sreenivasan, K.R. 2020 Classical $1/3$ scaling of convection holds up to $Ra = 10^{15}$. Proc. Natl Acad. Sci. USA 117, 7594.CrossRefGoogle Scholar
Kaczorowski, M., Chong, K.L. & Xia, K.Q. 2014 Turbulent flow in the bulk of Rayleigh–Bénard convection: aspect-ratio dependence of the small-scale properties. J. Fluid Mech. 747, 73–102.CrossRefGoogle Scholar
Kraichnan, R.H. 1962 Turbulent thermal convection at arbritrary Prandtl number. Phys. Fluids 5, 1374–1389.CrossRefGoogle Scholar
Lohse, D. & Xia, K.Q. 2010 Small-scale properties of turbulent Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 42, 335–364.CrossRefGoogle Scholar
Mellado, J.P. 2012 Direct numerical simulation of free convection over a heated plate. J. Fluid Mech. 712, 418–450.CrossRefGoogle Scholar
Moore, W.B. & Webb, A.A.G. 2013 Heat-pipe Earth. Nature 501, 501–505.CrossRefGoogle ScholarPubMed
Pandey, A. & Verma, M.K. 2016 Scaling of large-scale quantities in Rayleigh–Bénard convection. Phys. Fluids 28, 095105.CrossRefGoogle Scholar
van der Poel, E.P., Ostilla, R., Verzicco, R. & Lohse, D. 2014 Effect of velocity boundary conditions on the heat transfer and flow topology in two-dimensional Rayleigh–Bénard convection. Phys. Rev. E 90, 013017.CrossRefGoogle ScholarPubMed
van der Poel, E.P., Ostilla-Mónico, R., Verzicco, R., Grossmann, S. & Lohse, D. 2015 Logarithmic mean temperature profiles and their connection to plume emissions in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 115, 154501.CrossRefGoogle ScholarPubMed
van der Poel, E.P., Stevens, R.J.A.M. & Lohse, D. 2013 Comparison between two-and three-dimensional Rayleigh–Bénard convection. J. Fluid Mech. 736, 177–194.CrossRefGoogle Scholar
van der Poel, E.P., Stevens, R.J.A.M., Sugiyama, K. & Lohse, D. 2012 Flow states in two-dimensional Rayleigh–Bénard convection as a function of aspect-ratio and Rayleigh number. Phys. Fluids 24, 085104.CrossRefGoogle Scholar
Schlichting, H. & Gersten, K. 2000 Boundary Layer Theory, 8th edn. Springer.CrossRefGoogle Scholar
Shishkina, O., Horn, S., Emran, M.S. & Ching, E.S.C. 2017 Mean temperature profiles in turbulent thermal convection. Phys. Rev. Fluids 2, 113502.CrossRefGoogle Scholar
Shishkina, O., Horn, S., Wagner, S. & Ching, E.S.C. 2015 Thermal boundary layer equation for turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 114, 114302.CrossRefGoogle ScholarPubMed
Shishkina, O., Stevens, R.J.A.M., Grossmann, S. & Lohse, D. 2010 Boundary layer structure in turbulent thermal convection and its consequences for the required numerical resolution. New J. Phys. 12, 075022.CrossRefGoogle Scholar
Shishkina, O., Weiss, S. & Bodenschatz, E. 2016 Conductive heat flux in measurements of the Nusselt number in turbulent Rayleigh–Bénard convection. Phys. Rev. Fluids 1, 062301(R).CrossRefGoogle Scholar
Shraiman, B.I. & Siggia, E.D. 1990 Heat transport in high-Rayleigh number convection. Phys. Rev. A 42, 3650–3653.CrossRefGoogle ScholarPubMed
Spiegel, E.A. 1971 Convection in stars. Annu. Rev. Astron. Astrophys. 9, 323–352.CrossRefGoogle Scholar
Stevens, R.J.A.M., van der Poel, E.P., Grossmann, S. & Lohse, D. 2013 The unifying theory of scaling in thermal convection: the updated prefactors. J. Fluid Mech. 730, 295–308.CrossRefGoogle Scholar
Tritton, D.J. 1975 Internally heated convection in the atmosphere of Venus and in the laboratory. Nature 257, 110–112.CrossRefGoogle Scholar
Von Hardenberg, J., Goluskin, D., Provenzale, A. & Spiegel, E.A. 2015 Generation of large-scale winds in horizontally anisotropic convection. Phys. Rev. Lett. 115, 134501.CrossRefGoogle ScholarPubMed
Wang, Q., Chong, K.L., Stevens, R.J.A.M., Verzicco, R. & Lohse, D. 2020 a From zonal flow to convection rolls in Rayleigh–Bénard convection with free-slip plates. J. Fluid Mech. 905, A21.CrossRefGoogle Scholar
Wang, Q., Lohse, D. & Shishkina, O. 2021 Scaling in internally heated convection: a unifying theory. Geophys. Res. Lett. 48, e2020GL091198.Google Scholar
Wang, Q., Verzicco, R., Lohse, D. & Shishkina, O. 2020 b Multiple states in turbulent large-aspect-ratio thermal convection: what determines the number of convection rolls? Phys. Rev. Lett. 125, 074501.CrossRefGoogle ScholarPubMed
Wang, Y., He, X. & Tong, P. 2016 Boundary layer fluctuations and their effects on mean and variance temperature profiles in turbulent Rayleigh–Bénard convection. Phys. Rev. Fluids 1, 082301(R).CrossRefGoogle Scholar
Wang, Y., Xu, W., He, X., Yik, H., Wang, X., Schumacher, J. & Tong, P. 2018 Boundary layer fluctuations in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 840, 408–431.CrossRefGoogle Scholar
Weiss, S. & Ahlers, G. 2011 Turbulent Rayleigh–Bénard convection in a cylindrical container with aspect ratio $\varGamma =0.50$ and Prandtl number $Pr=4.38$. J. Fluid Mech. 676, 5–40.CrossRefGoogle Scholar
Weiss, S. & Ahlers, G. 2013 Effect of tilting on turbulent convection: cylindrical samples with aspect ratio $\varGamma =0.50$. J. Fluid Mech. 715, 314–334.CrossRefGoogle Scholar
Weiss, S., He, X., Ahlers, G., Bodenschatz, E. & Shishkina, O. 2018 Bulk temperature and heat transport in turbulent Rayleigh–Bénard convection of fluids with temperature-dependent properties. J. Fluid Mech. 851, 374–390.CrossRefGoogle Scholar
Wen, B., Goluskin, D., Leduc, M., Chini, G.P. & Doering, C.R. 2020 Steady Rayleigh–Bénard convection between stress-free boundaries. J. Fluid Mech. 905, R4.CrossRefGoogle Scholar
White, C.M. & Mungal, M.G. 2008 Mechanics and prediction of turbulent drag reduction with polymer additives. Annu. Rev. Fluid Mech. 40, 235–56.CrossRefGoogle Scholar
Xi, H.D. & Xia, K.Q. 2008 Flow mode transition in turbulent thermal convection. Phys. Fluids 20, 055104.CrossRefGoogle Scholar
Xu, W., Wang, Y., He, X., Wang, X., Schumacher, J., Huang, S. & Tong, P. 2021 Mean velocity and temperature profiles in turbulent Rayleight–Bénard convection at low Prandtl numbers. J. Fluid Mech. 918, A1.CrossRefGoogle Scholar
Yan, B., Shishkina, O. & He, X. 2021 Thermal boundary-layer structure in laminar horizontal convection. J. Fluid Mech. 915, R5.CrossRefGoogle Scholar
Zhang, Y.Z., Sun, C., Bao, Y. & Zhou, Q. 2018 How surface roughness reduces heat transport for small roughness heights in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 836, R2.CrossRefGoogle Scholar
Zwirner, L. & Shishkina, O. 2018 Confined inclined thermal convection in low-Prandtl-number fluids. J. Fluid Mech. 850, 984–1008.CrossRefGoogle Scholar
Zwirner, L., Tilgner, A. & Shishkina, O. 2020 Elliptical instability and multi-roll flow modes of the large-scale circulation in confined turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 125, 054502.CrossRefGoogle Scholar