Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-27T10:17:23.889Z Has data issue: false hasContentIssue false
  • English
  • Français

Thermal boundary layer structure in low-Prandtl-number turbulent convection

Published online by Cambridge University Press:  08 January 2021

Ambrish Pandey Open the ORCID record for Ambrish Pandey [Opens in a new window]
Ambrish Pandey*
Affiliation:
Center for Space Science, New York University Abu Dhabi, PO Box 129188 Abu Dhabi, UAE
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the structure of the thermal boundary layer (BL) in Rayleigh–Bénard convection for Prandtl number ($Pr$) 0.021 by conducting direct numerical simulations in a two-dimensional square box for Rayleigh numbers ($Ra$) up to $10^9$. The large-scale circulation in the flow divides the horizontal plates into three distinct regions, and we observe that the local thermal BL thicknesses in the plume-ejection region are larger than those in the plume-impact and shear-dominated regions. Moreover, the local BL width decreases as $Ra^{-\beta (x)}$, with $\beta (x)$ depending on the position at the plate. We find that the values of $\beta (x)$ are nearly the same in the impact and shear regions, and are larger in the ejection region. Thus, the local BL width decreases faster in the ejection region than in the shear and impact regions, and we estimate that the thermal BL structure would be uniform throughout the horizontal plate for $Ra \geq 8 \times 10^{12}$ in our low-$Pr$ convection. We compare the thermal BL profiles measured at various positions at the plate with the Prandtl–Blasius–Pohlhausen (PBP) profile and find deviations everywhere for all the Rayleigh numbers. However, the dynamically rescaled profiles, as suggested by Zhou & Xia (Phys. Rev. Lett., vol. 104, 2010, 104301), agree well with the PBP profile in the shear and impact regions for all the Rayleigh numbers, whereas they still deviate in the ejection region. We also find that, despite the growing fluctuations with increasing $Ra$, thermal boundary layers in our low-$Pr$ convection are transitional and not yet fully turbulent.

JFM classification

Boundary Layers: Boundary layer structure Turbulent Flows: Turbulent convection
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 910 , 10 March 2021 , A13
Check for updates
Copyright
© The Author(s), 2021. 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., 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
Bhattacharya, S., Samtaney, R. & Verma, M. K. 2019 Scaling and spatial intermittency of thermal dissipation in turbulent convection. Phys. Fluids 31 (7), 075104.CrossRefGoogle Scholar
Chandra, M. & Verma, M. K. 2013 Flow reversals in turbulent convection via vortex reconnections. Phys. Rev. Lett. 110, 114503.CrossRefGoogle ScholarPubMed
Chandrasekhar, S. 1981 Hydrodynamic and Hydromagnetic Stability. Dover.Google Scholar
Chen, X., Huang, S.-D., Xia, K.-Q. & Xi, H.-D. 2019 Emergence of substructures inside the large-scale circulation induces transition in flow reversals in turbulent thermal convection. J. Fluid Mech. 877, R1.CrossRefGoogle Scholar
Chillà, F. & Schumacher, J. 2012 New perspectives in turbulent Rayleigh–Bénard convection. Eur. Phys. J. E 35, 58.CrossRefGoogle ScholarPubMed
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
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. & Willis, G. E. 1967 Investigation of turbulent thermal convection between horizontal plates. J. Fluid Mech. 28 (4), 675–704.CrossRefGoogle Scholar
Fischer, P. F. 1997 An overlapping Schwarz method for spectral element solution of the incompressible Navier–Stokes equations. J. Comput. Phys. 133 (1), 84–101.CrossRefGoogle Scholar
Glazier, J., Segawa, T., Naert, A. & Sano, M. 1999 Evidence against ‘ultrahard’ thermal turbulence at very high Rayleigh numbers. Nature 398, 307–310.CrossRefGoogle Scholar
Grossmann, S. & Lohse, D. 2000 Scaling in thermal convection: a unifying theory. J. Fluid Mech. 407, 27.CrossRefGoogle Scholar
Hanasoge, S., Gizon, L. & Sreenivasan, K. R. 2016 Seismic sounding of convection in the sun. Annu. Rev. Fluid Mech. 48, 191–217.CrossRefGoogle Scholar
He, Y.-H. & Xia, K.-Q. 2019 Temperature fluctuation profiles in turbulent thermal convection: a logarithmic dependence versus a power-law dependence. Phys. Rev. Lett. 122, 014503.CrossRefGoogle ScholarPubMed
Horn, S., Shishkina, O. & Wagner, C. 2013 On non-Oberbeck–Boussinesq effects in three-dimensional Rayleigh–Bénard convection in glycerol. J. Fluid Mech. 724, 175–202.CrossRefGoogle Scholar
Kader, B. 1981 Temperature and concentration profiles in fully turbulent boundary layers. Intl J. Heat Mass Transfer 24 (9), 1541–1544.CrossRefGoogle Scholar
Landau, L. D. & Lifshitz, E. M. 1987 Fluid Mechanics. Pergamon.Google Scholar
Lui, S.-L. & Xia, K.-Q. 1998 Spatial structure of the thermal boundary layer in turbulent convection. Phys. Rev. E 57, 5494–5503.CrossRefGoogle Scholar
Niemela, J. J. & Sreenivasan, K. R. 2003 Rayleigh–number evolution of large-scale coherent motion in turbulent convection. Europhys. Lett. 62, 829–833.CrossRefGoogle Scholar
Ovsyannikov, M., Krasnov, D., Emran, M. S. & Schumacher, J. 2016 Combined effects of prescribed pressure gradient and buoyancy in boundary layer of turbulent Rayleigh–Bénard convection. Eur. J. Mech. B/Fluids 57, 64–74.CrossRefGoogle Scholar
Pandey, A., Scheel, J. D. & Schumacher, J. 2018 a Turbulent superstructures in Rayleigh–Bénard convection. Nat. Commun. 9, 2118.CrossRefGoogle ScholarPubMed
Pandey, A. & Verma, M. K. 2016 Scaling of large-scale quantities in Rayleigh–Bénard convection. Phys. Fluids 28, 095105.CrossRefGoogle Scholar
Pandey, A., Verma, M. K. & Barma, M. 2018 b Reversals in infinite-Prandtl-number Rayleigh–Bénard convection. Phys. Rev. E 98, 023109.CrossRefGoogle ScholarPubMed
Pandey, A., Verma, M. K., Chatterjee, A. G. & Dutta, B. 2016 Similarities between 2-D and 3-D convection for large Prandtl number. Pramana J. Phys. 87, 13.CrossRefGoogle Scholar
Pandey, A., Verma, M. K. & Mishra, P. K. 2014 Scaling of heat flux and energy spectrum for very large Prandtl number convection. Phys. Rev. E 89, 023006.CrossRefGoogle ScholarPubMed
Podvin, B. & Sergent, A. 2015 A large-scale investigation of wind reversal in a square Rayleigh–Bénard cell. J. Fluid Mech. 766, 172–201.CrossRefGoogle Scholar
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
Scheel, J. D., Emran, M. S. & Schumacher, J. 2013 Resolving the fine-scale structure in turbulent Rayleigh–Bénard convection. New J. Phys. 15, 113063.CrossRefGoogle Scholar
Scheel, J. D., Kim, E. & White, K. R. 2012 Thermal and viscous boundary layers in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 711, 281–305.CrossRefGoogle Scholar
Scheel, J. D. & Schumacher, J. 2014 Local boundary layer scales in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 758, 344–373.CrossRefGoogle Scholar
Scheel, J. D. & Schumacher, J. 2016 Global and local statistics in turbulent convection at low Prandtl numbers. J. Fluid Mech. 802, 147–173.CrossRefGoogle Scholar
Scheel, J. D. & Schumacher, J. 2017 Predicting transition ranges to fully turbulent viscous boundary layers in low Prandtl number convection flows. Phys. Rev. Fluids 2, 123501.CrossRefGoogle Scholar
Schlichting, H. & Gersten, K. 2004 Boundary Layer Theory. Springer.Google Scholar
Schmalzl, J., Breuer, M. & Hansen, U. 2004 On the validity of two-dimensional numerical approaches to time-dependent thermal convection. Europhys. Lett. 67, 390–396.CrossRefGoogle Scholar
Schumacher, J., Bandaru, V., Pandey, A. & Scheel, J. D. 2016 Transitional boundary layers in low-Prandtl-number convection. Phys. Rev. Fluids 1, 084402.CrossRefGoogle Scholar
Schumacher, J., Götzfried, P. & Scheel, J. D. 2015 Enhanced enstrophy generation for turbulent convection in low-Prandtl-number fluids. Proc. Natl Acad. Sci. USA 112, 9530–9535.CrossRefGoogle ScholarPubMed
Schumacher, J. & Sreenivasan, K. R. 2020 Colloquium: unusual dynamics of convection in the sun. Rev. Mod. Phys. 92, 041001.CrossRefGoogle Scholar
Shi, N., Emran, M. S. & Schumacher, J. 2012 Boundary layer structure in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 706, 5–33.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., 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. & Thess, A. 2009 Mean temperature profiles in turbulent Rayleigh–Bénard convection of water. J. Fluid Mech. 633, 449–460.CrossRefGoogle Scholar
Shishkina, O. & Wagner, C. 2008 Analysis of sheet-like thermal plumes in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 599, 383–404.CrossRefGoogle Scholar
Shraiman, B. I. & Siggia, E. D. 1990 Heat transport in high-Rayleigh–number convection. Phys. Rev. A 42, 3650–3653.CrossRefGoogle ScholarPubMed
Silano, G., Sreenivasan, K. R. & Verzicco, R. 2010 Numerical simulations of Rayleigh–Bénard convection for Prandtl numbers between $10^{-1}$ and $10^4$ and Rayleigh numbers between $10^5$ and $10^9$. J. Fluid Mech. 662, 409–446.CrossRefGoogle Scholar
Stevens, R., Lohse, D. & Verzicco, R. 2011 Prandtl and Rayleigh number dependence of heat transport in high Rayleigh number thermal convection. J. Fluid Mech. 688, 31–43.CrossRefGoogle Scholar
Stevens, R. J. A. M., Zhou, Q., Grossmann, S., Verzicco, R., Xia, K.-Q. & Lohse, D. 2012 Thermal boundary layer profiles in turbulent Rayleigh–Bénard convection in a cylindrical sample. Phys. Rev. E 85, 027301.CrossRefGoogle Scholar
Sugiyama, K., Ni, R., Stevens, R. J. A. M., Chan, T. S., Zhou, S.-Q., Xi, H.-D., Sun, C., Grossmann, S., Xia, K.-Q. & Lohse, D. 2010 Flow reversals in thermally driven turbulence. Phys. Rev. Lett. 105, 034503.CrossRefGoogle ScholarPubMed
Verma, M. K. 2018 Physics of Buoyant Flows. World Scientific.CrossRefGoogle Scholar
Verma, M. K., Kumar, A. & Pandey, A. 2017 Phenomenology of buoyancy-driven turbulence: recent results. New J. Phys. 19 (2), 025012.CrossRefGoogle Scholar
Wagner, S. & Shishkina, O. 2013 Aspect-ratio dependency of Rayleigh–Bénard convection in box-shaped containers. Phys. Fluids 25, 085110.CrossRefGoogle Scholar
Wagner, S., Shishkina, O. & Wagner, C. 2012 Boundary layers and wind in cylindrical Rayleigh–Bénard cells. J. Fluid Mech. 697, 336–366.CrossRefGoogle Scholar
Wang, J. & Xia, K.-Q. 2003 Spatial variations of the mean and statistical quantities in the thermal boundary layers of turbulent convection. Eur. Phys. J. B 32, 127–136.CrossRefGoogle Scholar
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.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
Werne, J. 1993 Structure of hard-turbulent convection in two dimensions: numerical evidence. Phys. Rev. E 48, 1020–1035.CrossRefGoogle ScholarPubMed
Yaglom, A. M. 1979 Similarity laws for constant-pressure and pressure-gradient turbulent wall flows. Annu. Rev. Fluid Mech. 11 (1), 505–540.CrossRefGoogle Scholar
Zhang, S., Xia, Z., Zhou, Q. & Chen, S. 2020 Controlling flow reversal in two-dimensional Rayleigh–Bénard convection. J. Fluid Mech. 891, R4.CrossRefGoogle Scholar
Zhang, Y., Huang, Y.-X., Jiang, N., Liu, Y.-L., Lu, Z.-M., Qiu, X. & Zhou, Q. 2017 a Statistics of velocity and temperature fluctuations in two-dimensional Rayleigh–Bénard convection. Phys. Rev. E 96, 023105.CrossRefGoogle ScholarPubMed
Zhang, Y., Zhou, Q. & Sun, C. 2017 b Statistics of kinetic and thermal energy dissipation rates in two-dimensional turbulent Rayleigh–Bénard convection. J. Fluid Mech. 814, 165–184.CrossRefGoogle Scholar
Zhou, Q., Stevens, R. J. A. M., Sugiyama, K., Grossmann, S., Lohse, D. & Xia, K.-Q. 2010 Prandtl–Blasius temperature and velocity boundary-layer profiles in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 664, 297–312.CrossRefGoogle Scholar
Zhou, Q., Sugiyama, K., Stevens, R. J. A. M., Grossmann, S., Lohse, D. & Xia, K.-Q. 2011 Horizontal structures of velocity and temperature boundary layers in two-dimensional numerical turbulent Rayleigh–Bénard convection. Phys. Fluids 23 (12), 125104.Google Scholar
Zhou, Q., Sun, C. & Xia, K.-Q. 2007 Morphological evolution of thermal plumes in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 98, 074501.CrossRefGoogle ScholarPubMed
Zhou, Q. & Xia, K. Q. 2010 Measured instantaneous viscous boundary layer in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 104, 104301.CrossRefGoogle ScholarPubMed
Zhou, Q. & Xia, K.-Q. 2013 Thermal boundary layer structure in turbulent Rayleigh–Bénard convection in a rectangular cell. J. Fluid Mech. 721, 199–224.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 ScholarPubMed
Zürner, T., Schindler, F., Vogt, T., Eckert, S. & Schumacher, J. 2019 Combined measurement of velocity and temperature in liquid metal convection. J. Fluid Mech. 876, 1108–1128.CrossRefGoogle Scholar
Zwirner, L., Khalilov, R., Kolesnichenko, I., Mamykin, A., Mandrykin, S., Pavlinov, A., Shestakov, A., Teimurazov, A., Frick, P. & Shishkina, O. 2020 The influence of the cell inclination on the heat transport and large-scale circulation in liquid metal convection. J. Fluid Mech. 884, A18.CrossRefGoogle Scholar

Pandey supplementary movie 1

See pdf file for movie caption

Download Pandey supplementary movie 1(Video)
Video 11.2 MB

Pandey supplementary movie 2

See pdf file for movie caption

Download Pandey supplementary movie 2(Video)
Video 15.7 MB

Pandey supplementary movie 3

See pdf file for movie caption

Download Pandey supplementary movie 3(Video)
Video 49.3 MB

Pandey supplementary movie 4

See pdf file for movie caption

Download Pandey supplementary movie 4(Video)
Video 75.3 MB

Pandey supplementary movie 5

See pdf file for movie caption

Download Pandey supplementary movie 5(Video)
Video 29 MB
Supplementary material: PDF

Pandey supplementary material

Captions for movies 1-5

Download Pandey supplementary material(PDF)
PDF 12.9 KB