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Radial boundary layer structure and Nusselt number in Rayleigh–Bénard convection

Published online by Cambridge University Press:  15 January 2010

RICHARD J. A. M. STEVENS ,
ROBERTO VERZICCO  and
DETLEF LOHSE
RICHARD J. A. M. STEVENS*
Affiliation:
Department of Science and Technology and J. M. Burgers Center for Fluid Dynamics, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands
ROBERTO VERZICCO
Affiliation:
Department of Mechanical Engineering, Universitá di Roma ‘Tor Vergata’, Via del Politecnico 1, 00133 Roma, Italy
DETLEF LOHSE
Affiliation:
Department of Science and Technology and J. M. Burgers Center for Fluid Dynamics, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands
*
†Email address for correspondence: r.j.a.m.stevens@tnw.utwente.nl
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Abstract

Results from direct numerical simulation (DNS) for three-dimensional Rayleigh–Bénard convection in a cylindrical cell of aspect ratio 1/2 and Prandtl number Pr=0.7 are presented. They span five decades of Rayleigh number Ra from 2 × 106 to 2 × 1011. The results are in good agreement with the experimental data of Niemela et al. (Nature, vol. 404, 2000, p. 837). Previous DNS results from Amati et al. (Phys. Fluids, vol. 17, 2005, paper no. 121701) showed a heat transfer that was up to 30% higher than the experimental values. The simulations presented in this paper are performed with a much higher resolution to properly resolve the plume dynamics. We find that in under-resolved simulations the hot (cold) plumes travel further from the bottom (top) plate than in the better-resolved ones, because of insufficient thermal dissipation mainly close to the sidewall (where the grid cells are largest), and therefore the Nusselt number in under-resolved simulations is overestimated. Furthermore, we compare the best resolved thermal boundary layer profile with the Prandtl–Blasius profile. We find that the boundary layer profile is closer to the Prandtl–Blasius profile at the cylinder axis than close to the sidewall, because of rising plumes close to the sidewall.

JFM classification

Turbulent Flows: Turbulent convection
Type
Papers
Information
Journal of Fluid Mechanics , Volume 643 , 25 January 2010 , pp. 495 - 507
Copyright
Copyright © Cambridge University Press 2010

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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.CrossRefGoogle Scholar
Amati, G., Koal, K., Massaioli, F., Sreenivasan, K. R. & Verzicco, R. 2005 Turbulent thermal convection at high Rayleigh numbers for a constant-Prandtl-number fluid under Boussinesq conditions. Phys. Fluids 17, 121701.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., Chilla, F., Castaing, B., Hebral, B., Chabaud, B. & Chaussy, J. 1997 Observation of the ultimate regime in Rayleigh–Bénard convection. Phys. Rev. Lett. 79, 3648–3651.CrossRefGoogle Scholar
Chavanne, X., Chilla, F., Chabaud, B., Castaing, B. & Hebral, B. 2001 Turbulent Rayleigh–Bénard convection in gaseous and liquid He. Phys. Fluids 13, 1300–1320.CrossRefGoogle Scholar
Emran, M. S. & Schumacher, J. 2008 Fine-scale statistics of temperature and its derivatives in convective turbulence. J. Fluid Mech. 611, 13–34.CrossRefGoogle Scholar
Funfschilling, D., Bodenschatz, E. & Ahlers, G. 2009 Search for the ‘ultimate state’ in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 103, 014503.CrossRefGoogle Scholar
Funfschilling, D., Brown, E., Nikolaenko, A. & Ahlers, G. 2005 Heat transport by turbulent Rayleigh–Bénard convection in cylindrical cells with aspect ratio one and larger. J. Fluid Mech. 536, 145–154.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. 2002 Prandtl and Rayleigh number dependence of the Reynolds number in turbulent thermal convection. Phys. Rev. E 66, 016305.CrossRefGoogle ScholarPubMed
Grötzbach, G. 1983 Spatial resolution for direct numerical simulations of Rayleigh–Bénard convection. J. Comput. Phys. 49, 241–264.CrossRefGoogle Scholar
Heslot, F., Castaing, B. & Libchaber, A. 1987 Transition to turbulence in helium gas. Phys. Rev. A 36, 5870–5873.CrossRefGoogle ScholarPubMed
Johnston, H. & Doering, C. R. 2009 Comparison of turbulent thermal convection between conditions of constant temperature and constant flux. Phys. Rev. Lett. 102, 064501.CrossRefGoogle ScholarPubMed
Kaczorowski, M. & Wagner, C. 2009 Analysis of the thermal plumes in turbulent Rayleigh–Bénard convection based on well-resolved numerical simulations. J. Fluid. Mech. 618, 89–112.CrossRefGoogle Scholar
Kraichnan, R. H. 1962 Turbulent thermal convection at arbitrary Prandtl number. Phys. Fluids 5, 1374–1389.CrossRefGoogle Scholar
Kunnen, R. P. J., Clercx, H. J. H., Geurts, B. J., Bokhoven, L. J. A., Akkermans, R. A. D. & Verzicco, R. 2008 A numerical and experimental investigation of structure function scaling in turbulent Rayleigh–Bénard convection. Phys. Rev. E 77, 016302.CrossRefGoogle ScholarPubMed
Lohse, D. & Toschi, F. 2003 The ultimate state of thermal convection. Phys. Rev. Lett. 90, 034502.CrossRefGoogle ScholarPubMed
Lohse, D. & Xia, K. Q. 2010 Small-scale properties of turbulent Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 42, 335–364.CrossRefGoogle Scholar
Niemela, J., Skrbek, L., Sreenivasan, K. R. & Donnelly, R. 2000 Turbulent convection at very high Rayleigh numbers. Nature 404, 837–840.CrossRefGoogle ScholarPubMed
Niemela, J., Skrbek, L., Sreenivasan, K. R. & Donnelly, R. J. 2001 The wind in confined thermal turbulence. J. Fluid Mech. 449, 169–178.CrossRefGoogle Scholar
Niemela, J. & Sreenivasan, K. R. 2003 Confined turbulent convection. J. Fluid Mech. 481, 355–384.CrossRefGoogle Scholar
Nikolaenko, A., Brown, E., Funfschilling, D. & Ahlers, G. 2005 Heat transport by turbulent Rayleigh–Bénard convection in cylindrical cells with aspect ratio one and less. J. Fluid Mech. 523, 251–260.CrossRefGoogle Scholar
Roche, P. E., Castaing, B., Chabaud, B. & Hebral, B. 2002 Prandtl and Rayleigh numbers dependences in Rayleigh–Bénard convection. Europhys. Lett. 58, 693–698.CrossRefGoogle Scholar
Schumacher, J., Sreenivasan, K. R. & Yeung, P. K. 2005 Very fine structures in scalar mixing. J. Fluid. Mech. 531, 113–122.CrossRefGoogle Scholar
Shang, X. D., Tong, P. & Xia, K.-Q. 2008 Scaling of the local convective heat flux in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 100, 244503.CrossRefGoogle ScholarPubMed
Shishkina, O. & Wagner, C. 2007 Local heat fluxes in turbulent Rayleigh–Bénard convection. Phys Fluids. 19, 085107.CrossRefGoogle Scholar
Shishkina, O. & Wagner, C. 2008 Analysis of sheetlike 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
Spiegel, E. A. 1971 Convection in stars. Annu. Rev. Astron. Astrophys. 9, 323–352.CrossRefGoogle Scholar
Sugiyama, K., Calzavarini, E., Grossmann, S. & Lohse, D. 2009 Flow organization two-dimensional in non-Oberbeck–Boussinesq Rayleigh–Bénard convection in water. J. Fluid Mech. 637, 105–135.CrossRefGoogle Scholar
Verzicco, R. & Camussi, R. 1999 Prandtl number effects in convective turbulence. J. Fluid Mech. 383, 55–73.CrossRefGoogle Scholar
Verzicco, R. & Camussi, R. 2003 Numerical experiments on strongly turbulent thermal convection in a slender cylindrical cell. J. Fluid Mech. 477, 19–49.CrossRefGoogle Scholar
Verzicco, R. & Orlandi, P. 1996 A finite-difference scheme for three-dimensional incompressible flow in cylindrical coordinates. J. Comput. Phys. 123, 402–413.CrossRefGoogle Scholar
Verzicco, R. & Sreenivasan, K. R. 2008 A comparison of turbulent thermal convection between conditions of constant temperature and constant heat flux. J. Fluid Mech. 595, 203–219.CrossRefGoogle Scholar
Zhong, J.-Q., Stevens, R. J. A. M., Clercx, H. J. H., Verzicco, R., Lohse, D. & Ahlers, G. 2009 Prandtl-, Rayleigh-, and Rossby-number dependence of heat transport in turbulent rotating Rayleigh–Bénard convection. Phys. Rev. Lett. 102, 044502.CrossRefGoogle ScholarPubMed

Stevens et al. supplementary movie

Temperature field close to the bottom plate for the simulations at Ra = 2 x 109, Pr = 0.7, and G = 0.5, with grid resolution 129 x 65 x 257 (heights: 0.0021L, 0.0049L, 0.0079L, 0.0111L, and here lqsl=0.0056L). The dimensionless time is indicated in the top of the movie. Note that in this low resolution simulation the smoothness of the solution is insufficient to represent all flow dynamics observed in a high resolution simulation.

Download Stevens et al. supplementary movie(Video)
Video 4.5 MB

Stevens et al. supplementary movie

Temperature field close to the bottom plate for the simulations at Ra = 2 x 109, Pr = 0.7, and G = 0.5, with grid resolution 193 x 65 x 257 (heights: 0.0021L, 0.0049L, 0.0079L, 0.0111L, and here lqsl=0.0059L). The dimensionless time is indicated in the top of the movie.

Download Stevens et al. supplementary movie(Video)
Video 4.5 MB

Stevens et al. supplementary movie

Temperature field close to the bottom plate for the simulations at Ra = 2 x 109, Pr = 0.7, and G = 0.5, with grid resolution 385 x 97 x 385 (heights: 0.0020L, 0.0044L, 0.0077L, 0.0112L, and here lqsl=0.0059L). The dimensionless time is indicated in the top of the movie.

Download Stevens et al. supplementary movie(Video)
Video 4 MB

Stevens et al. supplementary movie

Temperature field at mid height for the simulations at Ra = 2 x 109, Pr = 0.7, and G = 0.5 with grid resolution 129 x 65 x 257. The dimensionless time is indicated in the top of the movie. Note that in this low resolution simulation the smoothness of the solution is insufficient to represent all flow dynamics observed in a high resolution simulation.

Download Stevens et al. supplementary movie(Video)
Video 3.9 MB

Stevens et al. supplementary movie

Temperature field at mid height for the simulations at Ra = 2 x 109, Pr = 0.7, and G = 0.5 with grid resolution 193 x 65 x 257. The dimensionless time is indicated in the top of the movie.

Download Stevens et al. supplementary movie(Video)
Video 3.9 MB

Stevens et al. supplementary movie

Temperature field at mid height for the simulations at Ra = 2 x 109, Pr = 0.7, and G = 0.5 with grid resolution 385 x 97 x 385. The dimensionless time is indicated in the top of the movie.

Download Stevens et al. supplementary movie(Video)
Video 3.9 MB