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Prandtl and Rayleigh number dependence of heat transport in high Rayleigh number thermal convection

  • Richard J. A. M. Stevens (a1), Detlef Lohse (a1) and Roberto Verzicco (a1) (a2)
Abstract
Abstract

Results from direct numerical simulation for three-dimensional Rayleigh–Bénard convection in samples of aspect ratio and up to Rayleigh number are presented. The broad range of Prandtl numbers is considered. In contrast to some experiments, we do not see any increase in with increasing , neither due to an increasing , nor due to constant heat flux boundary conditions at the bottom plate instead of constant temperature boundary conditions. Even at these very high , both the thermal and kinetic boundary layer thicknesses obey Prandtl–Blasius scaling.

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Email address for correspondence: r.j.a.m.stevens@tnw.utwente.nl
References
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1. Ahlers G., Bodenschatz E., Funfschilling D. & Hogg J. 2009a Turbulent Rayleigh–Bénard convection for a Prandtl number of 0.67. J. Fluid Mech. 641, 157167.
2. Ahlers G., Funfschilling D. & Bodenschatz E. 2009b Transitions in heat transport by turbulent convection at Rayleigh numbers up to inline-graphic
$1{0}^{15} $
. New J. Phys. 11, 123001.
3. Ahlers G., Funfschilling D. & Bodenschatz E. 2011 Addendum to transitions in heat transport by turbulent convection at Rayleigh numbers up to inline-graphic
$1{0}^{15} $
. New J. Phys. 13, 049401.
4. Ahlers G., Grossmann S. & Lohse D. 2009c Heat transfer and large scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81, 503.
5. Ahlers G. & Xu X. 2001 Prandtl-number dependence of heat transport in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 86, 33203323.
6. Chavanne X., Chilla F., Chabaud B., Castaing B. & Hebral B. 2001 Turbulent Rayleigh–Bénard convection in gaseous and liquid He. Phys. Fluids 13, 13001320.
7. Chillà F., Rastello M., Chaumat S. & Castaing B. 2004 Long relaxation times and tilt sensitivity in Rayleigh–Bénard turbulence. Euro. Phys. J. B 40, 223227.
8. Cortet P., Chiffaudel A., Daviaud F. & Dubrulle B. 2010 Experimental evidence of a phase transition in a closed turbulent flow. Phys. Rev. Lett. 105, 214501.
9. Funfschilling D., Bodenschatz E. & Ahlers G. 2009 Search for the ‘ultimate state’ in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 103, 014503.
10. Grossmann S. & Lohse D. 2000 Scaling in thermal convection: a unifying view. J. Fluid Mech. 407, 2756.
11. Grossmann S. & Lohse D. 2001 Thermal convection for large Prandtl number. Phys. Rev. Lett. 86, 33163319.
12. Grossmann S. & Lohse D. 2002 Prandtl and Rayleigh number dependence of the Reynolds number in turbulent thermal convection. Phys. Rev. E 66, 016305.
13. Grossmann S. & Lohse D. 2004 Fluctuations in turbulent Rayleigh–Bénard convection: the role of plumes. Phys. Fluids 16, 44624472.
14. Grossmann S. & Lohse D. 2011 Multiple scaling in the ultimate regime of thermal convection. Phys. Fluids 23, 045108.
15. Hébert F., Hufschmid R., Scheel J. & Ahlers G. 2010 Onset of Rayleigh–Bénard convection in cylindrical containers. Phys. Rev. E 81, 046318.
16. Johnston H. & Doering C. R. 2009 Comparison of turbulent thermal convection between conditions of constant temperature and constant flux. Phys. Rev. Lett. 102, 064501.
17. Niemela J., Skrbek L., Sreenivasan K. R. & Donnelly R. 2000 Turbulent convection at very high Rayleigh numbers. Nature 404, 837840.
18. Niemela J., Skrbek L., Sreenivasan K. R. & Donnelly R. J. 2001 The wind in confined thermal turbulence. J. Fluid Mech. 449, 169178.
19. Niemela J. & Sreenivasan K. R. 2003 Confined turbulent convection. J. Fluid Mech. 481, 355384.
20. Niemela J. & Sreenivasan K. R. 2006 Turbulent convection at high Rayleigh numbers and aspect ratio 4. J. Fluid Mech. 557, 411422.
21. Niemela J. J. & Sreenivasan K. R. 2010 Does confined turbulent convection ever attain the ‘asymptotic scaling’ with inline-graphic
$1/ 2$
power?
New J. Phys. 12, 115002.
22. van der Poel E. P., Stevens R. J. A. M. & Lohse D. 2011 Connecting flow structures and heat flux in turbulent Rayleigh–Bénard convection, Phys. Rev. E. (in press).
23. Qiu X. L. & Xia K.-Q. 1998 Viscous boundary layers at the sidewall of a convection cell. Phys. Rev. E 58, 486491.
24. Roche P. E., Castaing B., Chabaud B. & Hebral B. 2001 Observation of the inline-graphic
$1/ 2$
power law in Rayleigh–Bénard convection
. Phys. Rev. E 63, 045303.
25. Roche P. E., Castaing B., Chabaud B. & Hebral B. 2002 Prandtl and Rayleigh numbers dependences in Rayleigh–Bénard convection. Europhys. Lett. 58, 693698.
26. Roche P.-E., Gauthier F., Kaiser R. & Salort J. 2010 On the triggering of the ultimate regime of convection. New J. Phys. 12, 085014.
27. 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.
28. Shishkina O. & Thess A. 2009 Mean temperature profiles in turbulent Rayleigh–Bénard convection of water. J. Fluid Mech. 633, 449460.
29. Stevens R. J. A. M., Clercx H. J. H. & Lohse D. 2010a Boundary layers in rotating weakly turbulent Rayleigh–Bénard convection. Phys. Fluids 22, 085103.
30. Stevens R. J. A. M., Verzicco R. & Lohse D. 2010b Radial boundary layer structure and Nusselt number in Rayleigh–Bénard convection. J. Fluid Mech. 643, 495507.
31. Sun C., Cheung Y. H. & Xia K. Q. 2008 Experimental studies of the viscous boundary layer properties in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 605, 79113.
32. Sun C., Xi H.-D. & Xia K.-Q. 2005 Azimuthal symmetry, Flow dynamics and Heat transport in turbulent thermal convection in a cylinder with an aspect ratio of 0.5. Phys. Rev. Lett. 95, 074502.
33. Verzicco R. & Camussi R. 1997 Transitional regimes of low-prandtl thermal convection in a cylindrical cell. Phys. Fluids 9, 12871295.
34. Verzicco R. & Camussi R. 2003 Numerical experiments on strongly turbulent thermal convection in a slender cylindrical cell. J. Fluid Mech. 477, 1949.
35. 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, 203219.
36. Weiss S. & Ahlers G. 2011 Turbulent Rayleigh–Bénard convection in a cylindrical container with aspect ratio inline-graphic
$\Gamma = 0. 5$
and Prandtl number inline-graphic
$\mathit{Pr}= 4. 38$
. J. Fluid Mech. 676, 540.
37. Xi H.-D. & Xia K.-Q. 2008 Flow mode transitions in turbulent thermal convection. Phys. Fluids 20, 055104.
38. Xia K.-Q., Lam S. & Zhou S. Q. 2002 Heat-flux measurement in high-Prandtl-number turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 88, 064501.
39. 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, 297312.
40. Zhou Q. & Xia K.-Q. 2010 Measured instantaneous viscous boundary layer in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 104, 104301.
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Type Description Title
VIDEO
Movies

Stevens et al. supplementary movies
Movie of the vertical velocity field in a vertical cut for the simulation at Ra = 2 x 1012 and Pr = 0.7 in an aspect ratio Γ = 0.5 sample. The dimensionless time is indicated in the top of the movie.

 Video (40.4 MB)
40.4 MB
VIDEO
Movies

Stevens et al. supplementary movies
Movie of the vertical velocity field in a vertical cut for the simulation at Ra = 2 x 1012 and Pr = 0.7 in an aspect ratio Γ = 0.5 sample. The dimensionless time is indicated in the top of the movie. This vertical cut is perpendicular to the plane shown in movie 2.

 Video (8.6 MB)
8.6 MB
VIDEO
Movies

Stevens et al. supplementary movies
Movie of the temperature field in a vertical cut for the simulation at Ra = 2 x 1012 and Pr = 0.7 in an aspect ratio Γ = 0.5 sample. The dimensionless time is indicated in the top of the movie. This vertical cut is perpendicular to the plane shown in movie 1.

 Video (8.6 MB)
8.6 MB
VIDEO
Movies

Stevens et al. supplementary movies
Movie of the temperature in three horizontal planes (0:25z/L, 0:50z/L, and 0:75z/L) for the simulation at Ra = 2 x 1012 and Pr = 0.7 in an aspect ratio Γ = 0.5 sample. The dimensionless time is indicated in the top of the movie.

 Video (50.7 MB)
50.7 MB
VIDEO
Movies

Stevens et al. supplementary movies
Movie of the vertical velocity field in a vertical cut for the simulation at Ra = 2 x 1012 and Pr = 0.7 in an aspect ratio Γ = 0.5 sample. The dimensionless time is indicated in the top of the movie.

 Video (9.9 MB)
9.9 MB
VIDEO
Movies

Stevens et al. supplementary movies
Movie of the temperature field in a vertical cut for the simulation at Ra = 2 x 1012 and Pr = 0.7 in an aspect ratio Γ = 0.5 sample. The dimensionless time is indicated in the top of the movie. This vertical cut is perpendicular to the plane shown in movie 1.

 Video (34.2 MB)
34.2 MB
VIDEO
Movies

Stevens et al. supplementary movies
Movie of the temperature field in a vertical cut for the simulation at Ra = 2 x 1012 and Pr = 0.7 in an aspect ratio Γ = 0.5 sample. The dimensionless time is indicated in the top of the movie.

 Video (39.7 MB)
39.7 MB
VIDEO
Movies

Stevens et al. supplementary movies
Movie of the temperature field in a vertical cut for the simulation at Ra = 2 x 1012 and Pr = 0.7 in an aspect ratio Γ = 0.5 sample. The dimensionless time is indicated in the top of the movie.

 Video (9.9 MB)
9.9 MB
VIDEO
Movies

Stevens et al. supplementary movies
Movie of the vertical velocity field in three horizontal planes (0:25z/L, 0:50z/L, and 0:75z/L) for the simulation at Ra = 2 x 1012 and Pr = 0.7 in an aspect ratio Γ = 0.5 sample. The dimensionless time is indicated in the top of the movie.

 Video (9.9 MB)
9.9 MB
VIDEO
Movies

Stevens et al. supplementary movies
Movie of the temperature in three horizontal planes (0:25z/L, 0:50z/L, and 0:75z/L) for the simulation at Ra = 2 x 1012 and Pr = 0.7 in an aspect ratio Γ = 0.5 sample. The dimensionless time is indicated in the top of the movie.

 Video (9.9 MB)
9.9 MB
VIDEO
Movies

Stevens et al. supplementary movies
Movie of the vertical velocity field in three horizontal planes (0:25z/L, 0:50z/L, and 0:75z/L) for the simulation at Ra = 2 x 1012 and Pr = 0.7 in an aspect ratio Γ = 0.5 sample. The dimensionless time is indicated in the top of the movie.

 Video (52.3 MB)
52.3 MB
VIDEO
Movies

Stevens et al. supplementary movies
Movie of the vertical velocity field in a vertical cut for the simulation at Ra = 2 x 1012 and Pr = 0.7 in an aspect ratio Γ = 0.5 sample. The dimensionless time is indicated in the top of the movie. This vertical cut is perpendicular to the plane shown in movie 2.

 Video (35.4 MB)
35.4 MB

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