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Thermal and viscous boundary layers in turbulent Rayleigh–Bénard convection

Published online by Cambridge University Press:  31 August 2012

J. D. Scheel ,
E. Kim  and
K. R. White
J. D. Scheel*
Affiliation:
Department of Physics, Occidental College, 1600 Campus Road, M21, Los Angeles, CA 90041, USA
E. Kim
Affiliation:
Department of Physics, Occidental College, 1600 Campus Road, M21, Los Angeles, CA 90041, USA
K. R. White
Affiliation:
Department of Applied Mathematics and Statistics, University of California Santa Cruz, Mail Stop SOE GRADS, 1156 High Street, Santa Cruz, CA 95064, USA
*
†Email address for correspondence: jscheel@oxy.edu
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Abstract

We present the results from numerical simulations of turbulent Rayleigh–Bénard convection for an aspect ratio (diameter/height) of 1.0, Prandtl numbers of 0.4 and 0.7, and Rayleigh numbers from to . Detailed measurements of the thermal and viscous boundary layer profiles are made and compared to experimental and theoretical (Prandtl–Blasius) results. We find that the thermal boundary layer profiles disagree by more than 10 % when scaled with the similarity variable (boundary layer thickness) and likewise disagree with the Prandtl–Blasius results. In contrast, the viscous boundary profiles collapse well and do agree (within 10 %) with the Prandtl–Blasius profile, but with worsening agreement as the Rayleigh number increases. We have also investigated the scaling of the boundary layer thicknesses with Rayleigh number, and again compare to experiments and theory. We find that the scaling laws are very robust with respect to method of analysis and they mostly agree with the Grossmann–Lohse predictions coupled with laminar boundary layer theory within our numerical uncertainty.

JFM classification

Boundary Layers: Boundary layer structure Turbulent Flows: Turbulent convection
Type
Papers
Information
Journal of Fluid Mechanics , Volume 711 , 25 November 2012 , pp. 281 - 305
Copyright
Copyright © Cambridge University Press 2012

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References

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, 157–167.CrossRefGoogle Scholar
2. Ahlers, G., Grossmann, S. & Lohse, D. 2009b Heat transfer and large-scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81.CrossRefGoogle Scholar
3. Bailon-Cuba, J., Emran, M. S. & Schumacher, J. 2010 Aspect ratio dependence of heat transfer and large-scale flow in turbulent convection. J. Fluid Mech. 655, 152–173.CrossRefGoogle Scholar
4. Belmonte, A., Tilgner, A. & Libchaber, A. 1994 Temperature and velocity boundary layers in turbulent convection. Phys. Rev. E 50, 269–279.CrossRefGoogle ScholarPubMed
5. Benzi, R., Toschi, F. & Tripiccione, R. 1998 On the heat transfer in Rayleigh–Bénard systems. J. Stat. Phys. 93, 901–918.CrossRefGoogle Scholar
6. Breuer, M., Wessling, S., Schmalzl, J. & Hansen, U. 2004 Effect of inertia in Rayleigh–Bénard convection. Phys. Rev. E 69, 026302.CrossRefGoogle ScholarPubMed
7. Brown, E., Nikolaenko, A. & Ahlers, G. 2005 Reorientation of the large-scale circulation in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 95, 084503.CrossRefGoogle ScholarPubMed
8. Burnishev, Y., Segre, E. & Steinberg, V. 2010 Strong symmetrical non-Oberbeck–Boussinesq turbulent convection and the role of compressibility. Phys. Fluids 22, 035108.CrossRefGoogle Scholar
9. 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
10. Chandra, M. & Verma, M. K. 2011 Dynamics and symmetries of flow reversals in turbulent convection. Phys. Rev. E 067303.CrossRefGoogle ScholarPubMed
11. Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.Google Scholar
12. 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
13. Chavanne, X., Chilla, F., Castaing, B., Hebral, B., Chabaud, B. & Chaussy, J. 2001 Turbulent Rayleigh–Bénard convection in gaseous and liquid He. Phys. Fluids 13, 1300–1320.CrossRefGoogle Scholar
14. Emran, M. S. & Schumacher, J. 2008 Finite-scale statistics of temperature and its derivatives in convective turbulence. J. Fluid Mech. 611, 13–34.CrossRefGoogle Scholar
15. Fischer, P. F., Lottes, J. W. & Kerkemeier, S. G. 2008nek5000 web page (http://nek5000.mcs.anl.gov).Google Scholar
16. Funfschilling, D. & Ahlers, G. 2004 Plume motion and large-scale circulation in a cylindrical Rayleigh–Bénard cell. Phys. Rev. Lett. 92, 194502.CrossRefGoogle Scholar
17. Funfschilling, D., Bodenschatz, E. & Ahlers, G. 2009 Search for the ultimate state in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 103, 014503.CrossRefGoogle Scholar
18. Grossmann, S. & Lohse, D. 2000 Scaling in thermal convection: a unifying theory. J. Fluid Mech. 407, 27–56.CrossRefGoogle Scholar
19. He, X., Funfschilling, D., Nobach, H., Bodenschatz, E. & Ahlers, G. 2012 Transition to the ultimate state of turbulent Rayleigh–Bénard convection. Phys. Rev. Lett 108, 024502.CrossRefGoogle Scholar
20. Kerr, R. M. 2001 Energy budget in Rayleigh–Bénard convection. Phys. Rev. Lett. 87, 244502.CrossRefGoogle ScholarPubMed
21. Kraichnan, R. H. 1962 Turbulent thermal convection at arbitrary Prandtl number. Phys. Fluids 5, 1374–1389.CrossRefGoogle Scholar
22. Lam, S., Shang, X.-D., Zhou, S.-Q. & Xia, K.-Q. 2002 Prandtl number dependence of the viscous boundary layer and the Reynolds numbers in Rayleigh–Bénard convection. Phys. Rev. E 65, 066306.CrossRefGoogle ScholarPubMed
23. 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
24. Maystrenko, A., Resagk, C. & Thess, A. 2007 Structure of the thermal boundary layer for turbulent Rayleigh–Bénard convection of air in a long rectangular enclosure. Phys. Rev. E 75, 066303.CrossRefGoogle Scholar
25. Niemela, J. J., Skrbek, L., Sreenivasan, K. R. & Donnelley, R. J. 2000 Turbulent convection at very high Rayleigh numbers. Nature 404, 837–840.CrossRefGoogle ScholarPubMed
26. du Puits, R., Resagk, C. & Thess, A. 2007a Mean velocity profile in confined turbulent convection. Phys. Rev. Lett. 99, 234504.CrossRefGoogle ScholarPubMed
27. du Puits, R., Resagk, C. & Thess, A. 2009 Structure of viscous boundary layers in turbulent Rayleigh–Bénard convection. Phys. Rev. E 80, 036318.CrossRefGoogle ScholarPubMed
28. du Puits, R., Resagk, C. & Thess, A. 2010 Thickness of the diffusive sublayer in turbulent convection. Phys. Rev. E 81, 016307.CrossRefGoogle ScholarPubMed
29. du Puits, R., Resagk, C. & Thess, A. 2012Thermal boundary layers in turbulent Rayleigh–Bénard convection at aspect ratio between 1 and 9. New J. Phys. (submitted).CrossRefGoogle Scholar
30. du Puits, R., Resagk, C., Tilgner, A., Busse, F. H. & Thess, A. 2007b Structure of thermal boundary layers in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 572, 231–254.CrossRefGoogle Scholar
31. Qiu, X.-L. & Xia, K.-Q. 1998 Spatial structure of the viscous boundary layer in turbulent convection. Phys. Rev. E 58, 5816–5820.CrossRefGoogle Scholar
32. van Reeuwijk, M., Jonker, H. J. J. & Hanjalic, K. 2008 Wind and boundary layers in Rayleigh–Bénard convection. II. Boundary layer character and scaling. Phys. Rev. E 77, 036312.CrossRefGoogle ScholarPubMed
33. Schlichting, H. & Gersten, K. 2000 Boundary Layer Theory, 8th edn. Springer.CrossRefGoogle Scholar
34. Shi, N., Emran, M. S. & Schumacher, J. 2012 Boundary layer structure in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 706, 5–33.CrossRefGoogle Scholar
35. 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
36. Shishkina, O. & Thess, A. 2009 Mean temperature profiles in turbulent Rayleigh–Bénard convection of water. J. Fluid Mech. 633, 449–460.CrossRefGoogle Scholar
37. Sreenivasan, K. R., Bershadskii, A. & Niemela, J. J. 2002 Mean wind and its reversal in thermal convection. Phys. Rev. E 65, 056306.CrossRefGoogle ScholarPubMed
38. Stevens, R. J. A. M., 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
39. Stevens, R. J. A. M., Verzicco, R. & Lohse, D. 2010 Radial boundary layer structure and Nusselt number in Rayleigh–Bénard convection. J. Fluid Mech. 643, 495–507.CrossRefGoogle Scholar
40. 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 027301.CrossRefGoogle Scholar
41. 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, 79–113.CrossRefGoogle Scholar
42. Urban, P., Musilova, V. & Skrbek, L. 2011 Efficiency of heat transfer in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 107, 014302.CrossRefGoogle ScholarPubMed
43. Verzicco, R. & Camussi, R. 1999 Prandtl number effects in convective turbulence. J. Fluid Mech. 383, 55–73.CrossRefGoogle Scholar
44. 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
45. 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
46. Wagner, S., Shishkina, O. & Wagner, C. 2012 Boundary layers and wind in cylindrical Rayleigh–Bénard cells. J Fluid Mech. 607, 336.CrossRefGoogle Scholar
47. Wu, X.-Z. & Libchaber, A. 1992 Scaling relations in thermal turbulence: the aspect-ratio dependence. Phys. Rev. A 45, 842–845.CrossRefGoogle ScholarPubMed
48. Xin, Y.-B., Xia, K.-Q. & Tong, P. 1996 Measured velocity boundary layers in turbulent convection. Phys. Rev. Lett. 77, 1266–1269.CrossRefGoogle ScholarPubMed
49. Zhou, Q., Stevens, R. J. A. M., Sugiyama, K., Grossmann, S., Lohse, D. & Xia, K.-Q. 2010 Prandtl–Blausius temperature and velocity boundary-layer profiles in turbulent Rayleigh–Bénard convection. J. Fluid Mech. 664, 297–312.CrossRefGoogle Scholar
50. Zhou, Q., Sugiyama, K., Stevens, R. J. A. M., Grossmann, S., Lohse, D. & Xia, K.-Q. 2011 Horizontal structures of velocity and temperature boundary-layer profiles in 2D numerical turbulent Rayleigh–Bénard convection. Phys. Fluids 23, 125104.CrossRefGoogle Scholar
51. 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
52. Zhou, Q. & Xia, K.-Q. 2010 Measured instantaneous viscous boundary layer in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 104, 104301.CrossRefGoogle ScholarPubMed