Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-14T22:47:22.688Z Has data issue: false hasContentIssue false
  • English
  • Français

Prandtl–Blasius temperature and velocity boundary-layer profiles in turbulent Rayleigh–Bénard convection

Published online by Cambridge University Press:  08 September 2010

QUAN ZHOU ,
RICHARD J. A. M. STEVENS ,
KAZUYASU SUGIYAMA ,
SIEGFRIED GROSSMANN ,
DETLEF LOHSE  and
KE-QING XIA
QUAN ZHOU
Affiliation:
Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China
RICHARD J. A. M. STEVENS
Affiliation:
Physics of Fluids Group, Department of Science and Technology, J. M. Burgers Centre for Fluid Dynamics, and Impact-Institute, University of Twente, 7500 AE Enschede, The Netherlands
KAZUYASU SUGIYAMA
Affiliation:
Physics of Fluids Group, Department of Science and Technology, J. M. Burgers Centre for Fluid Dynamics, and Impact-Institute, University of Twente, 7500 AE Enschede, The Netherlands Department of Mechanical Engineering, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bukyo-ku, Tokyo 113-8756, Japan
SIEGFRIED GROSSMANN
Affiliation:
Fachbereich Physik, Philipps-Universität Marburg, D-35032 Marburg, Germany
DETLEF LOHSE
Affiliation:
Physics of Fluids Group, Department of Science and Technology, J. M. Burgers Centre for Fluid Dynamics, and Impact-Institute, University of Twente, 7500 AE Enschede, The Netherlands
KE-QING XIA*
Affiliation:
Department of Physics, The Chinese University of Hong Kong, Shatin, Hong Kong, China
*
†Email address for correspondence: kxia@phy.cuhk.edu.hk
Get access Rights & Permissions [Opens in a new window]

Abstract

The shapes of the velocity and temperature profiles near the horizontal conducting plates' centre regions in turbulent Rayleigh–Bénard convection are studied numerically and experimentally over the Rayleigh number range 108 ≲ Ra ≲ 3 × 1011 and the Prandtl number range 0.7 ≲ Pr ≲ 5.4. The results show that both the temperature and velocity profiles agree well with the classical Prandtl–Blasius (PB) laminar boundary-layer profiles, if they are re-sampled in the respective dynamical reference frames that fluctuate with the instantaneous thermal and velocity boundary-layer thicknesses. The study further shows that the PB boundary layer in turbulent thermal convection not only holds in a time-averaged sense, but is most of the time also valid in an instantaneous sense.

JFM classification

Boundary Layers: Boundary Layers Convection: Convection
Type
Papers
Information
Journal of Fluid Mechanics , Volume 664 , 10 December 2010 , pp. 297 - 312
Copyright
Copyright © Cambridge University Press 2010

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., Brown, E., Fontenele Araujo, F., Funfschilling, D., Grossmann, S. & Lohse, D. 2006 Non-Oberbeck–Boussinesq effects in strongly turbulent Rayleigh–Bénard convection. J. Fluid Mech. 569, 409–445.CrossRefGoogle Scholar
Ahlers, G., Calzavarini, E., Fontenele Araujo, F., Funfschilling, D., Grossmann, S., Lohse, D. & Sugiyama, K. 2008 Non-Oberbeck–Boussinesq effects in turbulent thermal convection in ethane close to the critical point. Phys. Rev. E 77, 046302.Google Scholar
Ahlers, G., Fontenele Araujo, F., Funfschilling, D., Grossmann, S. & Lohse, D. 2007 Non-Oberbeck–Boussinesq effects in gaseous Rayleigh–Bénard convection. Phys. Rev. Lett. 98, 054501.CrossRefGoogle ScholarPubMed
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
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
Ching, E. S. C. 1997 Heat flux and shear rate in turbulent convection. Phys. Rev. E 55, 1189–1192.Google Scholar
Dubrulle, B. 2001 Logarithmic corrections to scaling in turbulent thermal convection. Eur. Phys. J. B 21, 295–304.Google Scholar
Dubrulle, B. 2002 Scaling in large Prandtl number turbulent thermal convection. Eur. Phys. J. B 28, 361–367.CrossRefGoogle Scholar
Grossmann, S. & Lohse, D. 2000 Scaling in thermal convection: a unifying theory. J. Fluid Mech. 407, 27–56.CrossRefGoogle Scholar
Grossmann, S. & Lohse, D. 2001 Thermal convection for large Prandtl numbers. Phys. Rev. Lett. 86, 3316–3319.CrossRefGoogle ScholarPubMed
Grossmann, S. & Lohse, D. 2002 Prandtl and Rayleigh number dependence of the Reynolds number in turbulent thermal convection. Phys. Rev. E 66, 016305.Google ScholarPubMed
Grossmann, S. & Lohse, D. 2004 Fluctuations in turbulent Rayleigh–Bénard convection: the role of plumes. Phys. Fluids 16, 4462–4472.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
Malkus, M. V. R. 1954 The heat transport and spectrum of thermal turbulence. Proc. R. Soc. Lond. A 225, 196–212.Google Scholar
du Puits, R., Resagk, C. & Thess, A. 2007 Mean velocity profile in confined turbulent convection. Phys. Rev. Lett. 99, 234504.CrossRefGoogle ScholarPubMed
Qiu, X.-L. & Xia, K.-Q. 1998 Viscous boundary layers at the sidewall of a convection cell. Phys. Rev. E 58, 486–491.Google Scholar
Schlichting, H. & Gersten, K. 2004 Boundary Layer Theory, 8th edn. Springer.Google Scholar
Shishkina, O. & Thess, A. 2009 Mean temperature profiles in turbulent Rayleigh–Bénard convection of water. J. Fluid Mech. 633, 449–460.CrossRefGoogle Scholar
Shraiman, B. I. & Siggia, E. D. 1990 Heat transport in high-Rayleigh-number convection. Phys. Rev. A 42, 3650–3653.CrossRefGoogle ScholarPubMed
Siggia, E. D. 1994 High-Rayleigh-number convection. Annu. Rev. Fluid Mech. 26, 137–168.CrossRefGoogle Scholar
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, 493–507.CrossRefGoogle Scholar
Sugiyama, K., Calzavarini, E., Grossmann, S. & Lohse, D. 2009 Flow organization in non-Oberbeck–Boussinesq Rayleigh–Bénard convection in water. J. Fluid Mech. 637, 105–135.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
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
Villermaux, E. 1995 Memory-induced low frequency oscillations in closed convection boxes. Phys. Rev. Lett. 75, 4618–4621.CrossRefGoogle ScholarPubMed
Xi, H.-D., Zhou, S.-Q., Zhou, Q., Chan, T.-S. & Xia, K.-Q. 2009 Origin of the temperature oscillation in turbulent thermal convection. Phys. Rev. Lett. 102, 044503.CrossRefGoogle ScholarPubMed
Xia, K.-Q., Sun, C. & Zhou, S.-Q. 2003 Particle image velocimetry measurements of the velocity field in turbulent thermal convection. Phys. Rev. E 68, 066303.Google ScholarPubMed
Zhou, Q., Xi, H.-D., Zhou, S.-Q., Sun, C. & Xia, K.-Q. 2009 Oscillations of the large-scale circulation in turbulent Rayleigh–Bénard convection: the sloshing mode and its relationship with the torsional mode. J. Fluid Mech. 630, 367–390.CrossRefGoogle Scholar
Zhou, Q. & Xia, K.-Q. 2010 Measured instantaneous viscous boundary layer in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 104, 104301.CrossRefGoogle ScholarPubMed