Journal of Fluid Mechanics

Scaling of hard thermal turbulence in Rayleigh-Bénard convection

Bernard  Castaing a1, Gemunu  Gunaratne a2, François  Heslot a3, Leo  Kadanoff a2, Albert  Libchaber a2, Stefan  Thomae a4, Xiao-Zhong  Wu a2, Stéphane  Zaleski a5 and Gianluigi  Zanetti a2
a1 CNRS-CRTBT, 25 Avenue des Martyrs-P.B. 166X, 38042 Grenoble Cedex, France
a2 The Research Institutes, The University of Chicago, 5640 S. Ellis Avenue, Chicago, IL 60637, USA
a3 Collège de France, Matiere Condensée, Place Marcellin Berthelot, 75005 Paris, France
a4 Institut für Festkörperforschung der KFA, Postfach 1913, D-5170 Jülich, W. Germany
a5 Laboratoire de Physique Statistique, ENS, 24 rue Lhomond, 75231 Paris Cedex 05, France

Article author query
castaing b   [Google Scholar] 
gunaratne g   [Google Scholar] 
heslot f   [Google Scholar] 
kadanoff l   [Google Scholar] 
libchaber a   [Google Scholar] 
thomae s   [Google Scholar] 
wu xz   [Google Scholar] 
zaleski s   [Google Scholar] 
zanetti g   [Google Scholar] 


An experimental study of Rayleigh-Bénard convection in helium gas at roughly 5 K is performed in a cell with aspect ratio 1. Data are analysed in a ‘hard turbulence’ region (4 × 107 < Ra < 6 × 1012) in which the Prandtl number remains between 0.65 and 1.5. The main observation is a simple scaling behaviour over this entire range of Ra. However the results are not the same as in previous theories. For example, a classical result gives the dimensionless heat flux, Nu, proportional to $Ra^{\frac{1}{3}}$ while experiment gives an index much closer to $\frac{2}{7}$. A new scaling theory is described. This new approach suggests scaling indices very close to the observed ones. The new approach is based upon the assumption that the boundary layer remains in existence even though its Rayleigh number is considerably greater than unity and is, in fact, diverging. A stability analysis of the boundary layer is performed which indicates that the boundary layer may be stabilized by the interaction of buoyancy driven effects and a fluctuating wind.

(Published Online April 26 2006)
(Received February 24 1988)
(Revised August 30 1988)