Scaling in Rayleigh-Bénard convection

Abstract

We consider the Nusselt-Rayleigh number problem of Rayleigh-Bénard convection and make the hypothesis that the velocity and thermal boundary layer widths, and, in the absence of a strong mean flow are controlled by the dissipation scales of the turbulence outside the boundary layers and, therefore, are of the order of the Kolmogorov and Batchelor scales, respectively. Under this assumption, we derive in the high limit, independent of the Prandtl number, and, where is the height of the convection cell. The scaling relations are valid as long as the Prandtl number is not too far from unity. For, we make a more general ansatz, where is the kinematic viscosity and assume that the dissipation scales as, where is a characteristic turbulent velocity. Under these assumptions we show that, implying that if were scaling as in a Blasius boundary layer and (with some logarithmic correction) if it were scaling as in a standard turbulent shear boundary layer. It is argued that the boundary layers will retain the intermediate scaling in the limit of high.