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, $\delta _u$ and $\delta _T$ , 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 $Nu \sim Ra^{1/3}$ in the high $Ra$ limit, independent of the Prandtl number, $\delta _T/L \sim Ra^{-1/3}$ and $\delta _u/L \sim Ra^{-1/3} Pr^{1/2}$ , where $L$ 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 $Pr \sim 1$ , we make a more general ansatz, $\delta _u \sim \nu ^{\alpha }$ , where $\nu$ is the kinematic viscosity and assume that the dissipation scales as $\sim u^3/L$ , where $u$ is a characteristic turbulent velocity. Under these assumptions we show that $Nu \sim Ra^{\alpha /(3-\alpha )}$ , implying that $Nu \sim Ra^{1/5}$ if $\delta _u$ were scaling as in a Blasius boundary layer and $Nu \sim Ra^{1/2}$ (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 $\alpha = 3/4$ in the limit of high $Ra$ .