Phenomenological Nusselt-Rayleigh scaling of turbulent thermal convection

Natural convection between the hot floor and the cool ceiling, so called Rayleigh-Benard convection, is pervasive and of both fundamental and industrial interests. One key issue is how heat transfer varies with increasing thermal potential, or equivalently how the Nusselt number (Nu) scales with the Rayleigh number (Ra). The overview of experimental findings remains to show the need of extra explanation complemental to the current theories. Here we present a model based on the phenomenological theory of turbulence, where the power-law spectral exponent of the energy spectrum is the only input parameter required. The goal aims to elucidate the unexplained aspect in the Nu-Ra scaling. We find that Kolmogorov turbulence in the current model leads to Nu similar to Ra-0.3, in good agreement with the modern experimental results. We hope that this model could stimulate the discussion as to the effects of the spectral phenomena on the Nu-Ra scaling, and thus augment our understanding of buoyancy-driven thermal convection