A New Parameterization of $Nu$-$Ra$ Relation in Turbulent Rayleigh-B\'{e}nard Convection*

Nusselt-Rayleigh relation is a key subject in the study of turbulent Rayleigh-B\'{e}nard convection (RBC). She et al. introduced Structural Ensemble Dynamics(SED) theory to study wall-bounded turbulence, which yields a multi-layer model of velocity and temperature profiles for RBC system. Here, we report a result of this study, i.e. a new parameterization of Nusselt number(Nu) as a function of Rayleigh number(Ra): $Nu=\alpha Ra^{1/7} \mathrm{exp}\left(\gamma Ra^\beta\right)$. The parameters ($\alpha$, $\beta$ and $\gamma$) are supposed to be slowly varying with Ra and other physical parameters, in particular Prandtl number(Pr). Analysis of a set of experimental data with $Ra=10^8\sim10^{12}$ and $Pr=0.7\sim7.0$ shows that this parameterization is efficient, yielding an accurate description of Nu-Ra with errors bounded within $1\%$. This parameterization surprisingly reveals two distinct states as $\alpha$ varies, with transition at $\alpha=1$. Then, an analytic model linking the variation of the three parameters is proposed, yielding a uniform description for the enormous empirical Nu-Ra data, significantly more accurate than the well-known Grossmann-Lohse (GL) model. In conclusion, the SED theory emphasizing the internal profiles provides a viable description of the RBC system.