Evaporating Rayleigh-B\'enard convection: prediction of interface temperature and global heat transfer modulation

We propose an analytical model to estimate the interface temperature $\Theta_{\Gamma}$ and the Nusselt number $Nu$ for an evaporating two-layer Rayleigh-B\'enard configuration in statistically stationary conditions. The model is based on three assumptions: (i) the Oberbeck-Boussinesq approximation can be applied to the liquid phase, while the gas thermophysical properties are generic functions of thermodynamic pressure, local temperature, and vapour composition, (ii) the Grossmann-Lohse theory for thermal convection can be applied to the liquid and gas layers separately, (iii) the vapour content in the gas can be taken as the mean value at the gas-liquid interface. We validate this setting using direct numerical simulations (DNS) in a parameter space composed of the Rayleigh number ($10^6\leq Ra\leq 10^8$) and the temperature differential ($0.05\leq\varepsilon\leq 0.20$), which modulates the variation of state variables in the gas layer. To better disentangle the variable property effects on $\Theta_\Gamma$ and $Nu$, simulations are performed in two conditions. First, we consider the case of uniform gas properties except for the gas density and gas-liquid diffusion coefficient. Second, we include the variation of specific heat capacity, dynamic viscosity, and thermal conductivity using realistic equations of state. Irrespective of the employed setting, the proposed model agrees very well with the numerical simulations over the entire range of $Ra-\varepsilon$ investigated.