What Rayleigh numbers are achievable under Oberbeck-Boussinesq conditions?

Abstract

The validity of the Oberbeck-Boussinesq (OB) approximation in Rayleigh-Benard (RB) convection is studied using the Gray & Giorgini (1976) criterion that requires that the residuals, i.e., the terms that distinguish the full governing equations from their OB approximations, are kept below a certain small threshold $\sigma$. This gives constraints on the temperature- and pressure-variations of the fluid properties (density, absolute viscosity, specific heat at constant pressure cp, thermal expansion coefficient, and thermal conductivity) and on the magnitudes of the pressure work and viscous dissipation terms in the heat equation, which all can be formulated as bounds regarding the maximum temperature difference in the system, $\Delta$, and the container height, L. Thus for any given fluid and $\sigma$ , one can calculate the OB-validity region (in terms of $\Delta$ and L) and also the maximum achievable Rayleigh number $Ra_{max}$, $\sigma$, and we did so for fluids water, air, helium and pressurized SF6 at room temperature, and cryogenic helium, for $\sigma$ = 5%, 10% and 20%. For the most popular fluids in high-Ra RB measurements, which are cryogenic helium and pressurized SF6, we have identified the most critical residual, which is associated with the temperature dependence of $c_p$. Our direct numerical simulations (DNS) showed, however, that even when the values of $c_p$ can differ almost twice within the convection cell, this feature alone cannot explain a sudden and strong enhancement in the heat transport in the system, compared to its OB analog.