Non-Oberbeck–Boussinesq effects in strongly turbulent Rayleigh–Bénard convection

Non-Oberbeck–Boussinesq (NOB) effects on the Nusselt number $Nu$ and Reynolds number $\hbox{\it Re}$ in strongly turbulent Rayleigh–Bénard (RB) convection in liquids were investigated both experimentally and theoretically. In the experiments the heat current, the temperature difference, and the temperature at the horizontal midplane were measured. Three cells of different heights $L$, all filled with water and all with aspect ratio $\Gamma$ close to 1, were used. For each $L$, about 1.5 decades in $Ra$ were covered, together spanning the range $10^8 \,{\le}\, Ra \,{\le}\, 10^{11}$. For the largest temperature difference between the bottom and top plates, $\Delta \,{=}\, 40$K, the kinematic viscosity and the thermal expansion coefficient, owing to their temperature dependence, varied by more than a factor of 2. The Oberbeck–Boussinesq (OB) approximation of temperature-independent material parameters thus was no longer valid. The ratio $\chi$ of the temperature drops across the bottom and top thermal boundary layers became as small as $\chi\,{=}\,0.83$, which may be compared with the ratio $\chi \,{=}\, 1$ in the OB case. Nevertheless, the Nusselt number $Nu$ was found to be only slightly smaller (by at most 1.4%) than in the next larger cell with the same Rayleigh number, where the material parameters were still nearly height independent. The Reynolds numbers in the OB and NOB case agreed with each other within the experimental resolution of about 2%, showing that NOB effects for this parameter were small as well. Thus $Nu$ and $\hbox{\it Re}$ are rather insensitive against even significant deviations from OB conditions. Theoretically, we first account for the robustness of $Nu$ with respect to NOB corrections: the NOB effects in the top boundary layer cancel those which arise in the bottom boundary layer as long as they are linear in the temperature difference $\Delta$. The net effects on $Nu$ are proportional to $ \Delta^2$ and thus increase only slowly and still remain minor despite drastic material-parameter changes. We then extend the Prandtl–Blasius boundary-layer theory to NOB Rayleigh–Bénard flow with temperature-dependent viscosity and thermal diffusivity. This allows calculation of the shift in the bulk temperature, the temperature drops across the boundary layers, and the ratio $\chi$ without the introduction of any fitting parameter. The calculated quantities are in very good agreement with experiment. When in addition we use the experimental finding that for water the sum of the top and bottom thermal boundary-layer widths (based on the slopes of the temperature profiles at the plates) remains unchanged under NOB effects within the experimental resolution, the theory also gives the measured small Nusselt-number reduction for the NOB case. In addition, it predicts an increase by about 0.5% of the Reynolds number, which is also consistent with the experimental data. By studying theoretically hypothetical liquids for which only one of the material parameters is temperature dependent, we are able to shed further light on the origin of NOB corrections in water: while the NOB deviation of $\chi$ from its OB value $\chi \,{=}\, 1$ mainly originates from the temperature dependence of the viscosity, the NOB correction of the Nusselt number primarily originates from the temperature dependence of the thermal diffusivity. Finally, we give predictions from our theory for the NOB corrections if glycerol were used as the operating liquid.\ud \u