Non-Oberbeck-Boussinesq Effects in Gaseous Rayleigh-Bénard Convection

Non-Oberbeck-Boussinesq (NOB) effects are measured experimentally and calculated theoretically for strongly turbulent Rayleigh-Bénard convection of ethane gas under pressure where the material properties strongly depend on the temperature. Relative to the Oberbeck-Boussinesq case we find a decrease of the central temperature as compared to the arithmetic mean of the top- and bottom-plate temperature and an increase of the Nusselt number. Both effects are of opposite sign and greater magnitude than those for NOB convection in liquids like water.

Figure 1

Pressure-temperature plane in reduced units. Star: critical point of ethane ($T_{*}=32.18°\mathrm{C}$, $P_{*}=48.718\mathrm{bars}$). Heavy line: liquid-vapor coexistence curve. Dotted line: critical isochore. The horizontal arrows show the maximum temperature intervals $\Delta$, centered about $T_m=39.98°\mathrm{C}$, that were used at each of the pressures $P_m/P_{*}=0.849$, 0.991, 1.026, 1.062, and 1.104 (bottom to top).

Figure 2

(a) Deviation $T_c-T_m$ of the center temperature $T_c$ from $T_m$ as function of $\Delta$ and (b) resulting ratio $\chi ={\Delta }_b/{\Delta }_t$ of the temperature drops across the bottom and the top boundary layers. Symbols: experiment. Lines: theory. As in Fig. 1, the pressure $P_m$ ranges from $0.849P_{*}$ to $1.104P_{*}$.

Figure 3

(a) Relative deviations $(X-X_m)/X_m$ of ethane properties from their values $X_m$ at $T_m=40°\mathrm{C}$ and $P_m=0.849P_{*}$. The curves are based on Ref. 17. (b)  Predicted $T_c-T_m$ for hypothetical gases, with one of the material parameters varying as function of temperature and all others fixed at their values $X_m$ at $T_m=40°\mathrm{C}$ and $P_m=0.849P_{*}$.

Figure 4

(a) ${\mathrm{Nu}}_{\mathrm{NOB}}$ and (b) ${\mathrm{Nu}}_{\mathrm{NOB}}/{\mathrm{Nu}}_{\mathrm{OB}}$ as function of Ra for $P/P_{*}=1.104$ (yellow triangles), 1.062 (blue diamonds), and 0.991 (purple squares). (c) ${\mathrm{Nu}}_{\mathrm{NOB}}/{\mathrm{Nu}}_{\mathrm{OB}}$ as a function of $\beta \Delta$. Dotted lines: ${\mathrm{Nu}}_{\mathrm{OB}}=0.1826{\mathrm{Ra}}^{0.292}$. Solid lines: ${\mathrm{Nu}}_{\mathrm{NOB}}/{\mathrm{Nu}}_{\mathrm{OB}}=1+0.125(\beta \Delta )+1.012(\beta \Delta {)}^2+1.150(\beta \Delta {)}^3$.