Non-Oberbeck-Boussinesq effects in turbulent thermal convection in ethane close to the critical point

As shown in earlier work [Ahlers et al., J. Fluid Mech. 569, 409 (2006)], non-Oberbeck-Boussinesq (NOB) corrections to the center temperature in turbulent Rayleigh-Bénard convection in water and also in glycerol are governed by the temperature dependences of the kinematic viscosity and the thermal diffusion coefficient. If the working fluid is ethane close to the critical point, the origin of non-Oberbeck-Boussinesq corrections is very different, as will be shown in the present paper. Namely, the main origin of NOB corrections then lies in the strong temperature dependence of the isobaric thermal expansion coefficient $\beta \left(T\right)$. More precisely, it is the nonlinear $T$ dependence of the density $\rho \left(T\right)$ in the buoyancy force that causes another type of NOB effect. We demonstrate this through a combination of experimental, numerical, and theoretical work, the last in the framework of the extended Prandtl-Blasius boundary-layer theory developed by Ahlers et al. as cited above. The theory comes to its limits if the temperature dependence of the thermal expension coefficient $\beta \left(T\right)$ is significant. The measurements reported here cover the ranges $2.1\lesssim \text{Pr}\lesssim 3.9$ and $5\times {10}^9\lesssim \text{Ra}\lesssim 2\times {10}^{12}$ and are for cylindrical samples of aspect ratios 1.0 and 0.5.

Figure 1

Pressure-temperature plane in reduced units. Star: critical point of ethane ($T_{\ast }=32.18°\text{C}$, $P_{\ast }=48.718\text{bars}$). Solid line: liquid-vapor coexistence curve. Dotted line: critical isochore. The horizontal arrows show the maximum temperature intervals $\Delta$. From bottom to top they correspond to (i) $P_m=49.98\text{bars}=1.026P_{\ast }$, centered at $T_m=40.00°\text{C}=1.025T_{\ast }$, (ii) $P_m=51.72\text{bars}=1.062P_{\ast }$, centered at $T_m=40.00°\text{C}$ (right of critical isochore) and at $T_m=27.00°\text{C}=0.983T_{\ast }$ (left of critical isochore), and (iii) $P_m=53.78\text{bars}=1.104P_{\ast }$, centered at $T_m=40.00°\text{C}$, and (iv) $P_m=55.17\text{bar}=1.132P_{\ast }$, centered at $T_m=41.00°\text{C}=1.029T_{\ast }$ (right of critical isochore) and at $T_m=35.00°\text{C}=1.009T_{\ast }$ (left of critical isochore).

Figure 2

(a) Temperature and (b) density profiles at $P_m=0.849P_{\ast }$, $T_m=40°\text{C}$, and $\Delta =15\text{K}$. The thermal slope thicknesses at the bottom or top ${\lambda }_{b,t}^{\text{sl}}/L=a_{b,t}/\sqrt{{\nu }_m/L{U}_c}$ are seen to have prefactors of about $a_b\approx 2.8$ and $a_t\approx 2.5$, i.e., ${\lambda }_b^{\text{sl}}>{\lambda }_t^{\text{sl}}$.

Figure 3

Deviation $T_c-T_m$ as function of $\Delta$, for $T_m=\left(\text{a}\right)40$ and (b) $27°\text{C}$.

Figure 4

(Color online) Schematic diagram (approximately to scale for $\Gamma =1$) of the high-pressure sample cell surrounded by the can containing ambient air and foam.

Figure 5

Nusselt number Nu as a function of the Rayleigh number Ra for $\Gamma =1.00$. Solid circles: $P=51.72\text{bars}$ and $T_m=40°\text{C}$ $\left(\text{Pr}=2.58\right)$. Open circles: $P=51.72\text{bars}$ and $T_m=27°\text{C}$ $\left(\text{Pr}=2.99\right)$. Solid squares: $P=51.72\text{bars}$ and $T_m=24°\text{C}$ $\left(\text{Pr}=2.71\right)$. Open squares: $P=51.72\text{bars}$ and $T_m=31°\text{C}$ $\left(\text{Pr}=3.85\right)$. Solid diamonds: $P=53.79\text{bars}$ and $T_m=40°\text{C}$ $\left(\text{Pr}=3.79\right)$. Open diamonds: $P=50.00\text{bars}$ and $T_m=40°\text{C}$ $\left(\text{Pr}=2.09\right)$. Stars from Ref. [6] for $\Gamma =0.5$ after correction for sidewall effects. Pluses from Ref. [7] for $\Gamma =0.5$.

Figure 6

(a) Ratio of the measured Nusselt number Nu to the estimate ${\text{Nu}}_{\text{OB}}$ of the Nusselt number under Boussinesq conditions as a function of the applied temperature difference $\Delta$. (b) Deviation of the center temperature $T_c$ from the mean temperature $T_m$ as a function of $\Delta$. All measurements were made at $T_m=40.00°\text{C}$ and a pressure of 51.72 bars $\left(P/P_{\ast }=1.062\right)$ where the Prandtl number is 2.58. Open symbols, $\Gamma =0.50$. Solid symbols, data from Ref. [46] with $\Gamma =1.00$.

Figure 7

Ratios $X/X_m$ for $P=51.72\text{bars}$ $\left(P/P_{\ast }=1.062\right)$ of several property values $X$ at temperatures $T-T_m$ to the value of $X$ at $T_m$ (based on Ref. [47]). (a),(c) $T_m=27.00°\text{C}$. (b),(d) $T_m=40.00°\text{C}$. (a),(b) Thermal conductivity $\Lambda$ (short-dashed line), density $\rho$ (dotted line), and dynamic viscosity $\eta$ (dash-dotted line). (c),(d) Thermal expansion coefficient $\beta$ (solid line), $\widehat{\beta }$ (long-dashed line), and heat capacity at constant pressure $c_P$ (double-dash-dotted line). The reference values $X_m$ for $27\left(40\right)°\text{C}$ are ${\Lambda }_m=0.07328\left(0.04343\right)\text{W}{\text{m}}^{-1}{\text{K}}^{-1}$, ${\rho }_m=331.12\left(123.26\right)\text{kg}{\text{m}}^{-3}$, ${\eta }_m=4.030\left(1.502\right)\times {10}^{-5}\text{kg}{\text{s}}^{-1}{\text{m}}^{-1}$, ${\beta }_m=0.01649\left(0.03815\right){\text{K}}^{-1}$, $c_{P,m}=5434\left(7452\right)\text{J}{\text{kg}}^{-1}{\text{K}}^{-1}$, and mean Prandtl number ${\text{Pr}}_m=2.99\left(2.58\right)$.

Figure 8

(a) Ratio of the measured Nusselt number Nu to the estimate ${\text{Nu}}_{\text{OB}}$ of the Nusselt number under Boussinesq conditions as a function of the applied temperature difference $\Delta$. (b) Deviation of the center temperature $T_c$ from the mean temperature $T_m$. These measurements were made for $\Gamma =1.00$ at a pressure of 51.72 bars $\left(P/P_{\ast }=1.062\right)$. Open symbols, $T_m=27.00°\text{C}$ $\left(\text{Pr}=2.99\right)$. Solid symbols, $T_m=40.00°\text{C}$ $\left(\text{Pr}=2.58\right)$. These two temperatures are on opposite sides of, but not equidistant from, the temperature $T_{\varphi }=34.97°\text{C}$ where the critical isochore is reached on this isobar.

Figure 9

Ratios $X/X_m$ for $P=55.17\text{bars}$ $\left(P/P_{\ast }=1.132\right)$ of several property values $X$ at temperatures $T-T_m$ to the value of $X$ at $T_m$ (based on Ref. [47]). (a),(c) $T_m=35.00°\text{C}$. (b),(d) $T_m=41.00°\text{C}$. (a),(b) Thermal conductivity $\Lambda$ (dashed line), density $\rho$ (dotted line), and dynamic viscosity $\eta$ (dash-dotted line). (c),(d) Thermal expansion coefficient $\beta$ (solid line), $\widehat{\beta }$ (long-dashed line), and heat capacity at constant pressure $c_P$ (double-dash-dotted line). The reference values $X_m$ for $35\left(41\right)°\text{C}$ are ${\Lambda }_m=0.06674\left(0.05163\right)\text{W}{\text{m}}^{-1}{\text{K}}^{-1}$, ${\rho }_m=282.48\left(153.43\right)\text{kg}{\text{m}}^{-3}{\text{K}}^{-1}$, ${\eta }_m=3.168\left(1.726\right)\times {10}^{-5}\text{kg}{\text{s}}^{-1}{\text{m}}^{-1}$, ${\beta }_m=0.04177\left(0.06876\right){\text{K}}^{-1}$, $c_{P,m}=9617\left(12534\right)\text{J}{\text{kg}}^{-1}{\text{K}}^{-1}$, and mean Prandtl number ${\text{Pr}}_m=4.56\left(4.20\right)$.

Figure 10

(a) Ratio of the measured Nusselt number Nu to the estimate ${\text{Nu}}_{\text{OB}}$ of the Nusselt number under Boussinesq conditions as a function of the applied temperature difference $\Delta$. (b) Deviation of the center temperature $T_c$ from the mean temperature $T_m$. These measurements were made for $\Gamma =0.50$ at a pressure of 55.17 bars $\left(P/P_{\ast }=1.132\right)$. Open symbols: the mean temperature $T_m=35°\text{C}$ $\left(\text{Pr}=4.56\right)$. Solid symbols: $T_m=41°\text{C}$ $\left(\text{Pr}=4.20\right)$. These two temperatures are on opposite sides of and nearly equidistant from the temperature $T_{\varphi }=38.06°\text{C}$ where the critical isochore is reached on this isobar.

Figure 11

(Color online) Deviation $T_c-T_m$ as function of $\Delta$, for $T_m=27°\text{C}$ and $P/P_{\ast }=1.062$. The symbols $(\times )$ correspond to experimental measurements. The dotted line is obtained from boundary-layer theory. The red symbols ($○$, $△$, and $\square$) with error bars correspond to the incompressible DNS results described in Sec. 5, measured at different $\text{Ra}={10}^6-{10}^8$. Note that though these Rayleigh numbers are smaller than in the experiments $\left[\text{Ra}=O\left({10}^9\right)-O\left({10}^{10}\right)\right]$, the comparison is still appropriate because the $T_c$ shift has proven to be rather independent of Ra for given $\Delta$, provided one is beyond the onset of the chaotic motion at $\text{Ra}\approx 2\times {10}^5$ [45, 60]. For further evidence for the weak Ra dependence of the center temperature we also refer to Table .

Figure 12

(Color online) Temperature dependences of the material properties $\nu$, $\kappa$, and $\rho$ in the relevant $T$ range. $d\rho /dT$ is also displayed. (a) Ethane around the temperature $T_m=27°\text{C}$ adapted from [47]. The pressure is $P/P_{\ast }=1.062$. (b),(c) Fuid properties for water (b) and glycerol (c), respectively, around the temperature $T_m=40°\text{C}$ (which has been studied in [45] and [60]). Note the significant variation of the density $\rho$ with $T$ in the case of ethane, as compared to the two liquids. In ethane, $\rho /{\rho }_m-1$ varies from about 0.07 for $T-T_m=-5\text{K}$ to about $-0.1$ for $T-T_m=+5\text{K}$, whereas in glycerol ${\rho }_m$ is practically independent of $T$. The strong variation of $\rho$ for ethane follows from the large $d\rho /dT$ (also shown).

Figure 13

(Color online) Normalized center temperature shift $\left(T_c-T_m\right)/\Delta$ versus the temperature difference $\Delta$ for several hypothetical liquids. We consider the following six hypothetical liquids: $\mathbf{\left(}\widehat{\beta }\left(T\right),\kappa \left(T\right),\nu \left(T\right)\mathbf{\right)}$, $\mathbf{\left(}{\beta }_m,\kappa \left(T\right),\nu \left(T\right)\mathbf{\right)}$, $\mathbf{\left(}\widehat{\beta }\left(T\right),\kappa \left(T\right),{\nu }_m\mathbf{\right)}$, $\mathbf{\left(}{\beta }_m,\kappa \left(T\right),{\nu }_m\mathbf{\right)}$, $\mathbf{\left(}\widehat{\beta }\left(T\right),{\kappa }_m,\nu \left(T\right)\mathbf{\right)}$, and $\mathbf{\left(}{\beta }_m,{\kappa }_m,\nu \left(T\right)\mathbf{\right)}$. In particular, in each panel we compare two cases, which differ only by their buoyancy’s $T$ dependence, i.e., $\widehat{\beta }\left(T\right)$ instead of ${\beta }_m$. The symbols indicate the simulation results at Rayleigh number $\text{Ra}={10}^6$. The circles $(○)$ correspond to the cases in which the full temperature dependence of the buoyancy $\left(T-T_m\right)\widehat{\beta }\left(T\right)$ is taken into account, while the triangles $(△)$ represent the cases where only the linear temperature dependence $\left(T-T_m\right){\beta }_m$ is considered. The solid line shows the prediction of boundary-layer theory with an incompressible flow assumption [43]. The dashed line stems from the solution with no convective flow ${\partial }_z\left[\kappa \left(T\right){\partial }_{z}T\right]=0$ with boundary conditions $T=T_b$ at $z=0$ and $T=T_t$ at $z=L$. See also Table .

Figure 14

Schematic plot of the mirror transformation $\widehat{\beta }\left(T\right)\to \widehat{\beta }\left(2T_m-T\right)$ of the thermal expansion coefficient $\widehat{\beta }$.

Figure 15

(Color online) Effect of the temperature dependence of the thermal expansion coefficient $\widehat{\beta }$ on the shift of the center temperature for several hypothetical fluids. DNS results for the normalized temperature difference $\left({\phantom{|}{T}_c|}_{\widehat{\beta }\left(T\right)}-{\phantom{|}{T}_c|}_{{\beta }_m}\right)/\Delta$ [or $-\left({\phantom{|}{T}_c|}_{\widehat{\beta }\left(2T_m-T\right)}-{\phantom{|}{T}_c|}_{{\beta }_m}\right)/\Delta$] with the same temperature dependence of the thermal diffusivity $\kappa$ and the kinematic viscosity $\nu$ are plotted; here ${\phantom{|}{T}_c|}_{\widehat{\beta }\left(T\right)}$ denotes the center temperature with the full temperature dependence of the buoyancy $g\left(1-\rho /{\rho }_m\right)$ as given in Table , and ${\phantom{|}{T}_c|}_{{\beta }_m}$ denotes that with the linear temperature dependence $g\left(1-\rho \left(T\right)/{\rho }_m\right)=C_1\left(T-T_m\right)$ only. The top panel shows $\left({\phantom{|}{T}_c|}_{\widehat{\beta }\left(T\right)}-{\phantom{|}{T}_c|}_{{\beta }_m}\right)/\Delta$ versus $\Delta$ at fixed Rayleigh number $\text{Ra}={10}^6$. The solid line shows the linear fit $1.473\times {10}^{-3}{\text{K}}^{-1}\times \Delta$ for the case $\widehat{\beta }\left(T\right),{\kappa }_m,{\nu }_m$ (see Table ). The middle panel shows $\left({\phantom{|}{T}_c|}_{\widehat{\beta }\left(T\right)}-{\phantom{|}{T}_c|}_{{\beta }_m}\right)/\Delta$ versus Ra at fixed temperature difference $\Delta =10\text{K}$. The bottom panel has the same parameters as the middle one, except it shows $-\left({\phantom{|}{T}_c|}_{\widehat{\beta }\left(2T_m-T\right)}-{\phantom{|}{T}_c|}_{{\beta }_m}\right)/\Delta$ for the case of mirror-transformed $\widehat{\beta }$.