Heat transfer and large scale dynamics in turbulent Rayleigh-Bénard convection

The progress in our understanding of several aspects of turbulent Rayleigh-Bénard convection is reviewed. The focus is on the question of how the Nusselt number and the Reynolds number depend on the Rayleigh number Ra and the Prandtl number Pr, and on how the thicknesses of the thermal and the kinetic boundary layers scale with Ra and Pr. Non-Oberbeck-Boussinesq effects and the dynamics of the large scale convection roll are addressed as well. The review ends with a list of challenges for future research on the turbulent Rayleigh-Bénard system.

Figure 1

(Color online) Plumes and flow field. (a) Shadowgraph visualization of rising and falling plumes at $\mathrm{Ra}=6.8\times {10}^8$, $\mathrm{Pr}=596$ (dipropylene glycol) in a $\Gamma =1$ cell. From 179. (b) Streak picture of temperature sensitive liquid crystal spheres taken near the top cold surface in a $\Gamma =1$ cell at $\mathrm{Ra}=2.6\times {10}^9$ and $\mathrm{Pr}=5.4$ (water), in order to visualize plume detachment. The view shows an area of $6.5\mathrm{cm}\times 4\mathrm{cm}$. From 75. (c) Time-averaged velocity-vector map in the plane of the LSC at $\mathrm{Ra}=7.0\times {10}^9$. Adapted from 199.

Figure 2

(Color online) Boundary-bulk partition. Sketch of the splitting of the kinetic (a) and thermal (b) dissipation rates on which the GL theory is based. In both figures the large scale convection roll with typical velocity amplitude $U$ is sketched. The typical width of the kinetic BL is ${\lambda }_u$, whereas the typical thermal BL thicknesses and the plume thicknesses are ${\lambda }_{\theta }$. Outside the BL/plume region is the background flow (bg).

Figure 3

(Color online) Phase diagram in Ra-Pr plane. (a) Phase diagram in the Ra-Pr plane according to 98, 99, 100, 102: The upper solid line means $\mathrm{Re}={\mathrm{Re}}_c$; the lower nearly parallel solid line corresponds to ${ϵ}_{u,\mathrm{BL}}={ϵ}_{u,\text{bulk}}$; the curved solid line is ${ϵ}_{\theta ,\mathrm{BL}}={ϵ}_{\theta ,\text{bulk}}$; and along the long-dashed line ${\lambda }_u={\lambda }_{\theta }$, i.e., $2a\mathrm{Nu}=\sqrt{\mathrm{Re}}$. The dotted line indicates where the laminar kinetic BL is expected to become turbulent, based on a critical shear Reynolds number ${\mathrm{Re}}_s^{*}\approx 420$ of the kinetic BL; cf. 128. Data points where Nu has been measured or numerically calculated have been included (for aspect ratios $\Gamma \approx 1∕2–1$): squares, 46; diamonds, 64; circles, 139; stars, 9; stars, 90, 148; triangles down, 230; triangles down, 197; triangles right, 80; triangles up, 214 (numerical simulations); squares, 123 (numerical simulations); triangles up, 10, 216 (numerical simulations). Note that some of the large Ra data probably are influenced by NOB effects. (b) An enlargement of part (a).

Figure 4

Nusselt number versus Rayleigh number. Reduced Nu for $\Gamma =1$, obtained using water $(\mathrm{Pr}\approx 4.4)$ and copper plates, as a function of Ra. Open symbols, uncorrected data. Solid symbols, after correction for the finite plate conductivity. Circles, 90. Squares, 196. The downwards and upwards triangles are upper and lower bounds on the actual Nusselt number at large Ra; the diamonds originate from an estimate, see the text for details. The solid line is the GL prediction (99).

Figure 5

(Color online) Nusselt number versus Prandtl number. (a) Nu(Pr) for $\mathrm{Ra}=6\times {10}^5$ and $\Gamma =1$ from numerical simulations by 214 (circles), from experiments with mercury by 174 (diamond), and from the experiments with sodium by 114 (square). The straight solid line is a fit to the numerical data with $\mathrm{Pr}<1$ (214), giving $\mathrm{Nu}=8.1{\mathrm{Pr}}^{0.14\pm 0.02}$. The exponent is in agreement with the low-Pr expectation $1∕8$ of the GL theory. The upper triangles are the numerical data for $\mathrm{Ra}={10}^7$ by 123, the dark circle results from the experimental data of 64 for the same $\mathrm{Ra}={10}^7$, and the diamonds are numerical results for $\mathrm{Ra}={10}^6$ by 24. The three solid lines are the results from the GL theory equations (21, 22) for the three Rayleigh numbers of the numerical data sets, namely, $\mathrm{Ra}=6\times {10}^5$, $\mathrm{Ra}={10}^6$, and $\mathrm{Ra}={10}^7$, bottom to top. (b) The reduced Nusselt number $\mathrm{Nu}{\mathrm{Ra}}^{-1∕4}$ as a function of the Prandtl number for the two Rayleigh numbers $1.78\times {10}^9$ (upper set) and $1.78\times {10}^7$ (lower set) in the large-Pr regime. Open circles, 9. Solid symbols, 230. Various organic fluids were used. From 230.

Figure 6

(Color online) High-precision data comparison of Nusselt versus Rayleigh numbers. (a) $\mathrm{Nu}∕{\mathrm{Ra}}^{{\gamma }_{\mathrm{eff}}}$ with ${\gamma }_{\mathrm{eff}}=0.323$ as a function of Ra. Solid circles, $\Gamma =0.5$ (139) after a correction for plate and sidewall effects (143). Open circles, $\Gamma =0.5$ (48). Solid squares, $\Gamma =1$ (141). Solid diamonds, $\Gamma =0.67$ and 0.43 (148). Open squares, $1\lesssim \Gamma \lesssim 3$, $\mathrm{Pr}=0.7$ (85). Stars in circles, numerical results for $\Gamma =1∕2$, $\mathrm{Pr}=0.7$ (190) for constant-temperature boundary conditions at the plates. Solid (dotted) line, the GL prediction for $\mathrm{Pr}=0.8$ $(\mathrm{Pr}=29)$. (b) The Prandtl numbers corresponding to the data in (a). In addition, Pr for the measurements for $\Gamma =1$ with water in Fig. 4 are shown as diamonds. The dashed and dotted lines in the bottom figure are estimates of the location of the transition to the Kraichnan regime for $\Gamma =1$, assuming critical boundary layer Reynolds numbers ${\mathrm{Re}}_s^{*}=220$ and 440, respectively (since the parameters of the GL theory have been determined only for $\Gamma =1$, a prediction of ${\mathrm{Ra}}^{*}$ for smaller $\Gamma$ is not available).

Figure 7

Profile and scaling of velocity. (a) Normalized horizontal velocity $u∕u_{\max }$ as a function of the vertical position ${\rho }_z=z∕L$. The measurements were made at $\mathrm{Ra}=3.08\times {10}^7$ (circles), $9.14\times {10}^7$ (triangles), and $3.7\times {10}^9$ (diamonds). From 161. Note that the data extend only up to ${\rho }_z\simeq 0.85$; thus the minimum near the sample top is not shown. (b) Reduced Reynolds numbers ${\mathrm{Re}}^u∕{\mathrm{Ra}}^{1∕2}$ (circles) and ${\mathrm{Re}}^{\omega }∕{\mathrm{Ra}}^{1∕2}$ (solid triangles), defined in the text, as functions of Ra. The measurements were done in water at $\mathrm{Pr}\sim 5.5$. From 162.

Figure 8

Parameter dependences of Reynolds number. (a), (b) Reduced Reynolds numbers $\mathrm{Re}∕{\mathrm{Ra}}^{1∕2}$. (a) ${\mathrm{Re}}^{\mathrm{pl}}$ for $\mathrm{Pr}=4.38$ (circles), $\mathrm{Pr}=5.55$ (up-pointing triangles), and $\mathrm{Pr}=3.32$ (down-pointing triangles), from 32. Stars, ${\mathrm{Re}}^{\omega }$ for $\mathrm{Pr}=28.9$, from 32. Pluses, ${\mathrm{Re}}^u$, from 162, $\mathrm{Pr}\simeq 5.4$. Dashed lines (from top to bottom), GL predictions for $\mathrm{Pr}=3.32$, 4.38, 5.55, and 28.9. (b) ${\mathrm{Re}}^{\mathrm{pl}}$ (solid symbols) and ${\mathrm{Re}}^{\omega }$ (open symbols) for $\mathrm{Pr}=4.38$ (circles), $\mathrm{Pr}=5.68$ (up-pointing triangles), and $\mathrm{Pr}=3.26$ (down-pointing triangles). Dashed lines (from top to bottom), GL predictions for $\mathrm{Pr}=3.26$, 4.38, and 5.68. (c) The ratio of Re to the GL prediction as a function of Pr for $\mathrm{Ra}⩽3\times {10}^9$. Open circles, ${\mathrm{Re}}^{\mathrm{pl}}$, $\mathrm{Pr}=4.38$. Stars, ${\mathrm{Re}}^{\omega }$, $\mathrm{Pr}=29.8$. Squares, ${\mathrm{Re}}^{\omega }$, from 127, $5.6⩽\mathrm{Pr}⩽1206$.

Figure 9

(Color online) Temperature and velocity profiles. (a) Local, only time-averaged, temperature, and temperature-fluctuation profiles at $\mathrm{Ra}=7.7\times {10}^{11}$ and $\Gamma =1.13$ as functions of the height $z$ (normalized by ${\lambda }_{\theta }^{\mathrm{sl}}$), measured below the center of the top plate. The fluid is air at atmospheric pressure, the cell height is $6.3\mathrm{m}$. From 80. (b) Rescaled mean velocity profiles $U\left(z\right)$ above the center of the plate for $\mathrm{Ra}=1.25\times {10}^9$ (shortest data set) up to $\mathrm{Ra}=2.02\times {10}^{10}$ (longest data set) for RB flow in water. $u$ is normalized by $u_{\max }$ and $z$ by the kinetic BL thickness ${\lambda }_u^{sl}(\ast )$. From 195.

Figure 10

Boundary layer thickness. Scaling of the kinetic BL thicknesses ${\lambda }_u^{\mathrm{sl}}(\ast )$ and ${\lambda }_u^m(\ast )$ as functions of (a) Ra and (b) Re, taken above the center of the lower plate. From 195. In (a) the scaling of the thermal BL thickness ${\lambda }_{\theta }^{\mathrm{sl}}(\ast )$ is also shown. It has been deduced from the temperature profile above the center of the lower plate. The lines are power-law fits.

Figure 11

Deviations from the Oberbeck-Boussinesq approximation. (a), (b) The ratio $\mathrm{Nu}∕{\mathrm{Nu}}_{\mathrm{OB}}$ as a function of $\Delta$ for fixed $T_m$. (c), (d) $T_c-T_m$ as a function of $\Delta$, also fixed $T_m$. (a), (c) Water at $T_m=40.0°\mathrm{C}$, $\Gamma =1.0$. Solid circles, $L=49.7\mathrm{cm}$. Open circles, $L=24.8\mathrm{cm}$. Solid squares, $L=9.2\mathrm{cm}$. (b), (d) Ethane at a pressure of $55.17\text{bars}$ for $\Gamma =0.5$. Open symbols, $T_m=35.0°\mathrm{C}$ below the critical isochore temperature $T_{\varphi }=38.06°\mathrm{C}$ at this pressure. Solid symbols, $T_m=41.0°\mathrm{C}$, above $T_{\varphi }$.

Figure 12

Behavior of LSC. Left: The LSC orientation as a function of time at $z=-L∕4$ (${\theta }_b$, dotted line), $z=0$ (${\theta }_0$, triangles), and $z=L∕4$ (${\theta }_t$, solid line). Right: The probability distribution $p({\theta }^{\prime }∕2\pi )$ as a function of ${\theta }^{\prime }∕2\pi$. Here ${\theta }^{\prime }={\theta }_t-{\theta }_0$ (open circles) and ${\theta }^{\prime }={\theta }_b-{\theta }_0$ (solid circles). Solid line, a Gaussian distribution. Adapted from 89.

Figure 13

Time dependence of LSC orientation. Top: A time series of ${\theta }_0\left(t\right)$ spanning about $11\text{days}$. Adapted from 26. Bottom left: The LSC orientation ${\theta }_0\left(t\right)$. Bottom right: The LSC azimuthal temperature amplitude $\delta \left(t\right)$ at the sidewall, during a cessation. Adapted from 34.