Dynamics of large-scale quantities in Rayleigh-Bénard convection

In this paper we estimate the relative strengths of various terms of the Rayleigh-Bénard equations. Based on these estimates and scaling analysis, we derive a general formula for the large-scale velocity $U$ or the Péclet number that is applicable for arbitrary Rayleigh number Ra and Prandtl number Pr. Our formula fits reasonably well with the earlier simulation and experimental results. Our analysis also shows that the wall-bounded convection has enhanced viscous force compared to free turbulence. We also demonstrate how correlations deviate the Nusselt number scaling from the theoretical prediction of ${\mathrm{Ra}}^{1/2}$ to the experimentally observed scaling of nearly ${\mathrm{Ra}}^{0.3}$.

Figure 1

The plane-averaged temperature profile $T_m\left(z\right)$ exhibits a near constant value in the bulk and steep variations in the thermal boundary layers whose thicknesses are ${\delta }_T$. The normalized temperature fluctuation is $\theta =T+z-1$; the figure also exhibits its plane-averaged profile ${\theta }_m\left(z\right)$.

Figure 2

Schematic of acceleration $\mathbf{a}$ of a fluid parcel that is rising against gravity: (a) in the turbulent regime, $\mathbf{a}$ is provided primarily by the pressure gradient $-\mathbf{\nabla }\sigma$; (b) in the viscous regime, $\mathbf{a}\approx 0$ with the buoyancy and viscous terms balancing each other. The directions of the forces are reversed for a descending fluid parcel. Note that buoyancy acts downward for the fluid parcels colder than the background.

Figure 3

Plots comparing the rms values of various terms of the momentum equation ${|\mathbf{u}·\mathbf{\nabla }\mathbf{u}|,|\left(\mathbf{\nabla }\sigma \right)}_{\mathrm{res}}|,\alpha g{\theta }_{\mathrm{res}}$, and $|\nu {\nabla }^2\mathbf{u}|$ as functions of Ra for (a) $\mathrm{Pr}=1$ and (b) $\mathrm{Pr}={10}^2$. In the turbulent regime [the shaded region of (a)], $|\mathbf{u}·\mathbf{\nabla }\mathbf{u}|\approx \left|\mathbf{\nabla }\sigma \right|\gg \alpha g{\theta }_{\mathrm{res}},\left|\nu {\nabla }^2\mathbf{u}\right|$; in the viscous regime (b), $\alpha g{\theta }_{\mathrm{res}},|\nu {\nabla }^2\mathbf{u}|\gg |\mathbf{u}·\mathbf{\nabla }\mathbf{u}|,|\mathbf{\nabla }\sigma |$.

Figure 4

Plots of $c_1$ (red squares), $c_2$ (green circles), $c_3$ (blue triangles), and $c_4$ (black diamonds) of Eqs. (3a)–(3d) as functions of the Pr for $\mathrm{Ra}=2\times {10}^7$. All $c_i$'s are weak functions of the Pr.

Figure 5

Plot of the normalized Péclet number $\mathrm{Pe}{\mathrm{Ra}}^{-1/2}$ as a function of the Ra for different Pr's. Our formula for the Pe [Eq. (5)] (shown as the continuous curves) fits reasonably well with the numerical results of ours ($\mathrm{Pr}=1$, red squares; $\mathrm{Pr}=6.8$, blue triangles; $\mathrm{Pr}={10}^2$, black diamonds), Scheel and Schumacher [34] ($\mathrm{Pr}=0.7$, green crosses), Reeuwijk et al. [23] ($\mathrm{Pr}=1$, red circles), and Silano et al. [24] ($\mathrm{Pr}={10}^3$, magenta pentagons), and the experimental results of Cioni et al. [15] ($\mathrm{Pr}\approx 0.022$, brown right-triangles) and Xin and Xia [14] ($\mathrm{Pr}\approx 6.8$, orange pluses). The agreement is deficient for the experimental result of Niemela et al. [18] ($\mathrm{Pr}\approx 0.7$, green down-triangles), possibly due to various factors discussed in the text.

Figure 6

The plots of normalized viscous dissipation rates $C_{{\epsilon }_u,1}$ (in the turbulent regime) and $C_{{\epsilon }_u,2}$ (in the viscous regime). $C_{{\epsilon }_u,1}\sim {\mathrm{Ra}}^{-0.22}$ and ${\mathrm{Ra}}^{-0.25}$ for $\mathrm{Pr}=1$ and 6.8, respectively, whereas $C_{{\epsilon }_u,2}\sim {\mathrm{Ra}}^{0.22}$ and ${\mathrm{Ra}}^{0.19}$ for $\mathrm{Pr}=6.8$ and ${10}^2$, respectively. These results are in qualitative agreement with model predictions (see Table 2).