Thermal Boundary Layer Equation for Turbulent Rayleigh–Bénard Convection

We report a new thermal boundary layer equation for turbulent Rayleigh–Bénard convection for Prandtl number $\mathrm{Pr}>1$ that takes into account the effect of turbulent fluctuations. These fluctuations are neglected in existing equations, which are based on steady-state and laminar assumptions. Using this new equation, we derive analytically the mean temperature profiles in two limits: (a) $\mathrm{Pr}\gtrsim 1$ and (b) $\mathrm{Pr}\gg 1$. These two theoretical predictions are in excellent agreement with the results of our direct numerical simulations for $\mathrm{Pr}=4.38$ (water) and $\mathrm{Pr}=2547.9$ (glycerol), respectively.

Figure 1

Normalized eddy thermal diffusivity $|{\kappa }_t/\kappa |$, calculated for ${\kappa }_t=(V\mathrm{\Theta }-⟨vT{⟩}_t)/{\partial }_z\mathrm{\Theta }$, and then averaged over horizontal cross sections, obtained in the DNS for $\mathrm{Pr}=4.38$ and $\mathrm{Ra}=10^7$ (diamonds), $10^8$ (triangles), $10^9$ (circles), and $10^{10}$ (pluses) together with a fit for $\mathrm{Ra}=10^9$ (solid line). It can be seen that close to the plate, ${\kappa }_t/\kappa \propto {\xi }^3$ holds. The dashed line shows the slope $\propto \xi$ that causes the logarithmical temperature profiles in the core part of the domain for sufficiently large Ra.

Figure 2

$\kappa /(\kappa +{\kappa }_t)$ (symbols) [${\kappa }_t$ is calculated as in Fig. 1], and ${\theta }_{\xi }$ (lines), averaged in time and over horizontal cross sections, as functions of the dimensionless vertical coordinate $\xi$ [see definition in Eq. (8)] obtained in the DNS for $\mathrm{Pr}=4.38$ and $\mathrm{Ra}=10^7$ (diamonds, dashed line), $10^8$ (triangles, dotted line), $10^9$ (circles, solid line), and $10^{10}$ (pluses, dot-dashed line).

Figure 3

Temperature profiles, averaged in time and over horizontal cross sections, obtained in the DNS for $\mathrm{Pr}=4.38$ (open symbols) and $\mathrm{Pr}=2547.9$ (filled symbols) and $\mathrm{Ra}=10^7$ (diamonds), $10^8$ (triangles), $10^9$ (circles), and $10^{10}$ (pluses). Excellent agreement with the predictions for $\mathrm{Pr}\gtrsim 1$ (31) (solid line) and $\mathrm{Pr}\gg 1$ (32) (dashed line) is demonstrated. An expanded view with the PBP prediction (17) for $\mathrm{Pr}=2547.9$ (dotted line) and $\mathrm{Pr}=4.38$ (long-dashed orange line, which almost coincides with the dotted line) for comparison is shown in the inset.