Mean temperature profiles in turbulent thermal convection

To predict the mean temperature profiles in turbulent thermal convection, the thermal boundary layer (BL) equation including the effects of fluctuations has to be solved. In Shishkina et al. [Phys. Rev. Lett. 114, 114302 (2015)], the thermal BL equation with the fluctuations taken into account as an eddy thermal diffusivity has been solved for large Prandtl-number fluids for which the eddy thermal diffusivity and the velocity field can be approximated, respectively, as a cubic and a linear function of the distance from the plate. In the present work, we make use of the idea of Prandtl's mixing length model and relate the eddy thermal diffusivity to the stream function. With this proposed relation, we can solve the thermal BL equation and obtain a closed-form expression for the dimensionless mean temperature profile in terms of two independent parameters for fluids with a general Prandtl number. With a proper choice of the parameters, our predictions of the temperature profiles are in excellent agreement with the results of our direct numerical simulations for a wide range of Prandtl numbers from 0.01 to 2547.9 and Rayleigh numbers from ${10}^7$ to ${10}^9$.

Figure 1

Normalized eddy thermal diffusivity $|{\kappa }_t/\kappa |$, calculated for ${\kappa }_t=\left(V\mathrm{\Theta }-{\langle vT\rangle }_t\right)/{\partial }_z\mathrm{\Theta }$, and then averaged over horizontal cross-sections, obtained in the DNS for $\text{Pr}=10$ and $\text{Ra}={10}^8$ (symbols) together with a fit for $|(l_v{l}_{\theta }/\kappa ){\partial }_{z}V|$ averaged over horizontal cross-sections (solid line). Here $l_v$ and $l_{\theta }$ are taken according to Eq. (34), i.e., $l_v{l}_{\theta }\sim z^2$. Thus, the symbols and the line represent, respectively, the magnitudes of the left- and right-hand sides of the assumption Eq. (33).

Figure 2

Temperature profiles, averaged in time and over horizontal cross sections, obtained in the DNS of RBC in a cylindrical container of the aspect ratio 1 for $\text{Pr}=0.1$ and different $\text{Ra}$ (symbols). One can see that the profiles converge with increasing $\text{Ra}$. Prandtl-Blasius predictions for $\text{Pr}\to \infty$ $\left(-·-\right)$ and $\text{Pr}\to 0$ (—) bound the gray region of Prandtl-Blasius predictions for all intermediate $\text{Pr}$.

Figure 3

Temperature profiles, averaged in time and over horizontal cross sections, obtained in the DNS of RBC in a cylindrical container of the aspect ratio 1 for $\text{Pr}=0.1,\text{Pr}=1$, and $\text{Pr}=2547.9$ and different $\text{Ra}$ (symbols) together with the predictions Eq. (43) (lines of the corresponding colours); see Table 1. The black dashed line corresponds to the simplification Eq. (48) for very large $\text{Pr}$, as reported in Ref. [37]. The Prandtl-Blasius region (gray) is as in Fig. 2.

Figure 4

Temperature profiles, averaged in time and over horizontal cross sections, obtained in the DNS of RBC in a cylindrical container of the aspect ratio 1 for $\text{Ra}={10}^7$ and different small Prandtl numbers (symbols) together with the predictions Eq. (43) from the present work (lines of the corresponding colours); see Table 1. The black dashed line corresponds to the limiting case Eq. (47) for the simplification Eq. (45) for large $\text{Pr}$. The Prandtl-Blasius region (gray) is as in Fig. 2.