Statistics of velocity and temperature fluctuations in two-dimensional Rayleigh-Bénard convection

We investigate fluctuations of the velocity and temperature fields in two-dimensional (2D) Rayleigh-Bénard (RB) convection by means of direct numerical simulations (DNS) over the Rayleigh number range ${10}^6\le \text{Ra}\le {10}^{10}$ and for a fixed Prandtl number $\text{Pr}=5.3$ and aspect ratio $\mathrm{\Gamma }=1$. Our results show that there exists a counter-gradient turbulent transport of energy from fluctuations to the mean flow both locally and globally, implying that the Reynolds stress is one of the driving mechanisms of the large-scale circulation in 2D turbulent RB convection besides the buoyancy of thermal plumes. We also find that the viscous boundary layer (BL) thicknesses near the horizontal conducting plates and near the vertical sidewalls, ${\delta }_u$ and ${\delta }_v$, are almost the same for a given Ra, and they scale with the Rayleigh and Reynolds numbers as $\sim {\text{Ra}}^{-0.26\pm 0.03}$ and $\sim {\text{Re}}^{-0.43\pm 0.04}$. Furthermore, the thermal BL thickness ${\delta }_{\theta }$ defined based on the root-mean-square (rms) temperature profiles is found to agree with Prandtl-Blasius predictions from the scaling point of view. In addition, the probability density functions of turbulent energy ${\varepsilon }_{u^{\prime }}$ and thermal ${\varepsilon }_{{\theta }^{\prime }}$ dissipation rates, calculated, respectively, within the viscous and thermal BLs, are found to be always non-log-normal and obey approximately a Bramwell-Holdsworth-Pinton distribution first introduced to characterize rare fluctuations in a confined turbulent flow and critical phenomena.

Figure 1

Ra dependence of the compensated Nusselt number ${\text{Nu/Ra}}^{1/3}$, the simulation results for 2D RB convection are indicated by solid circles, and the 3D numerical results conducted in a cylindrical convection cell by Lakkaraju et al. [20] ($\text{Pr}=5.2$) are marked with open circles. Note that the 3D data have been multiplied by a factor of 0.79, which agrees well with the constant 0.78 suggested by van der Poel et al. [26].

Figure 2

Instantaneous (a)–(c) and mean (d)–(f) flow fields at three different Rayleigh numbers $\text{Ra}={10}^6$ (left panel), ${10}^8$ (middle panel), and ${10}^{10}$ (right panel). The magnitudes of velocity and temperature are coded in arrows and color, respectively.

Figure 3

Root-mean-square values of horizontal velocity fluctuations $u_{\text{rms}}$ (a)–(c), vertical velocity fluctuations $v_{\text{rms}}$ (d)–(f), and temperature fluctuations ${\theta }_{\text{rms}}$ (g)–(i). Data are obtained at three different Rayleigh numbers (from left to right): $\text{Ra}={10}^6$ (left panel), ${10}^8$ (middle panel), and ${10}^{10}$ (right panel). The data used here are the same as those in Figs. 2–2.

Figure 4

Turbulent kinetic energy $\langle u^{\prime 2}\rangle +\langle v^{\prime 2}\rangle$ (a)–(c) and turbulent heat flux $\langle v^{\prime }{\theta }^{\prime }\rangle$ (d)–(f) fields for $\text{Ra}={10}^6$ (left panel), ${10}^8$ (middle panel), and ${10}^{10}$ (right panel).

Figure 5

Main component of the Reynolds stress $-\langle u^{\prime }{v}^{\prime }\rangle$ (a)–(c), turbulent energy production due to mean shear $P=-\langle u_i^{\prime }{u}_j^{\prime }\rangle \partial \langle u_i\rangle /\partial x_j$ ($i,j=1,2$) (d)–(f), and total turbulent energy production $P_0=P_{{\theta }^{\prime }}+P$ (g)–(i) for $\text{Ra}={10}^6$ (left panel), ${10}^8$ (middle panel), and ${10}^{10}$ (right panel).

Figure 6

Profiles of $u_{\text{rms}}$ (a), $v_{\text{rms}}$ (b), and ${\theta }_{\text{rms}}$ (c) near the plates for $\text{Ra}={10}^8$; the solid lines illustrate the definition of boundary layer thickness.

Figure 7

The normalized profiles for $u_{\text{rms}}$ (a), $v_{\text{rms}}$ (b), and ${\theta }_{\text{rms}}$ (c) at five different values of Ra. Normalized profiles compared between $u_{\text{rms}}$ and $v_{\text{rms}}$ for $\text{Ra}={10}^6$ (d), ${10}^8$ (e), and ${10}^{10}$ (f).

Figure 8

Ra and Re dependence of ${\delta }_u$, ${\delta }_v$, and ${\delta }_{\theta }$. The dashed lines correspond to the least-squares fit to the data and the scaling exponents are summarized in Table 3.

Figure 9

(a) The rms-velocity-based Reynolds numbers as a function of Ra. The dashed line has a slope of 0.5. (b) The dependence of ${\delta }_u/H$ on ${\text{Re}}_{u_{\text{rms}}^{\text{max}}}$ and (c) the dependence of ${\delta }_v/H$ on ${\text{Re}}_{v_{\text{rms}}^{\text{max}}}$. The dashed lines in (b) and (c) are the power-law fits to the corresponding data, which yield ${\delta }_u/H\sim {\text{Re}}_{u_{\text{rms}}^{\text{max}}}^{-0.52\pm 0.11}$ and ${\delta }_v/H\sim {\text{Re}}_{v_{\text{rms}}^{\text{max}}}^{-0.53\pm 0.08}$, respectively.

Figure 10

PDFs of turbulent dissipation rates ${\varepsilon }_{u^{\prime }}$ (a) and ${\varepsilon }_{{\theta }^{\prime }}$ (b) normalized, respectively, by their rms values ${\left({\varepsilon }_{u^{\prime }}\right)}_{\text{rms}}=\sqrt{{\langle {\varepsilon }_{u^{\prime }}^2\rangle }_{\text{BL},t}}$ and ${\left({\varepsilon }_{{\theta }^{\prime }}\right)}_{\text{rms}}=\sqrt{{\langle {\varepsilon }_{{\theta }^{\prime }}^2\rangle }_{\text{BL},t}}$ (here the notation ${\langle ·\rangle }_{\text{BL},t}$ denotes the average over BL parts and over time) for five different Rayleigh numbers. The solid lines are the best fits of stretched exponentials, as given by Eq. (9). The data are calculated within the respective viscous and thermal BLs and over a sequence of statistically independent snapshots.

Figure 11

PDFs of $[ln\left({\varepsilon }_{u^{\prime }}\right)-\mu ]/\sigma$ and $[ln\left({\varepsilon }_{{\theta }^{\prime }}\right)-\mu ]/\sigma$ for five different values of Ra. Here $\mu$ and $\sigma$ are the mean and standard deviation values of $ln\left({\varepsilon }_{u^{\prime }}\right)$ or $ln\left({\varepsilon }_{{\theta }^{\prime }}\right)$.