Transition to turbulence scaling in Rayleigh-Bénard convection

If a fluid flow is driven by a weak Gaussian random force, the nonlinearity in the Navier-Stokes equations is negligibly small and the resulting velocity field obeys Gaussian statistics. Nonlinear effects become important as the driving becomes stronger and a transition occurs to turbulence with anomalous scaling of velocity increments and derivatives. This process has been described by Yakhot and Donzis [Phys. Rev. Lett. 119, 044501 (2017)] for homogeneous and isotropic turbulence. In more realistic flows driven by complex physical phenomena, such as instabilities and nonlocal forces, the initial state itself, and the transition to turbulence from that initial state, is much more complex. In this paper, we discuss the Reynolds-number dependence of moments of the kinetic energy dissipation rate of orders 2 and 3 obtained in the bulk of thermal convection in the Rayleigh-Bénard system. The data are obtained from three-dimensional spectral element direct numerical simulations in a cell with square cross section and aspect ratio 25 by Pandey et al. [Nat. Commun. 9, 2118 (2018)]. Different Reynolds numbers $1\lesssim {\mathrm{Re}}_{\ell }\lesssim 1000$ which are based on the thickness of the bulk region $\ell$ and the corresponding root-mean-square velocity are obtained by varying the Prandtl number $\mathrm{Pr}$ from 0.005 to 100 at a fixed Rayleigh number $\mathrm{Ra}={10}^5$. A few specific features of the data agree with the theory. The normalized moments of the kinetic energy dissipation rate ${\mathcal{E}}_n$ show a nonmonotonic dependence for small Reynolds numbers before obeying the algebraic scaling prediction for the turbulent state. Implications and reasons for this behavior are discussed.

Figure 1

Transition from Gaussian to non-Gaussian statistics for two normalized dissipation moments ${\mathcal{E}}_n$ in a homogeneous isotropic turbulent flow with moment orders $n_1>n_2$. Panel (a) is the schematic when the large-scale Reynolds number Re is used, while (b) represents the situation when the Reynolds number is rescaled as described in the text.

Figure 2

Plane- and time-averaged vertical mean profiles of the convective heat current $j_{\mathrm{conv}}\left(x_3\right)$ (top) and conductive heat current $j_{\mathrm{cond}}\left(x_3\right)$ as defined in Eq. (28). Runs for $\mathrm{Pr}=0.005$, 0.7, 7, and 70 are displayed. The horizontal dashed line corresponds to Nusselt number Nu, as given in Table 1. We also indicate the bottom and top end positions, ${\ell }_1$ and ${\ell }_2$ (see Table 1), of the bulk volume that is used for statistical analysis.

Figure 3

Contour plots of the instantaneous kinetic energy dissipation rate $\epsilon (\mathit{x},t)$ at the midplane, $z=1/2$. Contour levels are displayed in units of the decadal logarithm. Top: $\mathrm{Pr}=0.005$. Bottom: $\mathrm{Pr}=70$. Only a quarter of the full cross section is shown.

Figure 4

Normalized moments ${\mathcal{E}}_n$ of kinetic energy dissipation rate vs Reynolds number ${\mathrm{Re}}_{\ell }$ for orders $n=2,3$. We have verified that these two moments converge. We have not shown data for $n>3$ because their convergence is doubtful. The solid line is a power law scaling of the theory [7]. The horizontal dashed lines indicate moments in correspondence with a Gaussian distribution, ${\mathcal{E}}_n=(2n-1)!!$. Error bars have been obtained as the difference of the normalized moments when calculated for the first and second halves of the corresponding data record.

Figure 5

Probability density functions of the temperature fluctuations in the bulk of the convection layer. Temperature amplitudes are normalized by the corresponding root-mean-square value. The Gaussian distribution is added as a dashed line for comparison. (a) All data sets are shown. The small box at the top center of panel (a) indicates the range that is shown in panels (b)–(d) where selected data sets are replotted. (b) $\text{Pr}=70$. (c) $\text{Pr}=0.7$. (d) $\text{Pr}=0.005$.

Figure 6

Rescaled energy dissipation moments vs Reynolds number ${\mathrm{Re}}_{\ell }$ for orders $n=2$ and 3. These data are obtained from the probability density functions of the temperature fluctuations in Fig. 5.