Plane-layer Rayleigh-Bénard convection up to $Ra=10^{11}$: Near-wall fluctuations and role of initial conditions

TL;DR

This study uses direct numerical simulations to explore plane-layer Rayleigh-Bénard convection over $Ra$ from $10^5$ to $10^{11}$ at $Pr=0.7$, focusing on height-dependent velocity and temperature fluctuations and a decomposition of the near-wall boundary layer into coherent (shear-dominated) and incoherent (plume-dominated) regions. It demonstrates persistent near-wall fluctuations across the range, with coherent regions occupying about $40 ext{ extpercent}$ of the wall and plume-dominated transport approaching the classical $Nu \\sim Ra^{1/3}$ in the appropriate regions; the global Nu and Re are robust to finite-amplitude initial perturbations, indicating relaxation to a single turbulent attractor. The study further probes weak and steady shear forcing, finding log-law-like near-wall layers that do not alter global transport, while a strong constant-pressure gradient drives a conventional turbulent boundary layer and enhances heat transfer. Overall, the near-wall RBC boundary layer in this setup is non-standard and exhibits strong resilience to perturbations, suggesting that achieving an ultimate regime may require higher $Ra$.

Abstract

We study turbulent Rayleigh-Bénard convection through direct numerical simulations in a three-dimensional plane layer of aspect ratio 4 for Rayleigh numbers $10^5 \leq Ra \leq 10^{11}$ and Prandtl number $Pr=0.7$. We summarize the height-dependent statistics of velocity and temperature fluctuations and corresponding scalings with the Rayleigh number. We include an analysis on the role of coherent and incoherent flow regions near the wall for global heat transfer. Furthermore, we investigate the dependence of turbulent transport on a finite-amplitude sinusoidal shear flow added at time $t=0$, which either freely decays in a long transient or remains existent when a steady sinusoidal volume forcing is added. In the latter case, weak logarithmic near-wall layers are formed, however, with von Kármán and offset constants that differ from standard values. The typical magnitude of both coefficients, and thus a full turbulent boundary layer of velocity and temperature, is re-established only for a switch from sinusoidal to constant pressure gradient driving of the flow. In all cases, except for the constant pressure gradient-driven flow, no enhancement of global turbulent heat and momentum transfer within error bars is detected, even though the sinusoidal amplitude is of the order of the characteristic free-fall velocity.

Figures (17)

• Figure 1: Contours of temperature (a, c) and vertical component of velocity (b, d) on a horizontal cross-section ($x$-$y$ plane) from a snapshot at $Ra=10^{11}$. The upper and lower rows shows the fields at $z=H/2$ and $\delta_T$ respectively. The structure of the flow is more sharply delineated at the thermal boundary layer where plumes dominate the flow. At the midplane in the bulk, the fields are thoroughly turbulent and well-mixed. Appropriate colorbar cut-offs have been chosen for increased contrast and clear visibility of structures.
• Figure 2: (a) Profiles of temperature fluctuations, $T_{\mathrm{rms}}$, averaged across area $A$ and runtime $t$, for all Rayleigh numbers. The symbols indicate the values of these profiles at $z=\delta_{T,\mathrm{rms}}$ (red circles), $z=2\delta_{T,\mathrm{rms}}$ (green squares), $z=4\delta_{T,\mathrm{rms}}$ (cyan diamonds), $z=8\delta_{T,\mathrm{rms}}$ (purple triangles), and $z=H/2$ (blue inverted triangles). (b) Scaling of the indicated values of $T_{\mathrm{rms}}$ in (a) versus $Ra$. (c) Corresponding profiles of velocity fluctuations, $U_{\mathrm{rms}}$, with symbols marked in the same manner as in (a). The heights are now multiples of the momentum boundary region thickness, $\delta_{U,\mathrm{rms}}$. (d) Scaling of the indicated values of $U_{\mathrm{rms}}$ in (c) versus $Ra$. Similarly, the fluctuation profiles and scaling of the vertical velocity component, $u_z$, are shown in panels (e) and (f) respectively.
• Figure 3: Probability density functions (PDF) of the temperature fluctuations $\theta$ (top row), velocity magnitude $U=|{\bm u}|$ (middle row) and vertical velocity component $u_z$ (bottom row), sampled at horizontal cross-sectional $x$-$y$ planes at increasing heights of $z = \delta_T,~2\delta_T,~4\delta_T, \mathrm{and}~8\delta_T$ (left to right) for all $Ra$ values. The $y$ axis is logarithmic in the bottom row; it is linear in the upper and middle rows. The skewness of these distributions is tabulated in table \ref{['tab:skew']}.
• Figure 4: Decomposition of the boundary region flow field into shear-dominated coherent sections and plume-dominated incoherent sections at the thermal boundary layer height $z = \delta_T$. Two methods are compared for $Ra$ ranging from $10^5$ to $10^9$ - thresholding the rms horizontal velocity, $U_h$ (top row), and the horizontal shear-stress, $S_h$ (middle row). The corresponding area fractions of the coherent regions are also indicated as $A^v_\mathrm{coh}$ and $A^s_\mathrm{coh}$ respectively. The cyclic colormap within the coherent regions indicates the angle of the flow in the wall-parallel plane, ranging from $-\pi$ to $\pi$. The agreement between the two methods is highlighted in the bottom row, where the coherent regions are plotted for velocity threshold in grey, shear-stress threshold in red, and the overlapping regions of both methods, $A^v_\mathrm{coh} \cap A^s_\mathrm{coh}$, in dark blue. The ratio between the overlapping area and $A^s_\mathrm{coh}$ is are 0.72, 0.78, 0.80, 0.80 and 0.84 for $Ra = 10^5, 10^6, 10^7, 10^8$ and $10^9$ respectively.
• Figure 5: Area fraction and Nusselt number scaling of coherent regions of the boundary region for $10^5 \leq Ra \leq 10^{11}$. (a) Area fraction $A_{\rm coh}$ versus $Ra$ for different decompositions of the whole plane, here given by the number $\sqrt{N_{\rm tiles}}$. (b) Nusselt number scaling $Nu(Ra)$ in double logarithmic plot conditioned to coherent and incoherent area fractions as well as for total area. The dotted lines are fits for $10^5\le Ra\le 10^8$, the dashed ones for $10^8\le Ra\le 10^{11}$ (c) Compensated linear-logarithmic replot of the data of panel (b). The contribution to global heat transport from the shear-dominated regions decreases with increasing $Ra$. In the high-Rayleigh-number regime for $Ra>10^8$, scaling of $Nu(Ra)$ is closest to $1/3$ in the plume dominated regions, whereas the shear-dominated regions display a smaller scaling exponent.