Universal fluctuations in the bulk of Rayleigh-Bénard turbulence

Abstract

We present an investigation of the root-mean-square (rms) temperature ${\sigma }_T$ and the rms velocity ${\sigma }_w$ in the bulk of Rayleigh-Bénard turbulence, using new experimental data from the current study and experimental and numerical data from previous studies. We find that, once scaled by the convective temperature ${\theta }_{\ast }$, the value of ${\sigma }_T$ at the cell centre is a constant, i.e. ${\sigma }_{T,c}/{\theta }_{\ast }\approx 0.85$, over a wide range of the Rayleigh number ($10^8\le Ra\le 10^{15}$) and the Prandtl number ($0.7\le Pr\le 23.34$), and is independent of the surface topographies of the top and bottom plates of the convection cell. A constant close to unity suggests that ${\theta }_{\ast }$ is a proper measure of the temperature fluctuation in the core region. On the other hand, ${\sigma }_{w,c}/w_{\ast }$, the vertical rms velocity at the cell centre scaled by the convective velocity $w_{\ast }$, shows a weak $Ra$-dependence ($\sim R{a}^{0.07\pm 0.02}$) over $10^8\le Ra\le 10^{10}$ at $Pr\sim 4.3$ and is independent of plate topography. Similar to a previous finding by He \& Xia ({\it Phys. Rev. Lett.,} vol. 122, 2019, 014503), we find that the rms temperature profile ${\sigma }_T(z)/{\theta }_{\ast }$ in the region of the mixing zone with a mean horizontal shear exhibits a power-law dependence on the distance $z$ from the plate, but now the universal profile applies to both smooth and rough surface topographies and over a wider range of $Ra$. The vertical rms velocity profile ${\sigma }_w(z)/w_{\ast }$ obey a logarithmic dependence on $z$. The study thus demonstrates that the typical scales for the temperature and the velocity are the convective temperature ${\theta }_{\ast }$ and the the convective velocity $w_{\ast }$, respectively. Finally, we note that ${\theta }_{\ast }$ may be utilised to study the flow regime transitions in the ultra-high-$Ra$-number turbulent convection.

12 pages, 6 figures; Accepted for publication in the Journal of Fluid Mechanics