Boundary layer structure in turbulent Rayleigh-Bénard convection in a slim box

Abstract

The logarithmic law of mean temperature profile has been observed in different regions in Rayleigh-Bénard turbulence. However, how thermal plumes correlate to the log law of temperature and how the velocity profile changes with pressure gradient are not fully understood. Here, we performed three-dimensional simulations of Rayleigh-Bénard turbulence in a slim-box without the front and back walls with aspect ratio, $L:D:H=1:1/6:1$, in the Rayleigh number $Ra=[1\times 10^8,1\times 10^{10}]$ for Prandtl number $Pr=0.7$. The velocity profile is successfully quantified by a two-layer function of a stress length, ${\ell }_u^{+}\approx {\ell }_0^{+}(z^{+}{)}^{3/2}{\left[1+{\left(z^{+}/z_{sub}^{+}\right)}^4\right]}^{1/4}$, as proposed by She et al. (She 2017), though neither a Prandtl-Blasius-Pohlhausen type nor the log-law is seen in the viscous boundary layer. In contrast, the temperature profile in the plume-ejecting region is logarithmic for all simulated cases, being attributed to the emission of thermal plumes. The coefficient of the temperature log-law, $A$ can be described by composition of the thermal stress length ${\ell }_{\hat{I}}^{\ast }¸$ and the thicknesses of thermal boundary layer $z_{sub}^{\ast }$ and $z_{buf}^{\ast }$, i.e. $A\simeq z_{sub}^{\ast }/\left({\ell }_{\hat{I}¸0}^{\ast }{z_{buf}^{\ast }}^{3/2}\right)$. The adverse pressure gradient responsible for turning the wind direction contributes to thermal plumes gathering at the ejecting region and thus the log-law of temperature profile. The Nusselt number scaling and local heat flux of the present simulations are consistent with previous results in confined cells. Therefore, the slim-box RBC is a preferable system for investigating in-box kinetic and thermal structures of turbulent convection with the large-scale circulation on a fixed plane.

16 pages, 37 figures