Heat transport and temperature boundary-layer profiles in closed turbulent Rayleigh–Bénard convection with slippery conducting surfaces

Abstract

We report direct numerical simulations (DNS) of the Nusselt number$Nu$, the vertical profiles of mean temperature$\varTheta (z)$and temperature variance$\varOmega (z)$across the thermal boundary layer (BL) in closed turbulent Rayleigh–Bénard convection (RBC) with slippery conducting surfaces ($z$is the vertical distance from the bottom surface). The DNS study was conducted in three RBC samples: a three-dimensional cuboid with length$L = H$and width$W = H/4$($H$is the sample height), and two-dimensional rectangles with aspect ratios$\varGamma \equiv L/H = 1$and$10$. The slip length$b$for top and bottom plates varied from$0$to$\infty$. The Rayleigh numbers$Ra$were in the range$10^{6} \leqslant Ra \leqslant 10^{10}$and the Prandtl number$Pr$was fixed at$4.3$. As$b$increases, the normalised$Nu/Nu_0$($Nu_0$is the global heat transport for$b = 0$) from the three samples for different$Ra$and$\varGamma$can be well described by the same function$Nu/Nu_0 = N_0 \tanh (b/\lambda _0) + 1$, with$N_0 = 0.8 \pm 0.03$. Here$\lambda _0 \equiv L/(2Nu_0)$is the thermal boundary layer thickness for$b = 0$. Considering the BL fluctuations for$Pr>1$, one can derive solutions of temperature profiles$\varTheta (z)$and$\varOmega (z)$near the thermal BL for$b \geqslant 0$. When$b=0$, the solutions are equivalent to those reported by Shishkinaet al.(Phys. Rev. Lett., vol. 114, 2015, 114302) and Wanget al.(Phys. Rev. Fluids, vol. 1, 2016, 082301(R)), respectively, for no-slip plates. For$b > 0$, the derived solutions are in excellent agreement with our DNS data for slippery plates.