Effect of slip boundary conditions on the heat flux and near-wall temperature equations in turbulent Rayleigh-B\'{e}nard convection

We present direct numerical simulations (DNS) of the heat transport and near-wall temperature profiles in turbulent Rayleigh-B\'{e}nard convection (RBC) with slip boundary conditions (BCs) on horizontal walls. The mean horizontal velocity on the wall is assumed as $\boldsymbol{u}_w=(b/L)(\partial{\boldsymbol{u}}/\partial{\boldsymbol{n}})|_w$. Here $L$ is the height of RBC sample, $b$ is the slip length with $b = 0$ for no-slip BC and $b \rightarrow \infty$ for free-slip BC. The simulations were for $0 \leq b/L \rightarrow \infty$ and the Prandtl numbers $Pr = 4.3$ in the Rayleigh-number range $10^{8} \lesssim Ra \lesssim \times 10^{10}$. As $b/L$ increases, we found that the ratio of dimensionless heat flux, as expressed by the Nusselt number follows $Nu/Nu_0 = 0.8\times tanh(100\times b/L) + 1$, where $Nu_0$ is the Nusselt number for $b = 0$. Considering the boundary layer fluctuations, we derived the equation $\Theta(\xi)=\int_0^{\xi}(1+p^x\eta^x)^{-n}d\eta$ for the mean temperature profile $\Theta(\xi)$ near the horizontal surface, where $p = \Gamma(1+1/x)\Gamma(n-1/x)/\Gamma(n)$ with $2 \leq x \leq 3$ depending on $b/L$ and $n>1$ for varying geometries of the convection sample.