Effect of slip boundary conditions on the heat flux and near-wall temperature equations in turbulent Rayleigh-Bénard convection

We present direct numerical simulations (DNS) of the heat transport and near-wall temperature profiles in turbulent Rayleigh-Bénard convection (RBC) with slip boundary conditions (BCs) on horizontal walls. The mean horizontal velocity on the wall is assumed as u_{w=(b / L) (∂ u / ∂ n) | w} . Here L is the height of RBC sample, b is the slip length with b = 0 for no-slip BC and b -> ∞ for free-slip BC. The simulations were for 0 <= b / L -> ∞ and the Prandtl numbers Pr = 4 . 3 in the Rayleigh-number range 10⁸ < Ra < ×10¹⁰ . As b / L increases, we found that the ratio of dimensionless heat flux, as expressed by the Nusselt number follows Nu / Nu₀ = 0 . 8 × tanh(100 × b / L) + 1 , where Nu₀ is the Nusselt number for b = 0 . Considering the boundary layer fluctuations, we derived the equation Θ(ξ) =∫₀^&xi;^{(1 +pxηx) - n} dη for the mean temperature profile Θ(ξ) near the horizontal surface, where p = Γ(1 + 1 / x) Γ(n - 1 / x) / Γ(n) with 2 <= x <= 3 depending on b / L and n > 1 for varying geometries of the convection sample.

Supported by NSFC Grant No. 11772111.