Thermal convection over fractal surfaces

We use well resolved numerical simulations with the lattice Boltzmann method to study Rayleigh-Bénard convection in cells with a fractal boundary in two dimensions for Pr = 1 and Ra ϵ [107, 1010], where Pr and Ra are the Prandtl and Rayleigh numbers. The fractal boundaries are functions characterized by power spectral densities S(k) that decay with wavenumber, k, as S(k) ∼ kp (p < 0). The degree of roughness is quantified by the exponent p with p <-3 for smooth (differentiable) surfaces and-3 ≤ p <-1 for rough surfaces with Hausdorff dimension Df = 12 ( p + 5). By computing the exponent β using power law fits of Nu ∼ Raβ, where Nu is the Nusselt number, we find that the heat transport scaling increases with roughness through the top two decades of Ra ϵ [108, 1010]. For p =-3.0,-2.0 and-1.5 we find β = 0.288 ± 0.005, 0.329 ± 0.006 and 0.352 ± 0.011, respectively. We also find that the Reynolds number, Re, scales as Re ∼ Ra?, where ζ ≈ 0.57 over Ra ϵ [107, 1010], for all p used in the study. For a given value of p, the averaged Nu and Re are insensitive to the specific realization of the roughness.