Thermal Convection over Fractal Surfaces

Abstract

We use well resolved numerical simulations to study Rayleigh-Bénard convection in cells with a fractal boundary in two dimensions for P r = 1 and Ra ∈ 10 7 , 2.15 × 10 9 . The fractal boundaries are functions characterized by power spectral densities S(k) that decay with wavenumber as S(k) ∼ k p (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 D f = 1 2 (p + 5). By computing the exponent β in power law fits N u ∼ Ra β , where N u and Ra are the Nusselt and the Rayleigh numbers, we observe that heat transport increases with roughness. For p = −3.0, −2.0 and −1.5 we find, respectively, β = 0.256, 0.281 and 0.306. For a given value of p we observe that the mean heat flux is insensitive to the specific realization of the roughness.