Thermal convection over a fractal surface

We use well resolved numerical simulations to study Rayleigh-B\'enard convection in cells with a fractal boundary in two dimensions for $Pr = 1$ and $Ra \in \left[10^7, 2.15 \times 10^9\right]$. The fractal boundaries are functions characterized by power spectral densities $S(k)$ that decay with wavenumber as $S(k) \sim k^{p}$ ($p < 0$). The degree of roughness is quantified by the exponent $p$ with $p < -3$ for smooth (differentiable) surfaces and $-3 \le p < -1$ for rough surfaces with Hausdorff dimension $D_f=\frac{1}{2}(p+5)$. By computing the exponent $\beta$ in power law fits $Nu \sim Ra^{\beta}$, where $Nu$ 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, $\beta = 0.256, 0.281$ and $0.306$. For a given value of $p$ we observe that the mean heat flux is insensitive to the details of the roughness.