Thermal Convection over Fractal Surfaces

We use well resolved numerical simulations with the Lattice Boltzmann Method to study Rayleigh-B'enard convection in cells with a fractal boundary in two dimensions for $Pr=1$ and $Ra\in [{10}^7,{10}^{10}]$. The fractal boundaries are functions characterized by power spectral densities $S(k)$ that decay with wavenumber, $k$, 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 $Nu\sim R{a}^{\beta }$, where $Nu$ and $Ra$ are the Nusselt and the Rayleigh numbers for $Ra\in [{10}^8,{10}^{10}]$, we observe that heat transport scaling increases with roughness over the top two decades of $Ra \in \left[10^8, 10^{10}\right]$\mathrm{.}For$p$ $=-3.0$, $-2.0$ and $-1.5$ we find $\beta = 0.288 \pm 0.005, 0.329 \pm 0.006$and$0.352 \pm 0.011$, respectively. We also observe that the Reynolds number, $Re$, scales as $Re\sim R{a}^{\xi }$, where $\xi \approx 0.57$ over $Ra\in [{10}^7,{10}^{10}]$, 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.