  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 $Ra \in \left[10^7, 10^{10}\right]$. 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 $Nu \sim Ra^{\beta}$, where $Nu$ and $Ra$ are the Nusselt and the
Rayleigh numbers for $Ra \in \left[10^8, 10^{10}\right]$, we observe that heat
transport scaling increases with roughness over the top two decades of $Ra \in
\left[10^8, 10^{10}\right]$. 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 $Re \sim Ra^{\xi}$,
where $\xi \approx 0.57$ over $Ra \in \left[10^7, 10^{10}\right]$, 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.
