  We model buoyancy-driven convection with chimneys -- channels of zero solid
fraction -- in a mushy layer formed during directional solidification of a
binary alloy in two-dimensions. A large suite of numerical simulations is
combined with scaling analysis in order to study the parametric dependence of
the flow. Stability boundaries are calculated for states of finite-amplitude
convection with chimneys, which for a narrow domain can be interpreted in terms
of a modified Rayleigh number criterion based on the domain width and
mushy-layer permeability. For solidification in a wide domain with multiple
chimneys, it has previously been hypothesised that the chimney spacing will
adjust to optimise the rate of removal of potential energy from the system. For
a wide variety of initial liquid concentration conditions, we consider the
detailed flow structure in this optimal state and derive scaling laws for how
the flow evolves as the strength of convection increases. For moderate
mushy-layer Rayleigh numbers these flow properties support a solute flux that
increases linearly with Rayleigh number. This behaviour does not persist
indefinitely, however, with porosity-dependent flow saturation resulting in
sub-linear growth of the solute flux for sufficiently large Rayleigh numbers.
Finally, we consider the influence of the porosity dependence of permeability,
with a cubic function and a Carmen-Kozeny permeability yielding qualitatively
similar system dynamics and flow profiles for the optimal states.
