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We model buoyancy-driven convection with chimneys -- channels of zero solidfraction -- in a mushy layer formed during directional solidification of abinary alloy in two-dimensions. A large suite of numerical simulations iscombined with scaling analysis in order to study the parametric dependence ofthe flow. Stability boundaries are calculated for states of finite-amplitudeconvection with chimneys, which for a narrow domain can be interpreted in termsof a modified Rayleigh number criterion based on the domain width andmushy-layer permeability. For solidification in a wide domain with multiplechimneys, it has previously been hypothesised that the chimney spacing willadjust to optimise the rate of removal of potential energy from the system. Fora wide variety of initial liquid concentration conditions, we consider thedetailed flow structure in this optimal state and derive scaling laws for howthe flow evolves as the strength of convection increases. For moderatemushy-layer Rayleigh numbers these flow properties support a solute flux thatincreases linearly with Rayleigh number. This behaviour does not persistindefinitely, however, with porosity-dependent flow saturation resulting insub-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 qualitativelysimilar system dynamics and flow profiles for the optimal states.

Nonlinear mushy-layer convection with chimneys: stability and optimal solute fluxes