Nonlinear interactions between an unstably stratified shear flow and an evolving phase boundary

Well resolved numerical simulations are used to study Rayleigh-B\'enard-Poiseuille flow over an evolving phase boundary for moderate values of P\'eclet ($Pe \in \left[0, 200\right]$) and Rayleigh ($Ra \in \left[2.15 \times 10^3, 10^6\right]$) numbers. The relative effects of mean shear and buoyancy are quantified using a bulk Richardson number: $Ri_b = Ra \cdot Pr/Pe^2$, where $Pr$ is the Prandtl number. For $Ri_b \ll 1$, we find that the Poiseuille flow inhibits convective motions, resulting in the heat transport being only due to conduction. In the opposite limit of $Ri_b \gg 1$, the flow properties and heat transport closely correspond to the purely convective case. We also find that for $Ri_b = O(1)$ there is a pattern competition for convection cells with a preferred aspect ratio. Furthermore, we find travelling waves at the solid-liquid interface when $Pe \neq 0$, in qualitative agreement with other sheared convective flows in the experiments of Gilpin \emph{et al.} (\emph{J. Fluid Mech} {\bf 99}(3), pp. 619-640, 1980) and the linear stability analysis of Toppaladoddi and Wettlaufer (\emph{J. Fluid Mech.} {\bf 868}, pp. 648-665, 2019).