  We study the convective patterns that arise in a nearly semi-cylindrical
cavity fed in with hot fluid at the upper boundary, bounded by a cold, porous
semi-circular boundary at the bottom, and infinitely extended in the third
direction. While this configuration is relevant to continuous casting processes
that are significantly more complex, we focus on the flow patterns associated
with the particular form of mixed convection that arises in it. Linear
stability analysis and direct numerical simulations (DNS) are conducted, using
the spectral element method to identify observable states. The nature of the
bifurcations is determined through Stuart--Landau analysis for completeness.
The base flow consists of two counter-rotating rolls driven by the baroclinic
imbalance due to the curved isothermal boundary. These are however suppressed
by the through-flow, which is found to have a stabilising influence as soon as
the Reynolds number $Re$ based on the through-flow exceeds 25. For a
sufficiently high Rayleigh number, this base flow is linearly unstable to three
different modes, depending on $Re$. For $Re\leq75$, the rolls destabilise
through a supercritical bifurcation into a travelling wave. For $100\leq Re
\leq 110$, a subcritical bifurcation leads to a standing oscillatory mode,
whereas for $Re\geq150$, the unstable mode is non-oscillatory and grows out of
a supercritical bifurcation. The direct numerical simulations confirm that in
all cases, the dominant mode returned by the linear stability analysis
precisely matches the topology and evolution of the flow patterns that arise
out of the fully nonlinear dynamics.
