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Mixed baroclinic convection in a cavity

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.