Stability of an air-water flow in a semispherical container

This numerical study analyzes the stability of a steady axisymmetric air-water flow driven by a rotating top disk in a sealed semispherical container. A motivation is possible applications in aerial bioreactors. The centrifugal force pushes the air to periphery near the disk, downward near sidewall, toward the axis near the interface, and upward near the axis. This meridional circulation of air drives the water counter-circulation while the centrifugal force tends to induce the water co-circulation. Their competition results in the development of a three-eddy pattern as the rotation intensifies. The air circulation and the water co-circulation are separated by a thin layer of water counter-circulation. It is shown that the time-oscillatory helical instability emerges when the three-eddy pattern is well formed. The azimuthal wave number is m=1 in the shallow-water case and m=2 otherwise. The analysis of flow patterns and critical-disturbance energy distributions indicates that the instability emerges in the air domain and likely is of the shear-layer type.