On the stability of boundary-layer flows over rotating spheroids

We study linear convective instabilities within of the boundary-layer flows over spheroids rotating in otherwise still fluids. Particular spheroids within the prolate and oblate families are considered, each characterized by an eccentricity parameter, . Viscous and streamline-curvature effects are included and local analyses conducted at latitudes between 10°–70° from the axis of rotation. Both travelling and stationary convective modes of type I (crossflow) and type II (streamline curvature) are found at each latitude within specific parameter spaces. The results of existing rotating-sphere investigations are reproduced at all latitudes in the limit of zero eccentricity.

In the prolate case, eccentricity is found to have a stabilizing effect on the type I mode at all latitudes and a destabilizing effect on the type II mode at latitudes above 50°. Eccentricity is therefore seen to be a destabilizing influence for low rotation rates (where instability occurs at high latitudes only and the type II mode dominates) and a stabilizing effect for high rotation rates (where instability occurs closer to the pole and the type I mode dominates). This effect is associated with the behavior of the local curvature of the prolate geometry as a function of eccentricity. In the oblate case, eccentricity is found to be universally stabilizing to both mode types at all latitudes. The oblate results demonstrate considerably lower sensitivity to eccentricity than the prolate results.

The most amplified modes are found to travel at 75% of the surface speed at each latitude and eccentricity for both spheroid families. This is consistent with existing theoretical studies of boundary-layer flows over the related geometries of the rotating-disk, -sphere and -cone, and leads to the prediction of slow vortices over highly-polished bodies.