Fluid instabilities in precessing spheroidal cavities

TLDR

Poincaré’s 1910 inviscid solution predicts uniform‑vorticity motion in precessing spheroidal cavities. We use direct numerical simulation to study the flow of an incompressible fluid in a slightly ellipsoidal precessing cavity. The study models shear arising from deviations from uniform vorticity, which triggers wave‑like instabilities propagating around the fluid’s rotation axis. Simulations reveal that boundary pressure and viscous torques reorient the average vorticity, produce axisymmetric shear layers consistent with Malkus experiments, and can destabilize Ekman layers.

Abstract

We study by direct numerical simulation the motion of incompressible fluid contained in an ellipsoid of revolution with ellipticity 0.1 or less which rotates about its axis of symmetry and whose rotation axis is executing precessional motion. A solution to this problem for an inviscid fluid given by Poincaré (1910) predicts motion of uniform vorticity. The simulations show how the orientation of the average vorticity of a real fluid is influenced by both pressure and viscous torques exerted by the boundaries. Axisymmetric shear layers appear which agree well with those observed experimentally by Malkus (1968). Shear caused by deviations from a velocity field with uniform vorticity triggers an instability consisting of waves propagating around the average rotation axis of the fluid. The Ekman layers at the boundaries may also become unstable.