Steady flow in a rapidly rotating spheroid with weak precession: Part 2

Abstract

Abstract This is a continuation of our companion paper (Kida 2020b Fluid Dyn. Res.) in which the steady flow in a precessing spheroid is studied in the strong spin and weak precession limit, and the elliptic flow with uniform vorticity, the direction of which is perpendicular to the spin axis on the plane spanned by the spin and the precession axes, is shown to be excited in the interior inviscid region. In this paper we examine the boundary-layer flow connected to this elliptic flow and also the higher-order inviscid flow modified by this boundary-layer flow. The explicit expression of the velocity and pressure fields is obtained for arbitrary aspect ratio c of the polar and equatorial radii of the spheroid. The boundary-layer approximation breaks down at two critical circles on which the boundary-layer thickness as well as the normal component of velocity diverge to infinity. The flow field in the neighborhood (called the critical regions) of the critical circles is analyzed to find the same structure, up to scales, as that for a sphere. The radial component of velocity in the critical region induces the conical shear layers along the characteristic cones in the inviscid region. They reflect on the spheroid surface, and the reflection takes place endlessly in general. On the other hand, it is proved that for a spheroid of which the aspect ratio is expressed as c = 3 cot [ ( π / 2 ) ( m / n ) ] (m/n being an irreducible fraction) the reflection terminates at (n − 2) times and the conical shear layers form a regular pattern. Since such special values of c are distributed densely over all the positive numbers, we may observe practically always a regular pattern of the conical shear layers. Such closed conical shear layers are clearly visualized by the pressure field numerically calculated in the asymptotic formulation. Furthermore, the total angular momentum of the steady flow is expressed explicitly up to the non-trivial leading order for all the three components over the whole range of the aspect ratio.