The unsteady flow due to an impulsively rotated sphere

Abstract

We consider the flow induced by a sphere, contained in an otherwise quiescent body of fluid, that is suddenly imparted with angular momentum. This classical problem is known to exhibit a finite-time singularity in the boundary-layer equations, due to the viscous boundary layer, induced by the sudden rotation, colliding at the sphere's equator. We consider this flow from the perspective of the post-collision dynamics, showing that the collision gives rises to a radial jet headed by a swirling toroidal starting vortex pair. The starting vortex propagates away from the sphere and, in doing so, loses angular momentum. The jet, in turn, develops an absolute instability which propagates back towards the sphere's equator. The starting vortex pair detaches from the jet and expands as a coherent (non-swirling) toroidal vortex pair. We also present results of some new experiments which show good qualitative agreement with our computational results.

On the non-parallel instability of the rotating-sphere boundary layer

“…The steadyflow solution has been improved significantly from the original formulation proposed by Howarth (1951) and solved by Banks (1965). It now better accounts for the boundarylayer eruption at the equator (Calabretto et al 2015) and also the inclusion of a number of viscous terms that were previously neglected in the Howarth and Banks solutions.…”

“…with ξ = β/δ (Riley 1962;Calabretto et al 2015). It should be noted that the characteristic variable ξ is constant in the radial direction †.…”

“…However, the determination of the coefficients of the series expansion becomes increasingly challenging as Re s is increased due to the growing importance of the non-linear interactions between the terms of the series. Numerical solutions of the rotating-sphere problem became recently available (see for instance Calabretto et al 2015) but the simulation of the flow is increasingly challenging as the Reynolds number is increased, since a boundary layer is formed near the sphere surface. Instead of seeking a solution of the full Navier-Stokes equations, a feasible alternative is provided by boundary-layer theory, giving a leading description of what happens close to the surface where the angular momentum transfer takes place.…”

“…The experiments of Bowden & Lord (1963); Hollerbach et al (2002); Calabretto et al (2015) and the numerical simulations of Dennis et al (1980) and Dennis & Duck (1988) suggest a more complicate structure where the two boundary layers coming from the poles collide close to the equator, generating a boundary-layer eruption that subsequently becomes a radial jet (Squire 1955;Riley 1962;Schwarz 1963). Stewartson (1958) proposed that the impinging region could be decomposed as an inviscid zone and as a very thin viscous region close to the equator plane and to the sphere surface, and described by planar Navier-Stokes equations.…”

“…All previous boundary-layer studies use the model due to Howarth (1951) and Banks (1965) in which no account is made for this eruption (and also a number of additional viscous effects are neglected) in the steady flow, and here we present a more sophisticated model. For a detailed study of the equatorial jet resulting from the eruption, see Calabretto et al (2015).…”