Unsteady flow in a rotating torus after a sudden change in rotation rate

Abstract

We consider the temporal evolution of a viscous incompressible fluid in a torus of finite curvature; a problem first investigated by Madden & Mullin (J. Fluid Mech., vol. 265, 1994, pp. 265-217). The system is initially in a state of rigid-body rotation (about the axis of rotational symmetry) and the container's rotation rate is then changed impulsively. We describe the transient flow that is induced at small values of the Ekman number, over a time scale that is comparable to one complete rotation of the container. We show that (rotationally symmetric) eruptive singularities (of the boundary layer) occur at the inner or outer bend of the pipe for a decrease or an increase in rotation rate respectively. Moreover, on allowing for a change in direction of rotation, there is a (negative) ratio of initial-to-final rotation frequencies for which eruptive singularities can occur at both the inner and outer bend simultaneously. We also demonstrate that the flow is susceptible to a combination of axisymmetric centrifugal and non-axisymmetric inflectional instabilities. The inflectional instability arises as a consequence of the developing eruption and is shown to be in qualitative agreement with the experimental observations of Madden & Mullin (1994). Throughout our work, detailed quantitative comparisons are made between asymptotic predictions and finite-(but small-) Ekman-number Navier-Stokes computations using a finite-element method. We find that the boundary-layer results correctly capture the (finite-Ekman-number) rotationally symmetric flow and its global stability to linearised perturbations.

On the formation of axial corner vortices during spin-up in a cylinder of square cross-section

“…For the BDF2 method, this composite solution is assumed at times of t = ∆t and t = 2∆t (where ∆t is the computational time step, typically 0.005) then computation determines the subsequent evolution for t = n∆t, n = 3, 4, · · · . This approach has been used in other comparable impulsive boundary-layer problems with some success, such as Hewitt et al (2011). Spatial adaptivity is employed to accurately resolve the fine near-wall structures and results presented herein have been confirmed to be converged by repeated computation over different spatial-temporal refinements.…”

The effect of seam imperfections on the unsteady flow within a fluid-filled torus

“…Eruptions resulting from finite-time singularities can also occur at both inner and outer radii, albeit at different times, if the direction of rotation is changed impulsively. Hewitt et al (2011) provided strong evidence that this finite-time singularity coincided with the development of an observable band in the top-down flow visualisation experiments of Madden & Mullin (1994).…”

“…For example, when the finite-time singularity manifests at point A in the Madden & Mullin (1994) experiment, a second laminar band is also observed at point B. Hewitt et al (2011) showed that the boundary layers on the opposite equator were subject to an axisymmetric centrifugal instability that could be triggered by an isolated wall perturbation. Furthermore, for typical laboratory flows, it was found that the time scales of this centrifugal instability and the boundary-layer eruption were similar.…”

“…Many of the experimentally observed features of this flow were subsequently explored using a mix of computational and asymptotic methods by Hewitt et al (2011) (also building upon further experimental work by del Pino et al 2008). In particular, Hewitt et al (2011) demonstrated that the boundary layers that develop within the flow, as a result of the impulsive change in rotation rate, collide and erupt as a result of a finite-time singularity.…”

“…In particular, Hewitt et al (2011) demonstrated that the boundary layers that develop within the flow, as a result of the impulsive change in rotation rate, collide and erupt as a result of a finite-time singularity. The position at which this phenomenon occurs depends upon the initial and final angular velocity.…”