Spin-up of a rapidly rotating heavy gas in a thermally insulated annulus

Abstract

The linear spin-up problem for a rapidly rotating viscous diffusive ideal gas is considered in the limit of vanishing Ekman number E. Particular attention is given to gases having a large molecular weight. The gas is enclosed in a cylindrical annulus, with flat top and bottom walls, which is rotating around its axis of symmetry with rotation rate Ω. The walls of the container are adiabatic. In a rotating gas (of any molecular weight), the Ekman layers on adiabatic walls are weak, which implies that there is no distinct non-diffusive response of the gas outside the Ekman and Stewartson boundary layers on the timescale E−1/2Ω−1 for spin-up of a homogeneous fluid. For the case of adiabatic walls, it is shown that the spin-up mechanisms due to viscous diffusion and Ekman suction are, from a formal point of view, equally strong. Therefore, the gas will adjust to the increased rotation rate of the container on the diffusive timescale E−1Ω−1. However, if E1/3 [Lt ] γ – 1 [Lt ] 1 and M [siml ] 1, which characterizes rapidly rotating heavy gases (where γ is the ratio of specific heats of the gas and M the Mach number), it is shown that the gas spins up mainly by Ekman suction on the shorter timescale (γ–1)2E−1Ω−1. In such cases, the interior motion splits up into a non-diffusive part of geostrophic character and diffusive boundary layers of thickness (γ – 1) outside the Ekman and Stewartson layers. The motion approaches the steady state of rigid rotation algebraically instead of exponentially as is usually the case for spin-up.

Energy transfer in the transient Ekman layer of a compressible fluid

“…This finding is in line with the results of preceding studies (e.g. Matsuda & Hashimoto 1976;Bark & Hultgren 1979;Lindbald et al 1994). Furthermore, the present discussion reinforces the usefulness of the energy flux content e in deducing physical rationalizations of the flow behaviour.…”

“…This important flow character has been pointed out, in a piecemeal fashion, in a few studies (e.g. Riley 1967;Sakurai & Matsuda 1974;Matsuda & Hashimoto 1976;Harada 1979;Hyun & Park 1992;Lindbald, Bark & Zahrai 1994).…”

Ekman pumping in compact astrophysical bodies

“…The spin-up of a compressible fluid has been studied in the context of rapidly rotating gases (Sakurai & Matsuda 1974;Bark, Meijer & Cohen 1978;Lindblad, Bark, & Zahrai 1994). These studies concluded that the basic structure for spin-up remained unchanged, but with some important and interesting differences.…”

SPIN-UP OF HOMOGENEOUS AND STRATIFIED FLUIDS

“…We also exclude from our survey the important area of the spin-up in a mixture of two fluids (see, for example, Amberg & Ungarish 1993). Finally, the considerable literature on unsteady processes in gas centrifuges (Hyun & Park 1992, Lindblad et al 1994 is also specifically excluded from our discussion.…”