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

Published online by Cambridge University Press:  26 April 2006

Ingemar A. A. Lindblad ,
Fritz H. Bark  and
Said Zahrai
Ingemar A. A. Lindblad
Affiliation:
The Aeronautical Research Institute of Sweden, S-161 11 Bromma, Sweden
Fritz H. Bark
Affiliation:
Department of Mechanics, Royal Institute of Technology, S-100 44 Stockholm, Sweden
Said Zahrai
Affiliation:
Department of Mechanics, Royal Institute of Technology, S-100 44 Stockholm, Sweden Present address: ABB Corporate Research. S-72178 Västerås, Sweden
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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.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 274 , 10 September 1994 , pp. 383 - 404
Copyright
© 1994 Cambridge University Press

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