The Taylor-Proudman column in a rapidly-rotating compressible fluid I. energy transports
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DOI 10.1007/s12206-014-0922-8
The Taylor-Proudman column in a rapidly-rotating compressible fluid I. energy transports† Jun Sang Park* School of Mechanical and Automotive Engineering, Halla University, Wonju, Kangwon-do, 220-712, Korea (Manuscript Received October 13, 2013; Revised June 11, 2014; Accepted June 24, 2014) ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Abstract A theoretical study is made of the steady flow of a compressible fluid in a rapidly rotating finite cylinder. Flow is generated by imposing mechanical and/or thermal disturbances at the rotating endwall disks. Both the Ekman and Rossby numbers are small. An examination is made of the energy budget for a control volume in the Ekman boundary layer. A combination of physical variables, which is termed the energy flux content, consisting of temperature and modified angular momentum, emerges to be relevant. The distinguishing features of a compressible fluid, in contrast to those of an incompressible fluid, are noted. A plausible argument is given to explain the difficulty in achieving the Taylor-Proudman column in a compressible rotating fluid. For the Taylor-Proudman column to be sustained, in the interior, it is shown that the net energy transport between the solid disk wall and the interior fluid should vanish. Physical rationalizations are facilitated by resorting to the concept of the afore-stated energy flux content. Keywords: Compressible rotating flow; Ekman number; Energy flux; Stewartson layer; Taylor-Proudman column ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
1. Introduction Flow of a fluid in a finite, closed cylinder (radius R∗ [≡ r H ∗ ], height H ∗ ), which rotates steadily about its longitudinal axis, has posed a classical problem. For most applications, the system Ekman number E[≡ μ/(ρ∗ Ω∗ H ∗ )]], where μ ∗ denotes the viscosity of fluid, ρ the reference density of ∗ fluid at the cylinder wall, Ω the representative rotation rate of the cylinder] is very small. When all the components of the solid walls of the cylindrical container rotate in unison, the fluid is in rigid-body rotation with rotation rate Ω∗ , and, therefore, no internal flows in the rotating frame exist. However, if there is difference ΔΩ∗ between the rotation rates of the components of the container walls, flows are generated. The departure from the rigid-body rotation is gauged by the Rossby number ε[≡ ∆Ω∗ /Ω∗ ]. Attention is focused to the practically interesting cases of a rapidly-rotating cylinder, E ≪ 1, ε ≪ 1, and studies have described salient characteristics of flows in the
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