Transient flows of a compressible fluid in a rapidly-rotating cylinder

Abstract

Firstly, the rayleigh problem of a gas contained in an infinitely long cylinder is considered. The time-dependent flow of the initially motionless gas driven by the rapidly rotating cylindrical sidewall is described. The flow field is determined by the two principal dynamic mechanisms: the compressible Rayleigh effect and the curvature effect. Unlike the Rayleigh problem about a flat plate, the centrifugal compression causes a small radially-outward normal velocity component in a narrow region very close to the sidewall. Also, due to the boundedness of the flow domain, the curvature effect relative to the compressible Rayleigh effect decreases as E.M decreases, where E and M denote respectively the Ekman number and the Mach number. Secondly, a numerical study is made of the transient flow of a gas contained in an infinite circular cylinder which impulsively starts rotating about the central longitudinal axis with angular frequency $\Omega$. The descriptions are given of the flows at moderate and large times leading to the steady state. Attention is confined to the situations in which the reference system Ekam number E is small, and the compressibility effect is substantial. Two types of the sidewall thermal boundary condition are considered: an isothermal wall, and an adiabatic wall. The steady-state flow structure is attained over the diffusive time scale, $O(E^{-1}\Omega^{-1})$, which is orders-of-magnitudes greater than the Rayleigh time scale $O(\Omega^{-1})$. The evolutionary profiles of the temperature fields show profound differences between the two sidewall thermal conditions. The evolutions of the radial velocities show fluctuations due to the presence of the thermo-acoustic disturbances. The global spin-up process is more effective under the isothermal wall than under the adiabatic wall. Diagnostic analyses are undertaken to elucidate the dominant balances of the dynamic ingredients. Thirdly, spin-up flows of a compressible gas in a finite, closed cyl...