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Boundary-layer separation on a sphere in a rotating flow

Published online by Cambridge University Press:  29 March 2006

John W. Miles
John W. Miles
Affiliation:
Institute of Geophysics and Planetary Physics and Department of Aerospace and Mechanical Engineering Sciences University of California, La Jolla
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Abstract

The velocity just outside the boundary layer and upstream of the separation ring on a sphere moving along the axis of a slightly viscous, rotating fluid is calculated through a least-squares approximation on the hypothesis of no upstream influence. A reverse flow is found in the neighbourhood of the forward stagnation point for k ≡ 2Ωa/U > k = 2·20 (Ω = angular velocity of fluid, U = translational velocity of sphere, a = radius of sphere) and is accompanied by a forwardseparation bubble, such as that observed by Maxworthy (1970) for k [gsim ] 1. Rotation also induces a downstream shift of the peak velocity; the estimated shift of the separation ring in the absence of forward separation increases with k to a maximum of 24°, in qualitative agreement with Maxworthy's observations.

The least-squares formulation is compared with that given by Stewartson (1958) for unseparated flow (Stewartson did not consider separation). Both formulations require truncation of an infinite set of simultaneous equations, but Stewartson's formulation yields a non-positive-definite matrix that may exhibit spurious singularities. The least-squares formulation yields a positive-definite matrix, albeit at the expense of slower convergence for fixed k, and is especially well suited for automatic computation.

An ad hoc incorporation of a cylindrical wave of strength [Uscr ], such that the maximum upstream axial velocity is [Uscr ]U, is considered in an appendix. It is found that k decreases monotonically from 2·2 to 0 as [Uscr ] increases from 0 to 1.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 45 , Issue 3 , 15 February 1971 , pp. 513 - 526
Copyright
© 1971 Cambridge University Press

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