Axisymmetric rotating flow past a circular disk

The steady, inviscid, axisymmetric, rotating flow past a circular disk in an unbounded liquid is determined on the hypothesis that all streamlines originate in a uniform flow far upstream of the body. The characteristic parameter for the flow is k = 2ωa/U, where ω and U are the angular and axial velocities of the basic flow and α is the radius of the disk. Forward separation is found to occur for k > k = 1.9, in agreement with observation (Orloff & Bossel 1971). The length of the upstream separation bubble is determined on the hypothesis that the previous solution remains valid for k > k, despite the existence of closed streamlines within the upstream separation bubble (which may, but do not necessarily, inva,lidate the solution). This length increases rapidly for k > 3, in qualitative agreement with observation. The hypothesis of unseparated flow implies a singularity at the rim of the disk, just as in potential flow. The strength of this singularity departs only slightly from its potential-flow value for 0 ≤ k ≤ 2, but increases rapidly with k for k > 3, which suggests that (quite apart from the difficulties implied by the existence of closed streamlines) the solution cannot remain valid for sufficiently large k.

Boundary-layer separation on a sphere in a rotating flow

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 fork≡ 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) fork[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 withkto 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 fixedk, and is especially well suited for automatic computation.Anad hocincorporation 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 thatkdecreases monotonically from 2·2 to 0 as [Uscr ] increases from 0 to 1.