Separation and free-streamline flows in a rotating fluid at low Rossby number

Abstract

The flow past a circular cylinder in a rotating frame is examined when the Rossby number Ro is O(E½), where E is the Ekman number. Previous studies of the configuration have shown that, provided the ratio Ro/E½ is less than a certain critical value, the flow around the cylinder is determined by the classical potential-flow solution. However, once Ro/E½ is greater than that critical value the E1/4 layer on the surface of the cylinder, which is rather like a boundary layer in a high-Reynolds-number non-rotating fluid, can separate from the cylinder and distort the potential flow. In this study the form of the flow once separation has occurred is examined using a method analogous to the Kirchhoff free-streamline theory in a non-rotating fluid. The results are compared with published experimental and numerical data on the flow for various values of Ro/E½.

The breakdown of jet flows in a low-Rossby-number rotating fluid

“…Whether this occurs depends crucially upon the initial momentum flux of the jet and, modelling it with a Bickley-type source of strength J, it is a simple matter to show that both the length of the jet and its entrained flux are multiplied by a factor of ()/ 2 when J $ 1. As a result, the distance before a singularity is encountered in the flow described above would be equal to A(J) 11 /2 which, when the obstacle is a circular cylinder, would be greater than the length of the separated region (see Page [3]) if J is larger than about one. Since much of the momentum of the shear layer would be lost by the time the reattachment point is reached, this would be unlikely, although it would need to be verified by more-detailed boundary-layer calculations for the separated shear layer.…”

“…One particular case where the results of this study can be applied immediately is to the proposal for the structure of separated flow past a cylindrical object in Page [3], where the lower part of the separated shear layer is assumed to be turned by 180°, forming into a jet along the axis of symmetry, in the manner of Smith [20]. Although this jet would be singular at its initial point, in the sense that its curvature is unlikely to be finite on the line of symmetry, it can be expected that it would undergo the same process as described in this paper, with its centreline velocity decreasing to zero before it reaches the rear of the cylinder.…”

“…Here, represents the scale thickness of the viscous shear layers in these flows, known as E 1 14 layers, and the final term contributes to the leading-order motion only in these regions. Elsewhere (2.10) implies that vorticity is removed from the fluid through the action of the Ekman layers, as for the inviscid flows considered by Page [3].…”

“…Phenomena such as separation (Walker and Stewartson [1], Page [2,3]) and wake formation (Page [4]) occur through similar mechanisms to their non-rotating counterparts, although there are some novel effects which arise through the effect of suction from the Ekman layers in these flows, where vortex stretching leads to the removal of vorticitity from the flow. Examples of this are the finite length of the separated region behind a cylindrical object, described in Page [3,5], and the rapid exponential decay of the vorticity in the wake of an obstacle. These effects have been noted in the experimental results of Boyer [6] and Boyer and Davies [7], and in the numerical calculations by Matsuura and Yamagata [8] and Becker [9].…”

“…Both the asymptotic structure and the resolution of the singularity are, however, applicable to the expected breakdown of any form of jet in rotating fluid under similar conditions. The consequences of this are discussed, particularly in relation to the separated-flow structure proposed for motion past a cylindrical obstacle in Page [3].…”