On the rotating-fluid flow near the rear stagnation point of a circular cylinder

Abstract

Low-Rossby-number flow past a circular cylinder in a rapidly rotating frame is studied when 1 < N < 2, where N is equal to E½/Ro in terms of the Ekman number E and Rossby number Ro. For this parameter range the E¼ boundary layer contains a singularity at the rear stagantion point. The asymptotic structure of this singularity is shown to consist of three distinct asymptotic regions, one of which is viscous while the others are inviscid. New accurate numerical solutions of the boundary-layer equation confirm this singularity structure. The use of Von Mises coordinates both simplifies the analysis, and enables numerical solutions to be found closer to the critical value N = 1, beneath which the flow separates upstream of the rear stagnation point.

On the unsteady low-Rossby number flow of a rotating fluid past a circular cylinder

“…Also the flow field is altered significantly as N increases from below to above unity and we expect this transition to have a similar effect in the unsteady case. In fact, many analogies can be drawn from the steady solution proposed by Page and Cowley 31 for 1рNϽ2 close to the rear stagnation point, with the unsteady solution proposed in the present work for the same range of N. In both cases the boundary layer consists of an inner, viscous region, along with either one ͑in the unsteady case͒ or two ͑in the steady case͒ inviscid outer regions and the analysis involves consideration of an exponentially small viscous edge layer. This similarity between the steady and unsteady cases is discussed in more detail in Sec.…”

“…The above condition will be satisfied for a range of values of 0ϽxϽ, if Nр1 but flow reversal in the usual sense cannot occur for NϾ1. Leibovich In summary, noting that Nϭ0 corresponds to the nonrotating case, as N increases the flow field for the steady case passes through the following regimes 0рNϽ1: Similar to the non-rotating case, with separation occurring away from the rear stagnation point, e.g., Page, 35 Becker, 36 Page and Duck;37 1рNϽ2: The solution becomes singular at the rear stagnation point and forms three asymptotic regions, e.g., Page, 34 Page and Cowley;31 2рNϽ3: The displacement thickness becomes infinite at the rear stagnation point and forms two asymptotic regions, e.g., Page and Cowley;31 Nу3: There exists a steady non-singular solution in the boundary layer, e.g., Leibovich,28 Buckmaster. 29,30 Little attention has been paid to the parameter regime 2ϽNр3.…”

“…In the study of two-dimensional flow of an electrically conducting viscous fluid past a circular cylinder in the presence of a magnetic field normal to the cylinder wall ͑for example Leibovich 28 and Buckmaster 29,30 ͒, two uncoupled equations govern the flow, the first of which is identical to the governing equation in our problem. Following the analysis for the steady case by Page and Cowley,31 and that for the magneto-hydrodynamic context, we use the parameter N in this paper ͑see Sec. II for a definition͒ to measure the relative importance of Coriolis effects on the boundary layer flow.…”

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

“…3 for sech 2 profiles. In particular, the decrease in the velocities in the jet in proportion to ( -x) and the increase in the thickness of the layer are reminiscent of features of the flow near the rear stagnation point of a circular cylinder, described in Page [18] and Page and Cowley [19], suggesting that the flow in the vicinity of 7s may be predominantly inviscid. As a result, a similar technique to that used in Page and Cowley [19] is used again here, transforming the inviscid form of (2.14) into Von Mises coordinates (, 5q) so that the flow near ix satisfies…”

“…1. In that case it can be expected that there will be an additional viscous layer close to y7= 0 on the approach to the singularity, rather like the viscous layer in Page [18] and Page and Cowley [19].…”