# Flow Past a Circular Cylinder on a $\beta$-Plane

D.L. Boyer, P. A. Davies

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### Abstract

With a view to obtaining a fuller understanding of the interactions between topography and large-scale geophysical flows, a series of laboratory investigations have been performed on the flow past a right circular cylinder in a rotating water channel. For large-scale flows on a spherical Earth the variation of the Coriolis parameter, F = 2$\Omega$ sin $\phi$, with latitude, $\phi$, is commonly written (Pedlosky 1979) as F = f + $\beta _{0}$y where f = 2$\Omega$ sin $\phi _{0}$, $\beta _{0}$ = 2$\Omega$ cos $\phi _{0}$/R$_{\text{E}}$, y is the distance to the north from the reference latitude $\phi _{0}$, and R$_{\text{E}}$ and $\Omega$(= 7.29 $\times$ 10$^{-5}$ s$^{-1}$) are the radius and rotation rate of the Earth respectively. In this paper we shall discuss laboratory experiments in which the variation of F can be simulated. We shall refer to those studies in which $\beta$ = 0 (i.e. the Coriolis parameter is uniform over the latitudinal extent of the region under investigation) as f-plane experiments. Models for which $\beta _{0}$ is non-zero will be referred to as $\beta$-plane experiments. In the experiments the $\beta$-effect has been simulated by tilting the upper and lower surfaces of the channel so that the depth of the fluid varies in the cross-stream direction. Flow patterns have been obtained over a range of five independent non-dimensional parameters: Rossby and Ekman numbers, cylinder aspect ratio, $\beta$-parameter and flow direction (eastward' or westward'). A dramatic difference in downstream behaviour is found between f-plane, $\beta$-plane westward and $\beta$-plane eastward flows. In particular, the $\beta$-plane eastward flows are characterized by bunching and pinching of streamlines in the wake region, the generation of damped stationary Rossby waves and downstream acceleration. Compared with f-plane flows the $\beta$-effect is shown to inhibit boundary layer separation from the cylinder for eastward flow and to enhance the separation for westward flow. Data are presented from all cases to show the asymmetry of the downstream flows and the transitions from fully attached to unsteady flows. Under otherwise identical conditions the downstream extent of the separated-bubble region is much greater for $\beta$-plane westward flow than, in turn, for f-plane and $\beta$-plane eastward flows. In addition, the data indicate that the size of the bubble increases with increasing Rossby number and decreases with increasing Ekman number and cylinder aspect ratio. For eastward flow the bubble size decreases with increasing $\beta$-parameter and for westward flow it increases with increasing $\beta$-parameter. Unsteady flows are investigated and instances of asymmetrical vortex shedding are presented.
Original language English 533-556 24 Philosophical Transactions of the Royal Society A - Mathematical Physical and Engineering Sciences 306 1496 https://doi.org/10.1098/rsta.1982.0094 Published - 15 Oct 1982