Michael A. Page

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

The flow past a cylinder in a rapidly rotating frame is described when the Rossby number Ro is O(E ½ ), where E is the Ekman number. Previous studies of the configuration have noticed the development of a singularity within the E ¼ layer at the rear stagnation point once the ratio Ro / E ½ is larger than a critical value, and concluded that the boundary-layer flow is unsteady. In this paper a description of a steady boundary-layer flow for this parameter range is presented, showing the development of flow separation as Ro / E ½ approaches a larger critical value. Details of the flow once the E ½ layer has separated from the cylinder are also described.

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

The flow past a circular cylinder in a rotating frame is examined when the Rossby number Ro is O ( E ½ ), where E is the Ekman number. Previous studies of the configuration have shown that, provided the ratio Ro / E ½ is less than a certain critical value, the flow around the cylinder is determined by the classical potential-flow solution. However, once Ro / E ½ is greater than that critical value the E 1/4 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 / E ½ .

Flow past a circular cylinder on a β-plane

This paper gives analytical and numerical solutions for both westward and eastward flows past obstacles on a beta-plane. The flows are considered in the quasi-geostrophic limit where nonlinearity and viscosity allow deviations from purely geostrophic flow. Asymptotic solutions for the layer structure in almost-inviscid flow are given for westward flow past both circular and more elongated cylindrical obstacles. Structures are given for all strengths of nonlinearity from purely linear flow through to strongly nonlinear flows where viscosity is negligible and potential vorticity conserved. These structures are supported by accurate numerical computations. Results on detraining nonlinear western boundary layers and corner regions in Page & Johnson (1991) are used to present the full structure for eastward flow past an obstacle with a bluff rear face, completing previous analysis in Page & Johnson (1990) of eastward flow past obstacles without rear stagnation points. Viscous separation is discussed and analytical structures proposed for separated flows. These lead to predictions for the size of separated regions that reproduce the behaviour observed in experiments and numerical computations on beta-plane flows.

Flow separation in a rotating annulus with bottom topography

The flow in a rotating annular cylinder, of finite depth, is examined when the Rossby number R o is O ( E ½ ), where E is the Ekman number, and when there is a topography of height O ( E ½ ) on the base of the container. The flow, relative to the rigid axial rotation, is forced by differential rotation of the lid and as it moves over the topography the streamlines are deflected parallel to the bottom surface. This induces O (1) velocity variations near the axial walls of the annulus to which the boundary layers there, of thickness O ( E ¼ ), respond. For sufficiently large values of a parameter γ ∝ R o/ E ½ the skin friction can vanish within these layers, with some similaritits to boundary-layer separation in a non-rotating fluid. In this study the interior flow, with horizontal viscous diffusion neglected, is calculated and used to provide a boundary condition for the, E ¼ layer flow. Once λ exceeds a finite critical value a singularity is encountered in the boundary layer corresponding to flow separation from the wall. This demonstrates that E ¼ layers in a rotating fluid, which for R o = 0 have little direct influence on the interior flow, can modify the gross properties of the flow for non-zero Rossby numbers, a conclusion also reached by Walker & Stewartson (1972) in a different context.

Flow separation and unsteadiness in a rotating sliced cylinder

Abstract The rotating sliced cylinder model has its origins in simulations of the large scale wind-driven ocean circulation. The model consists of a rotating circular cylinder with a sloping planar base and a lid which rotates at a rate slightly different from that of the background rotation. Numerical models for the inviscid flow, the boundary-layer flow and the viscous flow equations in this configuration are described and the flow patterns produced by them are compared. Laboratory experiments were also performed and the results are presented as validation of the numerical models. The unsteadiness of the flow in some parameter regimes which was noted by other authors (Beardsley, 1969, 1973a, b and Beardsley and Robbins, 1975) appears to be caused by instability of the separated boundary-layer flow. The separated flow has some unique features due to the effect of the driving force exerted on the flow by the rotating lid and these are also described.

A numerical study of detached shear layers in a rotating sliced cylinder

Abstract The low Rossby number flow in a rotating cylinder with an inclined bottom, of small slope, is examined when part of the lid of the container is rotating at a slightly different rate. The resulting flow is calculated numerically by solving the governing equations for the two-dimensional geostrophic motion which approximates the flow in most of the fluid including the inertially-modified E ¼ -layers. The presence of ageostrophic regions, on the container walls and beneath the velocity discontinuity on the lid, is accounted for in the governing equations and their boundary conditions. This study supplements previous work on this configuration, in which the zero Rossby number flow was calculated and experimental results were presented, by enabling a direct comparison to be made between the results of the low Rossby number theory and the experiments. The numerical results for a range of Rossby and Ekman numbers compare well with those from the experiments despite a severe limitation on the size of the...

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

Low-Rossby-number flow past a circular cylinder in a rapidly rotating frame is studied when 1 N N is equal to E ½ / Ro in terms of the Ekman number E and Rossby number Ro. For this parameter range the E ¼ 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.

Nonlinear western boundary current flow near a corner

Abstract This paper analyses a western boundary current striking a solid boundary. Interest is concentrated on the case where inertial effects are sufficient to modify the flow from its ‘Stommel-layer’ form with Ekman friction relatively unimportant in the interior. It is shown that, beyond a critical inflow speed, a complicated system of four asymptotic regions forms near the corner, turning some of the flow along the blocking boundary and then returning it westwards to rejoin the western boundary current. A comparison with the results of a simple ocean model shows that many of the features in that flow can be explained through the asymptotic theory.

The structure of separated flow past a circular cylinder in a rotating frame

Abstract This paper examines the detailed E 1/4-layer structure of separated flow past a circular cylinder in a low-Rossby-number rotating fluid as the Ekman number E tends to zero. This structure is based on an initial proposal by Page (1987) but with some modifications in response to further evidence, outlined both in this paper and elsewhere, on the behaviour of E 1/4-layer flows in this context. Numerical calculations for flow in an E 1/4 shear layer along the separated free streamline are described and the mass flux from this layer is then used to calculate the higher-order flow within the separation bubble. The flow structure is found to have two forms, depending on the value of the O(1) parameter λ, and these are compared with results from published “Navier-Stokes” type calculations for the flow at small but finite values of E.

Combined diffusion-driven and convective flow in a tilted square container

Asymptotic analytical solutions are derived for the configuration posed by Quon [Phys. Fluids 26, 632 (1983)] for diffusion-driven flow in a tilted square container when the diffusive parameter R is small. The key regions of the asymptotic structure are outlined and the leading-order solutions are determined in most of those regions. The analysis follows that in Page and Johnson [J. Fluid Mech. 629, 299 (2009)] and Page [Q. J. Mech. Appl. Math. (in press)] but includes an additional “R1/4-layer” region. Analytical solutions are compared with numerical results for small R and display excellent agreement. It is also shown that the solutions and flow structure are applicable over a wide range of Prandtl numbers.