On slow transverse flow past obstacles in a rapidly rotating fluid

Abstract

This paper describes an experimental and theoretical study of the complicated disturbance (Taylor column) due to the slow relative motion between a spherical, or short cylindrical, rigid object and an incompressible fluid of low viscosity in which the object is immersed, when the motion of the object is that of steady revolution with angular speed Ω rad/see about an axis (the Z-axis) whose perpendicular distance from the centre of the object,$\overline{R}$, is much greater than a typical linear dimension of the object,L, and the undisturbed fluid motion is one of steady rotation about the same axis with angular speed (Ω+Uϑ/R) and zero relative vorticity (i.e.d(UϑR)/dR= 0). It extends earlier experimental work on Taylor columns to systems of sufficiently large axial dimensions for Z variations in the disturbance pattern to be perceptible. Over the ranges of Rossby and Ekman numbers (based onL) covered by the experiments, namely ε = 1·89 × 10−3to 2·36 ×10−1and γ = 1·30 × 10−3to 2·03 × 10−2respectively, the axis of the Taylor column is found to trail in the downstream direction at a small angle ϕ = tan−1(Kε) to the line parallel to the Z-axis through the centre of the object, whereK= (1·54 ± 0·04) for a sphere. The variation with Z of the amplitude of the disturbance is roughly linear and the scale-length of this variation, Zc, is close toL/γ¼over the limited range of γ covered by the experiments.The experimental value ofKis remarkably close to the theoretical value derived by Prof. Lighthill in the appendix, where he applies his general linear theory of waves generated in a dispersive system by travelling forcing effects to the problem of describing a Taylor column at large distances from the moving object when the fluid is inviscid and unbounded.

Laboratory experiments with tilted convective plumes on a centrifuge: a finite angle between the buoyancy force and the axis of rotation

“…In the view 'west' (top right), it is seen that the spirals tend to be roughly aligned along the axis of rotation. The departure of the spirals from the perfect alignment is consistent with the result of Hide, Ibbetson & Lighthill (1968) that the motion of a particle perpendicular to the axis of rotation produces a deflected Taylor column with the angle of deflection proportional to the Rossby number.…”

Laboratory Experiments on Eddy Generation by a Buoyant Coastal Current Flowing over Variable Bathymetry*

“…These waves would have a wavelength U/f ϳ1 cm and would propagate a horizontal distance approximately equal to the depth of the fluid (e.g., Hide et al 1968). Because the fluid depth at the slope transition ranged from 0-15 cm, it is unlikely that these waves had a strong influence on the eddies formed far downstream of the slope transition.…”

On the slow motion of a spheroid in a rotating stratified fluid

“…If N = 0, the branch cut of § 4 extends from −if to if , leaving no space for a pole on the imaginary axis between the origin and the inertial-wave branch point, so there is no longer a slow mode and we have the problem and result of Stewartson (1953). This latter solution is only a local solution and, as pointed out by Lighthill (Hide, Ibbetson & Lighthill 1968), at large heights the disturbance decomposes into an inertial wave field (Cheng & Johnson 1982). The strength of the slow mode depends on the abruptness of the initiation of the motion and it has been shown that for sufficiently slowly accelerated bodies the mode can be arbitrarily weak.…”