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+U_ϑ/R) and zero relative vorticity (i.e. d(U_ϑR)/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 on L) covered by the experiments, namely ε = 1·89 × 10⁻³ to 2·36 ×10⁻¹ and γ = 1·30 × 10⁻³ to 2·03 × 10⁻² respectively, 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, where K = (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, Z_c, is close to L/γ^¼ over the limited range of γ covered by the experiments.

The experimental value of K is 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.