Detached shear layers in a rotating fluid

Abstract

The occurrence of detached shear layers should, according to straightforward theoretical arguments, often characterize hydrodynamical motions in a rapidly rotating fluid. Such layers have been produced and studied in a very simple system, namely a homogeneous liquid of kinematical viscosity v filling an upright, rigid, cylindrical container mounted coaxially on a turn-table rotating at Ω0 rad/s about a vertical axis, and stirred by rotating about the same axis at Ω1 rad/s a disk of radius a cm and thickness b’ cm immersed in the liquid with its plane faces parallel to the top and bottom end walls of the container. By varying Ω0, Ω1 and a, ranges of Rossby number, the modulus of ε ≡ (Ω1 + Ω0)/½ (Ω1 + Ω0), from 0·01 to 0·3, and Ekman number, E ≡ 2v/a2(Ω1 + Ω0), from 10−5 to 5 × 10−4 were attained. Although the apparatus was axisymmetric, only when |ε| did not exceed a certain critical value, |εT|, was the flow characterized by the same property of symmetry about the axis of rotation. Otherwise, when |ε| > |εT|, non-axisymmetric flow occurred, having the form in planes perpendicular to the axis of rotation of a regular pattern of waves, M in number, when ε was positive, and of a blunt ellipse when ε was negative.The axial flow in the axisymmetric detached shear layer, and the uniform rate of drift of the wave pattern characterizing the non-axisymmetric flow when ε is positive, depend in relatively simple ways on ε and E. The dependence of|εT| on E can be expressed by the empirical relationship |εT| = AEn, where A = 16·8 ± 2·2 and n = 0·568 ± 0·013 (= (4/7) × (1·000 − (0·005 ± 0·023))!), standard errors, 25 determinations. M does not depend strongly on E but generally decreases with increasing ε.