Free-surface effects on the spin-up of fluid in a rotating cylinder

Abstract

We present a theory for the decay of the relative motion of a homogenous fluid with a free surface in a rotating cylindrical tank with a flat bottom, induced by an abrupt change in the angular velocity. We then describe a set of laboratory experiments designed to test the predictions of the theory.At low rates of rotation the dynamics of the adjustment is well understood and measurements have verified the established theoretical results that the motion decays exponentially, with a timescale proportional to the rotation period divided by the square root of the Ekman number, and that the relative vorticity remains independent of radius. At higher rotation rates, however, the curvature and motion of the free surface complicate the dynamics, and have hindered the development of a more general theory.Both the theoretical predictions and the experiments show that at high rotation rates the decay of the relative vorticity is independent of radius and exponential in time, but with a decay timescale, τe, that increases linearly with the rotational Froude number F, i.e. . An analysis of the vorticity dynamics during spin-up indicates that, near the centre of the tank, this simple behaviour is the result of vigorous competition between the rate of vortex line stretching by Ekman-layer pumping and surface deformation. Near the boundary, these mechanisms cooperate, but are partially offset by the stretching produced by the secondary radial circulation.