The bathtub vortex in a rotating container

Abstract

We study the time-independent free-surface flow which forms when a fluid drains out of a container, a so-called bathtub vortex. We focus on the bathtub vortex in a rotating container and describe the free-surface shape and the complex flow structure using photographs of the free surface, flow visualizations, and velocity measurements. We find that the velocity field in the bulk of the fluid agrees with predictions from linear Ekman theory for the boundary layer at the bottom, and we discuss the limitations of linear Ekman theory for the source-sink flow in the experiment. We introduce a radial expansion approximation of the central vortex core and reduce the model to a single first-order equation. We solve the equation numerically and find that the axial velocity depends linearly on height whereas the azimuthal velocity is almost independent of height. We discuss the model of the bathtub vortex introduced by Lundgren (J. Fluid Mech. vol. 155, 1985, p. 381) and compare it with our experiment. We find that the measured velocities and surface profiles are described well by the model when Ekman upflow and surface tension effects are included.

Stationary bathtub vortices and a critical regime of liquid discharge

“…This modified model designated as the LABSRL-model, was applied to the interpretation of experimental whirlpool data obtained in a rotating cylindrical vessel with water circulating at a given flow rate. By adjusting the two fitting parameters contained in their theory, Andersen et al (2003Andersen et al ( , 2006 obtained fairly good agreement between their theoretical/numerical results and the experimental data for moderate flow conditions: drain rate through the vessel Q ∼ 1.8 × 10 −6 m 3 s −1 and vessel rotation rate Ω ∼ 1.26 rad s −1 (∼12 r.p.m.). It was also demonstrated in these experiments that surface tension greatly affected both the whirlpool shape and its dip.…”

“…However, the free-surface curvature can be presented in an exact differential form, so that the surface tension pressure is given by (cf. Andersen et al 2003Andersen et al , 2006:…”

“…The truncated equation should be solved simultaneously with (27) and (32) resulting in the complete set of equations after elimination of w r (cf. Andersen et al 2003Andersen et al , 2006Lautrup 2005):…”

“…Andersen et al 2003Andersen et al , 2006. Typical computational results are shown in figures 11 and 12 for R = 0.05, Q R = 10 6 , We = 3.4 × 10 4 , and K = 3.05 × 10 −3 .…”

“…A more consistent theory of stationary whirlpools in a rotating vessel was originally suggested by Lundgren (1985) and subsequently developed further by Andersen et al (2003Andersen et al ( , 2006 to include the surface tension and bottom upwelling owing to the effect of viscosity in the Ekman boundary layer near the outlet orifice. This modified model designated as the LABSRL-model, was applied to the interpretation of experimental whirlpool data obtained in a rotating cylindrical vessel with water circulating at a given flow rate.…”