Analysis of a stable bathtub vortex in a rotating container

Abstract

Rotating flows with free-surface vortices can be found in many engineering applications, such as pump and turbine intakes, vessels, and nuclear reactors. The need to address rather different flow regions existing in such flows, such as Ekman and Stewartson layers and the line vortex zone, in a coupled manner, makes modeling of free-surface rotating flows very challenging. In this work, the flow field of a free-surface vortex, created in a rotating cylinder with a drain hole in its bottom, is investigated numerically and analytically. Above the drain hole of the cylinder, a free-surface vortex, accompanied by axial velocity, is created. This axial velocity profile is governed by the Ekman boundary layer far from the axis and by the drainage in its proximity. The experiments of Andersen et al. [“Anatomy of a bathtub vortex,” Phys. Rev. Lett. 91(10), 104502 (2003a); “The bathtub vortex in a rotating container,” J. Fluid Mech. 556, 121–146 (2006)] on the so-called bathtub vortex are numerically modeled with the finite volume method. The simulations are validated with the available measurements from the experiments. Using the simulation results, self-similar and non-self-similar models, describing the velocity fields in the Ekman boundary layer, are compared and tested. It is shown that the self-similar model is more accurate than the non-self-similar model. It is also demonstrated that the analytical model of Andersen et al. [“Anatomy of a bathtub vortex,” Phys. Rev. Lett. 91(10), 104502 (2003a); “The bathtub vortex in a rotating container,” J. Fluid Mech. 556, 121–146 (2006)], when modified as suggested in the present study, is capable of predicting the free-surface profile for low rotation rates. However, for high rotation rates, only the numerical simulation can predict the relation between the flow field within the liquid and the free-surface profile.