Dependence of instability to induce a bathtub vortex in a rectangular vessel on the aspect ratio of the horizontal cross section

Abstract

Dependence of the linear instability to induce a bathtub vortex in a rectangular vessel on the horizontal aspect ratio of the vessel is investigated in the present paper. Water comes in through inlet sections on the top of two facing side walls, and is drained out through a pipe attached to the drain hole at the center on the bottom of the vessel. Water surface is assumed to be always flat without any dip even after instability yielded a large-scale vortex above the drain hole. The flow is steady and symmetric with respect to both vertical center planes along the streamwise direction and perpendicular to it at small Reynolds numbers, but becomes unstable to infinitesimal disturbances inevitably included in the flow at Reynolds numbers larger than a critical value. The steady symmetric flow is numerically obtained, and its stability is analyzed by numerically solving the linear disturbance equation. The instability is confirmed to lead to induction of a so-called bathtub vortex. It is found that there exists a threshold aspect ratio of the rectangular cross section for the instability, and the threshold value is determined.