Theory of the Bathtub Vortex

Abstract

The paper gives an analysis of the temporal growth of vorticity when water contained in a large vertical cylindrical tank flows out through a small hole in the base of the tank. The results indicate both an exponential growth of vorticity from the rotation of the frame of reference and an exponential growth of the residual vorticity in the water. Two main cases are considered; (a) the case when the initial swirl component does not vary in the vertical direction, (b) the case when the swirl component decreases downwards. The latter case simulates the effect of a tank bottom boundary layer or other vertical shear layer on the vorticity growth. While the analysis relates to a simplified model of the physical situation and also entails approximations, it nevertheless indicates that when a downwardly decreasing gradient of swirl velocity exists, a reversal from an initial clockwise residual swirl to a final counterclockwise rotation (in the Northern Hemisphere) may be anticipated. This is in accord with cited experimental results.