Bathtub Vortex Profile and Flow Field Model with Circulation.

Abstract

A new flow field model around a bathtub vortex was proposed, which was based on an exact solution of the Navier-Stokes equations called an expanding vortex flow. Velocity and pressure distributions were numerically calculated assuming that the downward velocity near the vortex center is proportional to a distance from the free surface and that the axial distribution of circulation is uniform. Radial distribution of circulation was found to be almost constant at a distance from the vortex center under some conditions, which was similar to the experimental result. Gas core profile, i.e. gas core radius and gas core depth, was analytically calculated with the new flow model. Gas core radius was defined to be a radial position where radial distribution of pressure is half. Gas core depth was calculated from the Bernoulli's equation. Analytical gas core profiles showed good agreement with experimental results. Formation of the gas core and the stagnant region around the vortex center was analytically discussed.