A Theoretical and Experimental Study of the Internal Circulation in Water Drops Falling at Terminal Velocity in Air

Abstract

Four theoretical approaches are presented for quantitatively determining the intensity of the internal circulation and the flow patterns inside and outside liquid water spheres falling at terminal velocity in air. The first approach assumes creeping flow outside and inside a water sphere, the second assumes potential flow outside and inviscid motion inside a water sphere, the third makes use of boundary layer theory, and the fourth approach uses a numerical method to solve the full Navier-Stokes equation of motion inside and outside a water sphere. The theoretical predictions are compared with data obtained from new quantitative wind tunnel experiments on spherical and deformed water drops. The results show that the creeping flow analysis greatly underestimates the strength of the internal velocity while the inviscid flow analysis greatly overestimates it. On the other hand, the results of the boundary layer approach and of the numerical approach agree reasonably well with the experimental data for drops with radii <500 ?. For larger drops the results of the boundary layer approach greatly overestimate the strength of the internal circulation and predict a completely wrong trend of the variation of the internal velocity with drop size, while the numerical results, although somewhat overestimating the circulation strength, predict the trend correctly. Reasonably good agreement is also found between the observed flow patterns inside the drop and those numerically predicted. In two appendices the effect of the internal circulation on drop shape and hydrodynamic drag is discussed.