Fluid drags on a sphere at finite Reynolds number--a classical problem examined afresh

Abstract

Evaluating the drag coefficient Cd for a sphere moving in a fluid at finite Reynolds number Re has become a classical research subject in fluid dynamics, with publications appearing continuously by numerous authors over the years due to the unavailability of closed-form mathematical solutions. To date, mathematical formulas for calculating Cd as a function of Re can only be obtained by correlation curve fitting to experimental data. The consequence is the coexistence of various formulas with different mathematical fitting functions and an increasing number of fitting parameters. The present study focuses on examining the accuracy of Klyachko's 1934 formula for Cd with two common fractions as fitting parameters, which could enable closed form integration of a simplified equation governing rectilinear motion of aerosol spheres. The computed values of root-mean-square relative error from recently developed formulas show reasonable accuracy of the Klyachko formula for Re less than 1000 and its extension to Re less than 3e5 as a piecewise continuous function, with the possibility of significant improvement through a slight modification. Hence, a few demonstrative examples are presented using the extended Klyachko formula in the study of rectilinear acceleration, the calculation of the terminal velocity of aerosol spheres, and so forth, to shed fresh light on how this classical problem can be studied.