A Numerical Study of the Drag on a Sphere at Low and Intermediate Reynolds Numbers

Abstract

Accurate solutions of the steady-state Navier-Stokes equations of motion have been obtained by means of a numerical method to determine the hydrodynamic drag on a rigid sphere falling at its terminal velocity in an unbounded fluid. The calculations were carried out for Reynolds numbers between 0.01 and 400. The numerical solutions were compared with the theoretical results of Stokes, Oseen, Goldstein, Proudman and Pearson, Jenson, Rimon and Cheng, and Carrier, and with the recent experimental data of Maxworthy, and Pruppacher and Steinberger. At the lowest Reynolds numbers the numerical solutions show closest agreement with the theory of Proudman and Pearson and at intermediate Reynolds numbers with the semi-theoretical relationship proposed by Carrier. At higher Reynolds numbers our present results agree well with the calculations of Hamielec et al. for Reynolds numbers of 40 and 100 and with the numerical results of Rimon and Cheng; they depart, however, significantly from the results of Jenson. Over the whole Reynolds number interval 0.01?400 our numerical results are in close agreement with the experimental data of Pruppacher, Pruppacher and Steinberger, and Beard and Pruppacher. It is concluded that our numerical study is unique in that it is able to predict theoretically accurate values for the drag on a sphere over a wide Reynolds number interval. The present study also confirms the findings of Maxworthy, and Pruppacher and Steinberger, that as the Reynolds number approaches zero the drag on a sphere approaches zero via the Oseen drag rather than via the Stokes drag. The significance of the present results to cloud physics is pointed out.