On the Existence of Logarithmic Terms in the Drag Coefficient and Nusselt Number of a Single Sphere at High Reynolds Numbers

Yousef El Hasadi, Johan Padding

Published 2020-07-20Version 1

In the beginning of the second half of the twentieth century, Proudman and Pearson (JFM,2(3), 1956, pp.237-262) suggested that the functional form of the drag coefficient of a single sphere subjected to uniform fluid flow consists of a series of logarithmic and power terms of the Reynolds number. In this paper, we will explore the validity of the above statement for Reynolds numbers up to $ 2\times 10^{5}$, by using a symbolic regression machine learning method.The algorithm is trained by using available experimental data, as well as data from a well-known correlation from the literature.The symbolic regression method finds the following expression for the drag coefficient $C_D = a+\frac{24}{Re}+f(\log(Re))$, where $Re$ is the Reynolds number, and the constituents of $f(\log(Re))$ are integer powers of $\log(Re)$. Interestingly, the value of $a$ resembles the value of $C_D$, at the point where laminar separation point occurs. We did the same analysis for the problem of heat transfer under forced convection around a sphere, and found that the logarithmic terms of $Re$ and Peclect number $Pe$ play an essential role in the variation of the Nusselt number $Nu$. The machine learning algorithm independently found the asymptomatic solution of Acrivos and Goddard (JFM, 23(2),pp.273-291).