Do logarithmic terms exist in the drag coefficient of a single sphere at high Reynolds numbers?

At the beginning of the second half of the twentieth century, Proudman and Pearson (J. Fluid. Mech.,2(3), 1956, pp.237–262) suggested that the functional form of the drag coefficient (C_D) of a single sphere subjected to uniform fluid flow consists of a series of logarithmic and power terms of the Reynolds number (Re). In this paper, we will explore the validity of the above statement for Reynolds numbers up to 10⁶ by using a symbolic regression machine learning method. The algorithm is trained by available experimental data and data from well-known correlations from the literature for Re ranging from 0.1 to 2×10⁵. Our results show that the functional form of C_D contains powers of log(Re), plus the Stokes term. The logarithmic C_D expressions can generalize (extrapolate) better beyond the training data than pure power series of Re and are the first in the literature to predict with acceptable accuracythe onset of the rapid decrease (drag crisis) of C_D at high Re, but also to follow the right behaviour towards zero Re. We also find a connection between the root of the Re-dependent terms in the C_D expression and the first point of laminar separation. The generalization behaviour of power-based drag coefficient equations is worse than logarithmic-based ones, especially towards the zero Re regime in which they give non-physical results. The logarithmic based C_D correctly describes the physics from the low Re regime to the onset of the drag crisis. Also, by applying a minor modification in the logarithmic based equations, we can predict the drag coefficient of an oblate spheroid in the high Re regime.