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

TL;DR

This study uses machine learning to analyze the functional form of the drag coefficient and Nusselt number for a sphere at high Reynolds numbers, revealing the significance of logarithmic terms and their connection to flow separation and heat transfer.

Contribution

It demonstrates that the drag coefficient and Nusselt number can be expressed with logarithmic terms of Reynolds and Peclet numbers, validated by machine learning and experimental data, extending understanding of high Reynolds number flows.

Findings

Logarithmic terms are present in the drag coefficient at high Re.

The model accurately predicts the drag crisis and flow separation points.

Logarithmic terms are crucial for modeling heat transfer around a sphere.

Abstract

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 $10^{6}$ 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 $2\times 10^5$.\ Our results show that the functional form of the $C_D$ contains powers of $\log(Re)$, plus the Stokes term, fulfilling partially the statement made above. The logarithmic $C_D$ expressions can generalize (extrapolate) beyond the training data and are the first in the literature to predict with acceptable accuracy the rapid decrease (drag crisis) of the $C_D$ at high $Re$.\ 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.\ 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 asymptotic solution of Acrivos and Goddard (J. Fluid. Mech., 23(2),1965, pp.273-291).