Friction factor correlations for laminar, transition and turbulent flow in smooth pipes

In this paper we derive an accurate composite friction factor vs. Reynolds number correlation formula for laminar, transition and turbulent flow in smooth pipes. The correlation is given as a rational fraction of rational fractions of power laws which is systematically generated by smoothly connecting linear splines in log–log coordinates with a logistic dose curve algorithm. This kind of correlation seeks the most accurate representation of the data independent of any input from theories arising from the researchers’ ideas about the underlying fluid mechanics. As such, these correlations provide an objective metric against which observations and other theoretical correlations may be applied. Our correlation is as accurate, or more accurate, than other correlations in the range of Reynolds numbers in which the correlations overlap. However, our formula is not restricted to certain ranges of Reynolds numbers but instead applies uniformly to all smooth pipe flow data for which data is available. The properties of the classical logistic dose response curve are reviewed and extended to problems described by multiple branches of power laws. This extended method of fitting which leads to rational fractions of power laws is applied to data of Marusic and Perry (1995) [1] for the velocity profile in a boundary layer on a flat plate with an adverse pressure gradient, to data of Nikuradse (1932) [2] and McKeon et al. (2004) [3] on friction factors for flow in smooth pipes and to the data of Nikuradse [4] for effectively smooth pipes.