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 as a problem of data mining 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 number 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 Marusic and Perry 1995 for the velocity profile in a boundary layer on a flat plate with an adverse pressure gradient, to data of Nikuradse 1932 and McKeon et al. 2004 on friction factors for flow in smooth pipes and to the data of Nikuradse 1933 for effectively smooth pipes. We argue that the transition from smooth to rough pipe flow occurs near a value of 13.6×10 in agreement with a similar, earlier and independent analysis of McKeon et al 2004.