A new universal model for friction factor in smooth pipes

Abstract

Friction factor models for turbulent flow in smooth pipes express friction factor λ as a function of the bulk Reynolds number ReD and may be broadly grouped into two categories: power-law models and log-law models. While the former stem from the spectral scaling arguments applied to eddy momentum transfer close to the wall, the latter are derived from the mean velocity log law and are known to be consistent with the attached eddy model of wall turbulence structure. Interestingly, none of these models individually describes the entire range of Reynolds numbers (Re) accessed to date, without requiring adjustment of coefficients and/or exponents, i.e., these models are not universal. In this work, we present a new semi-empirical universal model that combines, without introducing any additional empirical coefficients, the essence of both power-law and log-law models. Due to this, our model successfully describes the variation of friction factor over the entire range of Reynolds numbers (more than four decades in ReD) at once. The physical basis for our model is the observation that at finite Reynolds numbers, the flow appears to be a small perturbation of the so-called ultimate regime of smooth-pipe turbulence, as far as friction is concerned; in the ultimate regime, λ → 0 asymptotically as R e D → ∞. The new model has significant potential toward accurate estimation of friction factor or flow rate in smooth pipe flows.