Empirical implicit correlation for Darcy friction factor in non-smooth pipelines which works for non-laminar () flow:

\frac{1}{\sqrt{f}} = -2 \, \log \left( \frac{\epsilon}{3.7 \, d}  + \frac{2.51}{{\rm Re} \sqrt{f}} \right)

where

	Reynolds number of a pipe fluid flow
	Inner diameter of a pipe
	inner pipe walls roughness

There are numerous explicit approximations of Colebrook–White correlation, particularly (Monzon, Romeo, Royo, 2002):

f = 0.25 \, \left[ \log \left( \frac{\epsilon / d}{3.7065} - \frac{5.0272}{\rm Re} \log \Lambda \right)   \right]^{-2}

where – is dimensionless parameter:

\Lambda = \frac{(\epsilon/d)}{3.827} - \frac{4.657}{\rm Re} \log \Bigg[  \bigg( \frac{\epsilon/d}{7.7918} \bigg)^{0.9924} + \bigg( \frac{5.3326}{208.815+Re} \bigg)^{0.9345} \Bigg]

See also