Colebrook–White Friction Factor

Cyril Colebrook and Cedric White published this in 1937 after fitting Nikuradse's sand-grain roughness experiments to commercial pipe, and it is still the reference description of turbulent friction. The Moody diagram, which nearly every engineer has read off at some point, is nothing more than this equation plotted. Take a relative roughness of 0.0001 at $Re = 10^5$: iterate $x = -2\log_{10}(\arepsilon/3.7D + 2.51x/Re)$ from any sensible start and it settles on $x = 7.3494$ within five passes, giving $f = 1/x^2 = 0.01851$. That is a Moody chart read to the width of a pencil line.

The awkwardness is that $f$ appears on both sides. Before calculators this meant either a chart or a slide-rule iteration, which is why Swamee and Jain's explicit fit and half a dozen rivals exist. This page does the iteration properly rather than approximating it, so it agrees with the reference equation to machine precision instead of to about 1%. Going the other way is easier than it looks: once $f$ is known, $1/\sqrt f$ is known, the logarithm can be undone, and roughness, diameter or Reynolds number each fall out in closed form. That is why this page can solve for all four.

The surprise is at the two ends of the curve. At very high Reynolds number the $2.51/(Re\sqrt f)$ term vanishes and friction stops depending on velocity at all, which is the fully rough regime where the Moody curves go flat. Ask this page for a Reynolds number in that region and it will tell you honestly that none can be recovered. At the other end, below about $Re = 4000$, the equation simply does not apply; use $f = 64/Re$ instead, and treat the 2300 to 4000 transition band as territory no correlation describes and no prudent designer operates in.