Colebrook-White equation for smooth and rough pipes

Colebrook-White equation for turbulent flow in commercial pipes:

$$\frac{1}{\sqrt f} = -2\log_{10}\left(\frac{k}{3.7D} + \frac{2.51}{Re\sqrt f}\right)$$

where $f$ is the Darcy friction factor, $k$ the equivalent roughness, $D$ the diameter and $Re$ the Reynolds number.

Smooth pipes ($k \to 0$, so $k/D \to 0$)

The roughness term vanishes:

$$\frac{1}{\sqrt f} = -2\log_{10}\frac{2.51}{Re\sqrt f} = 2\log_{10}\frac{Re\sqrt f}{2.51} = 2\log_{10}(Re\sqrt f) - 2\log_{10}2.51$$

Since $2\log_{10}2.51 = 0.80$:

$$\frac{1}{\sqrt f} = 2\log_{10}(Re\sqrt f) - 0.8$$

This is Prandtl's universal resistance law for smooth pipes.

Rough pipes (large $Re$, fully rough)

As $Re \to \infty$ the second term $\dfrac{2.51}{Re\sqrt f} \to 0$:

$$\frac{1}{\sqrt f} = -2\log_{10}\frac{k}{3.7D} = 2\log_{10}\frac{D}{k} + 2\log_{10}3.7$$

Since $2\log_{10}3.7 = 1.14$:

$$\frac{1}{\sqrt f} = 2\log_{10}\frac{D}{k} + 1.14$$

This is the von Karman-Prandtl law for rough pipes, where $f$ is independent of $Re$.

For intermediate values both terms in the bracket matter, so the equation describes the transition zone, and it covers the whole turbulent range with one formula.