Work in the first half of the 20^{th} century led to the development of the Colebrook-White equation which is the most accurate equation for both rough turbulent and smooth turbulent flow.

Work by Prandtl and Nikuradse on smooth and artificially roughened pipes revealed three zones of turbulent flow :

- Smooth turbulent zone in which the friction factor f is a function of Reynolds number only and expressed by :

# $\frac{1}{\sqrt{f}}=2\mathrm{log}\frac{{R}_{e}\sqrt{f}}{2.51}$

- Transitional turbulent zone in which the friction factor f is a function of both relative roughness (k/D) and Reynolds number.

- Rough turbulent zone in which the friction factor f is a function of relative roughness (k/D) only and expressed by :

# $\frac{1}{\sqrt{f}}=2\mathrm{log}\frac{3.7}{k/D}$

The previous two equations are known as Karman-Prandtl equations. Colebrook and White found that the function resulting from addition of the rough and smooth equations in the following form :

# $\frac{1}{\sqrt{f}}=-2\mathrm{log}\left[\frac{k/D}{3.7}+\frac{2.51}{{R}_{e}\sqrt{f}}\right]$

However, the Colebrook-White equation has no direct solution and requires an iterative approach. The implicit nature of the equation has limited the application of the full Colebrook-White equation, particularly before the advent of modern computers.

Where :

f : Friction coefficient

k : Roughness height of pipeline (meters)

D : Diameter of pipeline (meters)

R_{e} : Reynolds number (dimensionless)

Read more about other equations to calculate the friction factor f :

Swamee-Jain Equation

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