Many formulas to calculate the head losses of an hydraulic flow are available in the literature. Anyhow before write about this, we need to understand the development of an hydraulic flow.

The head losses and the state of motion of a flow, are a function of the velocity and can change from laminar flow to smooth pipe flow, to transient flow, to absolute turbulent flow, depending on the conditions.

In detail:

Laminar flow (a) is typical in very low velocity flow and small pipe diameters. The fluid flows in parallel layers, with no disruption between the layers, and the head losses are proportional to the Reynolds number (Re).

Turbulent flow (b)  is typical of all other condition. It is characterized by chaotic property changes, and it is divided into the following states:
Smooth pipe condition, typical of low velocity or not high diameter. The boundary layer thickness is greater than the roughness and the head losses don’t depend on the pipe roughness.
Transient flow condition, intermedia between smooth pipe condition and rough pipe condition. The boundary layer thickness is less than the roughness and the head losses depend partly on pipe roughness.
Rough pipe condition or completely turbulent flow, typical of high velocity and high diameter. The roughness is fully exposed outside of the boundary layer and the head losses depends only on pipe roughness.


 Colebrook - Flussi

For each of this condition and depending on the fluid, a number of different formulas have been written including: Hazen-Williams, Manning, Gaukler-Strickler, Blasius, Kutter, Marchetti, Scimemi-Veronese, Orsi, etc. Each of those is valid in a specific condition and only for a specific fluid (typically water). When fluid is water and we know the state of flow, the previous equations are available.

Colebrook-White equation is the natural evolution of Nikuradse studies, and represents the culmination of all the previous study in hydraulic rough pipes head looses. With the laminar flow equation, it allows to calculate any type of flow and fluid, at any temperature.

Before the computer era, it was very hard to calculate head losses by using Colebrook-White formula, because its implicit form.

To use Colebrook-White approach we have to start from Darcy-Weisbach equation (1).

Colebrook - Darcy(1)

For the symbols see literature.

In the laminar flow, we can calculate friction factor as follow. This equation is valid for Re<2300. (2)

 Colebrook - laminare(2)

In the turbulent flow, the friction factor depend on relative pipe roughness and Re, and we can calculate it as follow. This equation is valid for Re > 4000. (3)

Colebrook - Colebrook(3)

Between 2000<Re<4000 the flow is not stable and it changes from the one to the other state (Transition region).

Re is defined as follow. (4)

Colebrook - Re (4)

The Colebrook-White equation is in implicit form, so we cannot resolve it in closed form. Combining the Darcy-Weisbach and Colebrook-White equation the problem it can be solved in a numerical way. (5)

Colebrook - V(5)

To solve it we can use bisection method. It is a simple method than, in several iteration, find the solution. For more details see relevant literature.

The Moody Diagram relates the Darcy-Weisbach friction factor, Re and relative roughness for fully developed flow in a circular pipe.


Colebrook - Moody

Several observations.

In case of complete turbulent flow the head losses are proportional to V2. In this region are valid Manning and Gaukler-Strikler equation, typically used in rivers and cannels calculations.

In case of smooth pipe condition (Blasius equation) the head losses are proportional to V1,75.

In transient region  the head losses are proportional to Va, where a = 1,75 – 2. In this region are valid Hazen-Williams formula and many monomial formula.