The entry length of a turbulent flow is much shorter than that of a laminar flow, J. Nikuradse determined that a fully developed profile for turbulent flow can be observed after an entry length of 25 to 40 diameters. We shall focus to fully developed turbulent flow in this section. - Considering a fully developed turbulent pipe flow (Fig. 34.3) we can write
or
 | | (34.19) |
Fig. 34.3 Fully developed turbulent pipe flow
It can be said that in a fully developed flow, the pressure gradient balances the wall shear stress only and has a constant value at any  . However, the friction factor ( Darcy friction factor ) is defined in a fully developed flow as
 | | (34.20) |
Comparing Eq.(34.19) with Eq.(34.20), we can write
 | | (34.21) |
H. Blasius conducted a critical survey of available experimental results and established the empirical correlation for the above equation as
 where  | (34.22) |
- It is found that the Blasius's formula is valid in the range of Reynolds number of Re ≤105. At the time when Blasius compiled the experimental data, results for higher Reynolds numbers were not available. However, later on, J. Nikuradse carried out experiments with the laws of friction in a very wide range of Reynoldsnumbers, 4 x 103 ≤ Re ≤ 3.2 x 106. The velocity profile in this range follows:
 | | (34.23) |
where  is the time mean velocity at the pipe centre and  is the distance from the wall . The exponent n varies slightly with Reynolds number. In the range of Re ~ 105, n is 7.
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