Both laminar and turbulent pipe flow produce velocity profiles that are symmetric about the axis of the pipe with a maximum velocity at the centre of the pipe.

Laminar pipe flow yields a parabolic velocity profile as shown in the illustration. The mean velocity V is half the magnitude of the centre-line velocity and the profile is:

u / u_max = 1 - (r / R)^2

where u is the local velocity value.

Turbulent pipe flow yields a velocity profile that is much flatter across the core of the flow, which can be approximated quite well with a power law of the form

u / u_max = ( 1 - r / R )^(1 / n)

where n depends on the friction factor such that

1 / n = sqrt( f ) for f < 0.1

The result is usually n around 7 and referred to as the 1/7 Power Law. This power law gives a good general description of the shape of the turbulent core velocity profile, even though it fails in regions very close to the wall. The mean velocity V is much closer to the centre-line velocity in turbulent flow. For more detail see White, from which this figure is taken.