Fully developed velocity profiles in a pipe.
The velocity profile in the fully developed region of a steady, laminar, incompressible pipe flow is found by solving the force balance on the fluid. In this region the fluid particles move along constant streamlines, and the radial velocity profile u(r) does not change in the axial direction.
Balancing the forces on a small fluid element of thickness dr and length dx gives the parabolic laminar profile:
u(r) = Umax [ 1 − (r / R)2 ]
When the Reynolds number in a pipe exceeds about 4000 the flow is turbulent. Fully developed turbulent profiles are flatter than laminar ones, but are not uniform "plug" flow. The empirical power-law profile describes them simply:
u(r) = Umax ( 1 − r / R )1/n
where the exponent n depends on the Reynolds number, based on the average and maximum velocity (the maximum is at the pipe centre).
