Fundamentals of Haemodynamics - Computational Hemodynamics: Theory, Modelling and Applications - page 79

where

2

dP R

u (0) = U =−

(4.12)

max

dx 4 µ

The velocity profile is parabolic in shape with a maximum at the centreline where
r = 0. At the wall where r = R , then the velocity becomes zero. If we integrate
Eq. 4.11 from r = 0 to r = R we obtain the average velocity which gives

2

r

ur () 2 = U 1 − (4.13)

avg

2

R

This means that

2

1 1 dP R

U = U =−

(4.14)

avg max

2 2 dx 4 µ

This means that the average velocity in the fully developed region of a laminar pipe
flow is half that of its maximum velocity.
When the Reynolds number is greater than 4000 in pipes, turbulent flow behav-
iour is present. Fully developed turbulent velocity profiles are flatter than laminar
flow profiles, and not uniform nor a “plug” flow. The power-law empirical velocity
profile expression provides a simple method to describe the profiles,

1/ n

ur () r

=− 1

(4.15)

U R

max

where the exponent n is a constant value dependent on the Reynolds number shown
in Fig. 4.10 , but can also be calculated based on the average and maximum velocity
(which occurs in the pipe centre) of the flow,

2

U 2 n

ave

= (4.16)

U n ( + 1)( 2 n + 1)

max

The laminar profiles develop into a parabolic shape because the fluid inertia can-
not overcome the viscous forces sufficiently causing a gradual decrease in velocity
away from the zero velocity at the wall. The turbulent profile has much greater flow
inertia and this energy allows greater mixing between each of the fluid particles.
This results in a more even distribution of velocity throughout the profile. This mix-
ing is referred to as diffusion of fluid inertia or its momentum (Fig. 4.11 ).