8. Flow in Pipes
The flow in a pipe is a viscous flow, and there
exist 2 distinct regions: laminar and turbulent
region, connected by a transition region.
•
Laminar flow
:
dye remains a
fine filament
y
x
•
Turbulent flow
:
y
x
dye is rapidly
dispersed randomly
The shear stress,
, is a function of the velocity gradient.
The greater the change in
U
with respect to
y
, the greater
is the shear stress. Therefore, the shear stress and the
frictional losses are higher in turbulent flow.
Whether the flow is laminar or turbulent depends on the
Reynolds number. Transition from laminar to turbulent
flow occurs at
Re
=2300. However, if conditions such as
pipe smoothness and velocity are well controlled,
transition can be delayed up to
Re
=10
5
for incompressible
flow.
74
As the flow enters a pipe uniformly, the velocity
at the walls must decrease to zero (noslip
Boundary Condition). Continuity dictates that
the velocity at the center must increase. Thus,
the velocity profile is changing continuously
from the pipe entrance until it reaches fully
developed conditions. This distance,
L
, is called
the entrance length.
fullydeveloped
profile
L
8.1. Poiseuille Flow
Consider the steady, fully developed laminar
flow in a straight pipe of circular cross section
having a constant diameter,
D
.
τ
τ
y
x
b
The condition of equilibrium in
x
direction
requires that the pressure force (
p
1

p
2
)π
y
2
acting
on the faces of the cylinder be equal to the shear
stress 2π
yb
acting on the circumferential area:
= (
p
1

p
2
)
y
/2
b
75
In accordance with the law of friction
(Newtonian fluid), we have:
=
d
u
/d
y
.
Since
u
decreases with
y
, therefore:
d
u
d
y
= –
p
1
–
p
2
μ
b
y
2
and upon integration:
4
d
d
1
4
)
(
2
2
2
1
y
C
x
p
y
C
b
p
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 Fall '08
 ZOHAR
 Fluid Dynamics, Incompressible Flow, Turbulent Flow