Objective

To describe the velocity distribution in unidirectional
flows
Contents





Conditions at a solid boundary
Laminar flow velocity profile
Turbulent flow velocity profile
Boundary layers
Entry length
1
Normal component of velocity
The normal component of (relative) velocity is zero
Tangential components of velocity
Most fluids ‘stick’ to boundaries. The tangential components
of (relative) velocity is zero at the boundary. This is called the:
No-slip condition: u = 0 at the wall
[Some complex fluids (suspension, pastes, etc.) show
apparent slip at the wall]
Shear stress at the wall
From Newton’s law of viscosity:
du
w  
dy y 0
y
u(y)
tangent
y=0
u=0
2
In laminar flow, layers of fluid slide smoothly over each other
Unidirectional flow
All motion is in the same direction, e.g. flow along a pipe
Velocity profile in a pipe
The velocity is zero at each wall,
and maximum at the centre
The velocity profile is parabolic:
u(r )  (1  [r / R ])U MAX
2
2
Shear stress
The shear stress is largest at the walls and zero at the centre
3
• The slower moving fluid near the wall is mixed with the
faster moving fluid near the centre
• The velocity still goes to zero at the wall (‘no-slip’)
The mean velocity profile in turbulent flow is almost uniform
Velocity profile
Laminar
Turbulent
Uniform
Stresses
There are additional stresses, called ‘ Reynolds stresses ’
caused by fluctuations and mixing, so Newton ’ s law of
viscosity is not sufficient
4
A thin layer of fluid adjacent to a boundary where viscous
effects are important and the velocity changes sharply
Flow over a flat plate
U
Boundary
layer
y
Laminar
xT
Turbulent
x
The laminar boundary layer grows thicker in the flow direction
 Transition occurs at a critical Reynolds number:
ReT = UxT/ in the range: 5  105 to 5  106
depending on the plate roughness, etc.
 A turbulent velocity profile is flatter than a laminar one
 A turbulent boundary layer is thicker and continues to grow

5
Near the beginning of a pipe, the flow is not ‘fully-developed’.
Given a uniform velocity profile at the entrance, how far into a
pipe must we go before entrance effects become negligible?
D
Entry length
Laminar flow:
Turbulent flow:
LE / D  Re/ 16
LE / D  10  50
Fully developed flow
The flow no longer changes in the flow-direction (x-direction)
6
Show that the velocity profile in laminar pipe flow is
parabolic (to be done in Lecture 14):
u(r )  (1  [r / R ])U MAX
2
2
7