BOUNDARY LAYERS
Boundary Layer Approximation
Viscous effects confined to within
some finite area near the
boundary → boundary layer
In unsteady viscous flows at low Re
(impulsively started plate) the
boundary layer thickness δ grows
with time
 4 t
In periodic flows, it remains constant
 
2



Du
2
  g  p   u
Can derive δ from Navier-Stokes equation: 
Dt
  2u  2u 
u
u
1 p

w

  

Within δ : u
2
2 
x
z
 x
z 
 x
http://nomel.org/post/210363522/ideaelectrostatic-boundary-layer-reduction
U∞
http://media.efluids.com/galleries/boundary?medium=260
δ
L
http://web.cecs.pdx.edu/~gerry/class/ME322/notes/
U∞
Boundary layers
Streamlines of
inviscid flow
δ
Airfoil
Wake
L
http://web.cecs.pdx.edu/~gerry/class/ME322/notes/
 2u  U 
u U 2

~
u
~
2
z
2
x
L
2
U
 U

If viscous = advective
~
L
2
L
 ~
U
Will now simplify momentum
equations within δ
U∞
The behavior of w within δ can
be derived from continuity:
δ
u w

0
x
z
L
http://web.cecs.pdx.edu/~gerry/class/ME322/notes/
U
w
~
L

u  w
U
w ~
L



x
z

u w
~
x
z
p
u
~

u
Assuming that pressure forces are of the order of inertial forces:
x
x
p ~ U2
 ~
L
U
w ~
U
L
p ~ U2
x
Nondimensional variables in the boundary layer
x'
L
(to eliminate small terms in momentum equation):
u' 
u
w
w' 
U
 U L
p' 
p
 U 2
Re 
z' 
z

UL

The complete equations of motion in the boundary layer in terms of these
nondimensional variables:
2
2
u '
u '
p'
1  u'  u'
u'
 w'



2
x '
z '
x ' Re x '
z '2
1  w '
w ' 
p'
1  2w '
1  2w '
 w'


 u'

2
2
Re  x '
z ' 
z ' Re x '
Re z '2
u '  w '

 0 @ Re  
x '
z '
?
u '
u '
p'  2u '
u'
 w'


x '
z '
x ' z '2
u
u
1 p
 2u
u
w


x
z
 x
z 2
0
p '
z '
p
g
z
u '  w '

0
x '
z '
u w

0
x
z
ux ,0  0 w x ,0  0 ux ,   U x  ux0 , z   U0 z 
Boundary Conditions
U∞
Initial Conditions
Diffusion in x << Diffusion in z
Pressure field can be found from
irrotational flow theory
δ
L
http://web.cecs.pdx.edu/~gerry/class/ME322/notes/
Other Measures of Boundary Layer Thickness
 4 t
 99
@ u  0.99U
 95
@ u  0.95U
 
arbitrary
2

 ~
L
U
Velocity profile measured
at St Augustine inlet on
Oct 22, 2010
 99
 95
Another measure of the boundary layer thickness
Displacement Thickness δ*
Distance by which the boundary would need to be displaced in a
hypothetical frictionless flow so as to maintain the same mass flux as in the
actual flow
z
z
U
U
H
δ*
H
 udz  U H   *
0

u

 *    1  dz
U
0
Displacement Thickness δ*

u

 *    1  dz
U
0
Velocity profile measured
at St Augustine inlet on
Oct 22, 2010
 *
Velocity profile measured
at St Augustine inlet on
Oct 22, 2010
 *
Another measure of the boundary layer thickness
Momentum Thickness θ
Determined from the total momentum in the fluid, rather than the total
mass, as in the case of δ*
Momentum flux = velocity times mass flux rate (same dimensions as force)
from Kundu’s book
H
z
Momentum flux
2

U
H
across A
(per unit width)
Momentum flux
across B
H  *
H
0
0
2
2
2

u
dz


u
dz



*
U


The loss of momentum caused by the boundary layer is then the
difference of the momentum flux between A and B:

U 2H

H
 u
H
dz    * U
0



A
2
U 2  U 2H   u 2dz    * U 2
2
0
B
H
substituting  *   1  u dz U 2 
0  U 

u
u
    1  dz
U
U
0
Replaced H by ∞ because
u = U for z > H
 U
H
0
u

 u 2 dz  U 2   1  dz
U
0
H
2
from Kundu’s book
H
z
BOUNDARY LAYERS
 4 t
Boundary Motion
2
 

L
 ~
 99
Boundary Fixed
From Stokes’ Second Problem
Scaling Advection-Diffusion Equation
U
@ u  0.99U
 95
@ u  0.95U


0
 *    1
 
From Stokes’ First Problem

u
dz
U
Arbitrary
Displacement Thickness
(mass flux)
u
u
Momentum Thickness
1

dz
0 U  U 
(momentum flux)