BOUNDARY LAYERS
description
Transcript of BOUNDARY LAYERS
BOUNDARY LAYERS
upgDt
uD
2
2
2
2
21zu
xu
xp
zuw
xuu
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
Can derive δ from Navier-Stokes equation:
Boundary Layer Approximation
In periodic flows, it remains constant
t 4
2
Within δ :
http://media.efluids.com/galleries/boundary?medium=260
http://nomel.org/post/210363522/idea-electrostatic-boundary-layer-reduction
http://web.cecs.pdx.edu/~gerry/class/ME322/notes/
U∞
L
δ
http://web.cecs.pdx.edu/~gerry/class/ME322/notes/
U∞
L
δ
LU
xuu
2
~
22
2
~
Uzu
If viscous = advective2
2
~
UL
U
UL
~
Streamlines ofinviscid flow
Airfoil Wake
Boundary layers
http://web.cecs.pdx.edu/~gerry/class/ME322/notes/
U∞
L
δ
The behavior of w within δ can be derived from continuity:
0
zw
xu
zw
xu
zxwu
~
w
LU ~
LUw ~
Will now simplify momentum equations within δ
xuu
xp
~Assuming that pressure forces are of the order of inertial forces:
2~ Up
UL
~L
Uw ~ 2~ Up
Lxx 'Nondimensional variables in the boundary layer
(to eliminate small terms in momentum equation): zz '
Uuu'
LUww
' 2'
Upp
The complete equations of motion in the boundary layer in terms of these nondimensional variables:
2
2
2
2
''
''
Re1
''
'''
'''
zu
xu
xp
zuw
xuu
2
2
2
2
2 ''
Re1
''
Re1
''
'''
'''
Re1
zw
xw
zp
zww
xwu
0''
''
zw
xu
?Re@
LURe
2
2
''
''
'''
'''
zu
xp
zuw
xuu
''0
zp
0''
''
zw
xu
2
21zu
xp
zuw
xuu
zpg
0
zw
xu
http://web.cecs.pdx.edu/~gerry/class/ME322/notes/
U∞
L
δ
zUzxuxUxuxwxu 00 ,,00,00,
Boundary Conditions Initial Conditions
Diffusion in x << Diffusion in zPressure field can be found fromirrotational flow theory
Velocity profile measured at St Augustine inlet on Oct 22, 2010
Other Measures of Boundary Layer Thickness
t 4 2
UL
~
Uu 99.0@99
Uu 95.0@95 arbitrary
95
99
Displacement Thickness δ*Another measure of the boundary layer thickness
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 zU U
δ*
H
H
HUudz0
* dzUu
0
1*
Velocity profile measured at St Augustine inlet on Oct 22, 2010
dzUu
0
1*
Displacement Thickness δ*
Velocity profile measured at St Augustine inlet on Oct 22, 2010
*
*
Momentum Thickness θAnother measure of the boundary layer thickness
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)
Momentum flux across A(per unit width)
HU 22
0
2*
0
2 *UdzudzuHH
Momentum flux across B
H
z
from Kundu’s book
The loss of momentum caused by the boundary layer is then the difference of the momentum flux between A and B:
A
HU 2
B
H
Udzu 2
0
2 * 2
0
222 *UdzuHUUH
dzUuH
0
1*substituting
HH
dzUuUdzuUU
0
2
0
222 1
0
1 dzUu
Uu
H
z
Replaced H by ∞ becauseu = U for z > H
from Kundu’s book
dzUu
0
1* Displacement Thickness(mass flux)
0
1 dzUu
Uu Momentum Thickness
(momentum flux)
t 4 From Stokes’ First Problem
2
From Stokes’ Second Problem
UL
~ Scaling Advection-Diffusion Equation
Uu 99.0@99
Uu 95.0@95 Arbitrary
BOUNDARY LAYERS
Boundary Motion
Boundary Fixed