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