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BOUNDARY LAYERS

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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 δgrows with time

In periodic flows, it remains constant

Can derive δfrom Navier-Stokes equation:

Within δ:

http://nomel.org/post/210363522/idea-electrostatic-boundary-layer-reduction

U∞

δ

http://media.efluids.com/galleries/boundary?medium=260

L

http://web.cecs.pdx.edu/~gerry/class/ME322/notes/

Boundary layers

Streamlines of

inviscid flow

Airfoil

Wake

U∞

δ

L

If viscous = advective

http://web.cecs.pdx.edu/~gerry/class/ME322/notes/

Will now simplify momentum equations within δ

The behavior of w within δ can be derived from continuity:

U∞

δ

Assuming that pressure forces are of the order of inertial forces:

L

http://web.cecs.pdx.edu/~gerry/class/ME322/notes/

Nondimensional variables in the boundary layer

(to eliminate small terms in momentum equation):

The complete equations of motion in the boundary layer in terms of these nondimensional variables:

U∞

Boundary Conditions

Initial Conditions

Diffusion in x << Diffusion in z

δ

Pressure field can be found from

irrotational flow theory

L

http://web.cecs.pdx.edu/~gerry/class/ME322/notes/

Other Measures of Boundary Layer Thickness

Velocity profile measured at St Augustine inlet on Oct 22, 2010

arbitrary

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*

Displacement Thickness δ*

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

from Kundu’s book

H

z

Momentum flux

across A

Momentum flux

across B

The loss of momentum caused by the boundary layer is then the difference of the momentum flux between A and B:

substituting

from Kundu’s book

H

z

Replaced H by ∞ because

u = U for z > H