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

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

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/

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

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

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 http://nomel.org/post/210363522/idea-electrostatic-boundary-layer-reductionBoundary Layer Thickness

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

arbitrary

Another measure of the boundary layer thicknesshttp://nomel.org/post/210363522/idea-electrostatic-boundary-layer-reduction

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

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

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

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