BOUNDARY LAYERS Boundary Layer Approximation Viscous effects confined

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 In periodic flows, it remains constant Can derive δ from Navier-Stokes equation: Within δ :

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/ME 322/notes/

U∞ Boundary layers Streamlines of inviscid flow δ Airfoil L http: //web. cecs. pdx. edu/~gerry/class/ME 322/notes/ If viscous = advective Wake

Will now simplify momentum equations within δ U∞ δ The behavior of w within δ can be derived from continuity: L http: //web. cecs. pdx. edu/~gerry/class/ME 322/notes/ Assuming that pressure forces are of the order of inertial forces:

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:

Boundary Conditions U∞ Initial Conditions Diffusion in x << Diffusion i Pressure field can be found irrotational flow theory δ L http: //web. cecs. pdx. edu/~gerry/class/ME 322/notes/

Other Measures of Boundary Layer Thickness arbitrary Velocity profile measured at St Augustine inlet on Oct 22, 2010

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

Another measure of the boundary layer thickness Momentum Thickness θ Determined from the total momentum in the fluid, rather than the total mass, 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 Replaced H by ∞ because u = U for z > H H z