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BOUNDARY LAYERS

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.

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BOUNDARY LAYERS

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  1. 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:

  2. 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/

  3. Boundary layers Streamlines of inviscid flow Airfoil Wake U∞ δ L If viscous = advective http://web.cecs.pdx.edu/~gerry/class/ME322/notes/

  4. 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/

  5. 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:

  6. 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/

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

  8. 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*

  9. 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

  10. 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 (same dimensions as force) from Kundu’s book H z Momentum flux across A (per unit width) Momentum flux across B

  11. 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

  12. BOUNDARY LAYERS From Stokes’ First Problem From Stokes’ Second Problem Boundary Motion Scaling Advection-Diffusion Equation Arbitrary Boundary Fixed Displacement Thickness (mass flux) Momentum Thickness (momentum flux)

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