BOUNDARY LAYERS 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 Within δ :

electrostatic-boundary-layer-reduction U∞U∞ L δ

U∞U∞ L δ If viscous = advective Streamlines of inviscid flow Airfoil Wake Boundary layers

U∞U∞ L δ The behavior of w within δ can be derived from continuity: Will now simplify momentum equations within δ 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:

U∞U∞ L δ Boundary Conditions Initial Conditions Diffusion in x << Diffusion in z Pressure field can be found from irrotational flow theory

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

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 z U U δ*δ* H

Velocity profile measured at St Augustine inlet on Oct 22, 2010 Displacement Thickness δ* Velocity profile measured at St Augustine inlet on Oct 22, 2010

Momentum Thickness θ Another measure of the boundary layer thickness total momentumtotal mass 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) 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: substituting H z Replaced H by ∞ because u = U for z > H from Kundu’s book

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