Ghosh - 550Page 112/29/2018

Boundary Layer Thicknesses

In the definition of the edge of boundary layer, y =  was rather arbitrarily chosen (we know that y = , the edge of the boundary layer, could very well be chosen when ue = .999U or .9999U). Now we would introduce some boundary layer thicknesses that have more physical significance. Velocity boundary layers are regions near the solid wall in which most of the fluid shear (or, rotation) are confined. The gradual decrease of shear values from the maximum (yx = max on wall) to a minimum value of shear is reflected in the shape of the velocity vector profile. When an actual velocity profile is compared with an ideal velocity profile over a flat plate, we note the velocity deficit = U – u.

Since velocity inside the boundary layer falls short of the ideal flow profile, it is obvious that the mass flux or, momentum flux will also fall short of the ideal flow mass or, momentum flux. Hence, we introduce two boundary layer thicknesses as defined below.

Displacement Thickness

Displacement thickness is defined to facilitate the use of an ideal flow velocity profile in matching the mass flux in the real flow boundary layer. Since there exists a velocity deficit (and consequently a mass flux deficit) in the viscous boundary layer, we may be able to match the real in if the wall was allowed to grow a little as shown in the sketch below.

Here we have shown the growth of the wall by a distance *, called the displacement thickness, such that the true in the boundary layer is the same as the displaced ideal flow’s mass flux . This results in (by equating the deficit)

i.e.,

Example of Useof *

Momentum Thickness

This is similar to the displacement thickness, except now we are interested in matching the real momentum flux deficit,  in the boundary layer. Thus the momentum thickness is defined as the growth of a wall necessary in an ideal flow such that the real flow’s momentum flux deficit may be duplicated. Once again, equating the deficit momentum flux in the two cases we get

or,

Other boundary layer thickness may be defined such as the energy thickness, shear thickness, etc. They all convey the similar ideas of modeling real fluid flows using ideal flow models. However for this course we shall omit them.

Momentum ThicknessExample Continue