# Three Dimensional Boundary Layers

Consider a flat wing, away from the tips, at zero angle of attack in incompressible flow. The boundary layer equations are

du dv d w

dx + d y + dz

The same result can be found that ~ 0, hence the pressure is given by the inviscid flow solution.

The boundary conditions at the solid surface are u = v = w = 0.

At the outer edge of the boundary layer, u = ui and w = wi. The v-component at the edge carries the influence of the boundary layer on the inviscid flow. It is not known a priori and is part of the solution.

The initial condition, upstream of the wing is simply uniform flow u = UOT.

For infinite swept wings with the z-axis in the direction of span (Fig. 8.12)

d p du dw

dz = dz = dz

hence, the boundary layer equations reduce to (p = const.)

du dv

dx + d y ^

dw dw d2w

pu dx + pv dy = ‘‘ay?

Note that u and v are independent of w, the so-called independence principle. This is true only for incompressible flow where p = const.

The general three-dimensional boundary layers are covered in Howarth [4] and Stewartson [6]

Finally, the sources of the fundamental materials in this section are reviewed. The analysis of the relative motion near a point is discussed by Batchelor [7].

The theory of boundary layer is covered in Schlichting [3], Rosenhead [8] and White [9].

Applications to low speed aerodynamics are discussed in Moran [10]. Compress­ible viscous flows are studied in Liepmann and Roshko [2] and in Stewartson [6].

Stability of viscous flows as well as transitional and turbulent flows are not covered here and the reader is referred to literature for these topics.