Four alternative theoretical treatments of ‘displacement thickness’, and, generally, of the influence of boundary layers and wakes on the flow outside them, are set out, first for two-dimensional, and then for three-dimensional, laminar or turbulent, incompressible flow. They may be called the methods of ‘flow reduction’, ‘equivalent sources’, ‘velocity comparison’ and ‘mean vorticity’.

The principal expression obtained for the displacement thickness δ1 in three-dimensional flow may be written $\delta_1 = \delta_x - \frac{1}{Uh_y} \frac {\partial}{\partial y}\int^x_0 \delta_y dx,$ if, as orthogonal coordinates (x, y) specifying position on the surface, we choose x as the velocity potential of the external flow, and y as a coordinate, constant along the external-flow streamlines, such that hydy is the distance between (x, y) and (x, y + dy); and if also δx and δy are the streamwise and transverse ‘volume-flow thicknesses’ $\delta_x = \frac {1}{U}\int _0^\infty (U - u)\;dz,\;\;\;\;\;\;\delta_y = \frac {1}{U}\int ^\infty_0 v\; dz,$z is the distance from the surface, u and v are the x and y components of velocity, and u takes the value U just outside the boundary layer.