"Similar" Three-Dimensional Boundary Layers

In two-dimensional boundary-layer flows, almost exclusively considered up to the present time, all flow parameters are independent of the third coordinate. In three-dimensional flow processes, in contrast, the outer potential flow is a function of two space coordinates parallel to the wall surrounded by the flow, whereas the boundary-layer flow shows all three velocity components which, in the most general case, are dependent on all three space coordinates. Besides the addition of a new coordinate and velocity component, several novel phenomena arise in the three-dimensional case: If, for instance, the streamlines of the outer potential flow are rectilinear but convergent or divergent, there occurs, essentially, only a variation of the boundary-layer thickness, compared to the two-dimensional flow. In the case of lateral curvature of the outer potential streamlines, in contrast, a so-called "secondary flow" appears. The reason for the latter is: In the external flow the pressure gradient normal to the flow direction and the centrifugal force are in equilibrium. The same pressure gradient prevails inside the boundary layer as outside, whereas the centrifugal force is reduced due to the lower velocity so that the fluid particles can more easily yield to the transverse-pressure gradient. Thus a transport of fluid toward the concave side of the potential streamlines is produced. The appearance of this "secondary" or "transverse flow" causes the direction of the streamlines directly at the wall to deviate sometimes quite considerably from the direction of the outer potential streamlines.