3D Boundary Layers

Three dimensional boundary layer flows can be quite complex, and a complete treatment of 3D boundary layer theory is far out of scope here. Only a general overview of the key new effects will be given here. See McLean [18] for a much more comprehensive discussion.

4.9.1 Streamwise and crossflow profiles

A 3D boundary layer features non-planar velocity profiles V(n), such as the one shown in Figure 4.18. Traditional notation uses s,s2,n as the local cartesian coordinates (instead of s,£,n), with si parallel to the edge velocity vector Ve, and s2 perpendicular to it. Within the boundary layer we then have

V(n) — ui si + U2 S2 (4.72)

where u1(n) is the streamwise profile, and u2(n) is the crossflow profile which appears only in 3D boundary layers. The presence of crossflow means that the wall shear stress vector Tw is in general not parallel to the local edge velocity. The lines parallel to Tw are called wall streamlines, which differ from the usual potential-flow streamlines which are parallel to Ve.

Crossflow is typically generated by a transverse pressure gradient dp/ds2 which is felt by the fluid over the entire boundary layer thickness. The slower-moving fluid within the boundary layer curves in response more strongly than does the outer potential flow, and thus forms the crossflow profile some distance downstream, as shown in Figure 4.18 on the right. This is essentially the same mechanism as the one shown in Figure 4.6, but here it occurs in the transverse direction.