Three-dimensional boundary layers with short spanwise scales

We investigate three-dimensional (laminar) boundary layers that include a spanwise scale comparable to the boundary-layer thickness. A forcing of short spanwise scales requires viscous dissipation to be retained in the two-dimensional cross section, perpendicular to the external flow direction, and in this respect the flows are related to previous work on corner boundary layers. We use two examples to highlight the main features of this category of boundary layer: (i) a flat plate of narrow (spanwise) width, and (ii) a narrow (spanwise) gap cut into an otherwise infinite flat plate; in both cases the plate is aligned with a uniform oncoming stream. We find that a novel feature arises in connection with the external flow; the presence of a narrow gap/plate (or indeed any comparable short-scale feature of long streamwise extent) necessarily modifies the streamwise mass flux in that vicinity, which in turn induces an associated boundary-layer transpiration on the same short spanwise length scale. This (short-scale) transpiration region leads to a half-line-source/sink correction to the outer inviscid, irrotational flow. Crucially, the volumetric-flux associated with this line-source/sink must be explicitly included as part of the computational procedure for the leading-order boundary layer, and as such there is a weak interaction between the outer (inviscid) flow and the boundary layer. This is a generic feature of boundary layers that are forced through the presence of short-scale spanwise variations.