The vortex-entrainment sheet

A vortex sheet is a well-known inviscid model of high Reynolds number viscous boundary and shear layers. The circulation in the layer is preserved via a jump in the harmonic potential across the sheet; there is also a jump in the tangential velocity that defines the vortex sheet strength. However, viscous layers necessarily contain mass and momentum, in addition to vorticity, which are entrained by a normal velocity at the edge of the layer. Hence, a more complete model of a viscous layer or wake that is collapsed to an infinitely thin sheet should account for the entrainment as well as the vorticity. We propose such a model, termed a vortex-entrainment sheet, which is characterized by jumps in the harmonic potential and the stream function or, equivalently, by jumps in both the tangential and normal components of velocity. For separation at a sharp edge, the singular pressure gradient is neutralized, in accordance with the normal momentum balance, by delivering an instantaneous impulse to the fluid. A non-tangential shedding angle of the sheet from the sharp edge necessarily requires non-zero entrainment.