The vortex-entrainment sheet in an inviscid fluid: theory and separation at a sharp edge

Abstract

In this paper a model for viscous boundary and shear layers in three-dimensions is introduced and termed a vortex-entrainment sheet. The vorticity in the layer is accounted for by a conventional vortex sheet. The mass and momentum in the layer are represented by a two dimensional surface having its own internal tangential flow. Namely, the sheet has a mass density per-unit-area making it dynamically distinct from the surrounding outer fluid. The mechanism of entrainment is represented by a discontinuity in the normal component of velocity across the sheet. The sheet mass is able to support a pressure jump, which in turn may cause additional entrainment. This feature was confirmed when the model was used to represent the Falkner-Skan boundary layers. The velocity field induced by the vortex-entrianment sheet is given by a generalized Birkhoff-Rott equation with a complex sheet strength. The model was also applied to the case of separation at a sharp edge. There is no requirement for an explicit Kutta condition in the form of a singularity removal as this condition is inherently satisfied through an appropriate balance of normal momentum with the pressure jump across the sheet. A pressure jump at the edge results in the generation of new vorticity. The shedding angle is dictated by the normal impulse of the intrinsic flow inside the bound sheets as they merge to form the free sheet. When there is zero entrainment everywhere the model reduces to the conventional vortex sheet with no mass. Consequently, the pressure jump must be zero and the shedding angle must be tangential so that the sheet simply convects off the wedge face. Lastly, the vortex-entrainment sheet model was demonstrated on two shedding example problems.