The vortex doublet as a generating mechanism of inviscid vortex sheets. Part I: separation at a sharp edge.
In two-dimensional flow, the vortex sheet corresponding to inviscid flow separation at a sharp edge is generated by a vortex doublet, which arises as a result of representing the solid surfaces as vortex doublet sheets. Namely, the inviscid limit of the attached boundary layer and its image layer inside the surface comprise the doublet sheet. The sheet strength represents the amount of circulation in the layer and the net strength of interacting layers at the sharp edge relates to the shed circulation. This net strength is a global quantity that represents communication of flow changes to the shedding point. The shedding of a vortex sheet is interpreted as the doublet sheet being `torn apart,' such that one layer is the vortex sheet shed into the fluid and the other layer is the image sheet. The unsteady Kutta condition is manifested by requiring that the strength of the doublet induce a flow that instantaneously and mutually neutralizes itself with the singular pressure gradient of the flow attempting to navigate around the sharp edge. The neutralization of the pressure gradient is accomplished by the inviscid generation of vorticity at the edge and is also the mechanism that tears apart the doublet. These results are obtained at the level of the Euler equation (momentum) instead of Bernoulli's equation (energy). As such, there is a finite force exactly at the sharp edge that is associated with the inviscid generation of vorticity and is proportional to the change of the doublet strength. Furthermore, this force corresponds to an `acceleration reaction' of the fluid that is impulsively accelerated as it passes the sharp edge, which in turn communicates an instantaneous change in the total kinetic energy of the fluid to infinity. For a finite body, this force is finite even for an impulsive acceleration. Example simulations are presented for validation of the derived shedding equations.
Publisher URL: http://arxiv.org/abs/1801.07346