The large-scale structure of unsteady self-similar rolled-up vortex sheets

Abstract

Two problems involving the unsteady motion of two-dimensional vortex sheets are considered. The first is the roll-up of an initially plane semi-infinite vortex sheet while the second is the power-law starting flow past an infinite wedge with separation at the wedge apex modelled by a growing vortex sheet. In both cases well-known similarity solutions are used to transform the time-dependent problem for the sheet motion into an integro-differential equation. Finite-difference numerical solutions to these equations are obtained which give details of the large-scale structure of the rolled-up portion of the sheet. For the semi-infinite sheet good agreement with Kaden's asymptotic spiral solution is obtained. However, for the starting-flow problem distortions in the sheet shape and strength not predicted by the leading-order asymptotic solutions were found to be significant.