A double-scale investigation of the asymptotic structure of rolled-up vortex sheets

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Abstract
A double scale technique is used to determine the asymptotic behaviour of a rolled-up vortex sheet. The technique relies on a process of averaging out the saw-tooth-like behaviour of the flow variables, which generates a continuous solution having the structure of a vortex filament. The fine-scale behaviour of the flow is described and includes concentrated vorticity on the sheet. Application to the conical vortex sheet allows the solution of Mangler & Weber (1967) to be rederived. A further application, to Kaden's problem, is worked out and the results are in complete agreement with Moore's asymptotic formulae for the shape of the spiral.

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