Computation of Vortex Sheet Roll-up

In this article we shall review some recent developments for computing vortex sheet roll-up. A vortex sheet is an asymptotic model of a free shear layer in which the transition region between the two fluid streams is approximated by a surface across which the tangential velocity component is discontinuous. A common theme in fluid dynamics is that the vortex sheet model can be useful in understanding the dynamics of coherent vortex structures observed in laminar and turbulent flows. If this goal is to be realized, reliable methods for computing vortex sheet evolution must be developed. At present, numerical methods are available for studying the initial value problem in two space dimensions. For example, detailed analytical phenomena such as singularity formation in the shape of an evolving periodic vortex sheet can be studied with existing methods. The complex roll-up process and the interaction of several spiral vortices has also been investigated numerically. These calculations have been stimulated by recent theoretical results about vortex sheets and by progress in the convergence theory of general vortex methods. First, results for the periodic vortex sheet will be reviewed. Then an application to some vortex sheet problems occurring in aerodynamics will be discussed in more detail. Finally, some open questions and directions for further research will be summarized.