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On some analytic and computational aspects of two dimensional vortex sheet evolution

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1435)

Abstract

This survey paper gives an account of recent analytic and numerical results of the initial value problem:

$$\begin{gathered}\frac{{\partial \bar Z}}{{\partial t}}\left( {\gamma ,t} \right) = \frac{1}{{2\pi i}}f_{ - \infty }^\infty \frac{{d\gamma '}}{{Z\left( {\gamma ,t} \right) - Z\left( {\gamma ',t} \right)}}, \hfill \\Z\left( {\gamma ,0} \right) = \gamma + S\left( \gamma \right), \hfill \\\end{gathered}$$

which is the Birkhoff-Rott equation for the evolution of a slightly perturbed flat vortex sheet. We will indicate some open problems of current research and propose a new physically desingularized Vortex sheet equation, which agrees with the finite thickness vortex layer equations in the localized approximation.

Keywords

  • Singularity Formation
  • Point Vortex
  • Vortex Sheet
  • Roundoff Error
  • Vortex Layer

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The author was economically supported by FONDECYT (grant number 235, 1987–88) and Universidad Técnica Federico Santa María (grants 88.12.08, 1988 and 89.12.08) 1989).

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© 1990 Springer-Verlag

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Orellana, O.F. (1990). On some analytic and computational aspects of two dimensional vortex sheet evolution. In: Ruscheweyh, S., Saff, E.B., Salinas, L.C., Varga, R.S. (eds) Computational Methods and Function Theory. Lecture Notes in Mathematics, vol 1435. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0087904

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  • DOI: https://doi.org/10.1007/BFb0087904

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