Analytical and Computational Challenges of Incompressible Flows at High Reynolds Number

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Calculation of Complex Singular Solutions to the 3D Incompressible Euler Equations

Mike Siegel

                            Department of Mathematics, New Jersey Institute of Technology

Abstract:   We describe an approach for the construction of singular solutions to the 3D Euler equations for complex initial data. The approach is based on a numerical simulation of complex traveling wave solutions with imaginary wave speed, originally developed by Caflisch for axisymmetric flow with swirl. Here, we simplify and generalize this construction to calculate traveling wave solutions in a fully 3D (nonaxisymmetric) geometry. We also discuss a semi-analytic approach to the problem of Euler singularities based on numerical computation of the complex traveling wave solutions, followed by perturbation construction of a real solution. The perturbation analysis depends on a small amplitude of the singularity in the traveling wave solution; techniques for producing such a small amplitude are described. This is joint work with Russel Caflisch.
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