Technical Report CS0542

Title: A Product Integration Method for Vortex Dynamics in Two and Three Dimensions
Authors: M. Israeli and M.J. Shelley
Abstract: A new, high order, spectral method for evaluating the multi-dimensional integrals appearing in the vorticity formulation of inviscid, incompressible fluid dynamics is presented. This formulation is most relevant for vortex dominated flows which are of major importance in science and engineering, and can be modified to account for visCous effects in high Reynolds number flows. The present method, unlike discrete vortex methods, reduces the singularity in the ( Biot-Savart ) integrand analytically and does not require the introduction of a smoothing finite core function ( "vortex blob" ) , and its associated nonphysical length scale. The support of a general compact vorticity distribution in two or three dimensions is covered by a coordinate system periodic in (at least) one direction. By evaluating the integrals in an iterated fashion, the integrals over the periodic coordinate are evaluated with speetral accuracy ( i.e. the error decays faster then any power of the mesh distance). Special analytic treatment is used for very close contours. The resulting scheme yields good accuracy even at moderate resolution and is expected to improve both short and long time solutions of time dependent flows.
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