A Lagrangian Vorticity Collocation Method for Viscous, Axisymmetric Flows with and without Swirl

ISSN： 0021-9991

Source： Journal of Computational Physics, Vol.138, Iss.2, 1997-12, pp. : 302-330

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Abstract

This paper presents a new Lagrangian vorticity collocation method for viscous, axisymmetric fluid flows with or without swirl. The velocity calculation is performed using a representation of the vorticity field in terms of Gaussian vortex rings for off-center points and Gaussian vortex blobs along the axis of symmetry. A matrix equation for the element "amplitudes" is obtained via a collocation approach, by fitting the vorticity representation to the known vorticity values at the control points, prior to each velocity evaluation. This matrix equation is solved using an iterative procedure, which both speeds the calculation and filters out "noise" in the element amplitudes. Viscous diffusion is accomplished with use of a diffusion velocity method, in which control points are moved with the sum of the local fluid velocity and an additional "diffusion velocity" that accounts for the effect of viscosity on the spread of vorticity support. Derivatives are obtained by locally fitting a polynomial function to control points about a given control point via a least-square formulation and then differentiating the polynomial. This method is found to maintain high accuracy even for very irregularly spaced control points. The method introduced for viscous diffusion in the paper can also be used for three-dimensional vortex methods in general. Copyright 1997 Academic Press.