#
A Fast Adaptive Vortex Method in Three Dimensions

### Thomas Buttke

### Abstract

We present a fast vortex method for incompressible flow
in three dimensions. It is based on two ideas. The first is an extension
of Anderson's Method of Local Corrections (MLC) algorithm, a particle-particle
particle-mesh method, from two spatial dimensions to three.
In this approach, the calculation of the velocity field
induced by a collection of vortices is split into two parts:
(i) a finite difference velocity field calculation using a fast Poisson solver,
the results of which are used to
represent the velocity field induced by vortices far from the evaluation
point; and (ii) an N-body calculation to compute the velocity
field at a vortex induced by nearby vortices. The second idea is the use of
adaptive mesh refinement in the finite difference calculation.
Calculations with a vortex ring in three dimensions show
the break-even point between the MLC with AMR and the
direct method is at N approximately 3000 on a Cray Y-MP; for
N approximately 64,000,
MLC with AMR can be twelve times faster than the direct method.
Results from calculations of two colliding inviscid vortex rings are
presented; these results demonstrate the increased resolution which can
be obtained using fast methods.