| dc.description.abstract | Vortex-element methods are often used to efficiently simulate
incompressible flows using Lagrangian techniques. Use of the FMM (Fast
Multipole Method) allows considerable speed up of both velocity
evaluation and vorticity evolution terms in these methods. Both
equations require field evaluation of constrained (divergence free)
vector valued quantities (velocity, vorticity) and cross terms from
these. These are usually evaluated by performing several FMM accelerated
sums of scalar harmonic functions.
We present a formulation of the vortex methods based on the
Lamb-Helmholtz decomposition of the velocity in terms of two scalar
potentials. In its original form, this decomposition is not invariant
with respect to translation, violating a key requirement for the FMM.
One of the key contributions of this paper is a theory for translation
for this representation. The translation theory is developed by
introducing "conversion" operators, which enable the representation to
be restored in an arbitrary reference frame. Using this form, extremely
efficient vortex element computations can be made, which need evaluation
of just two scalar harmonic FMM sums for evaluating the velocity and
vorticity evolution terms. Details of the decomposition, translation and
conversion formulae, as well as sample numerical results are presented. | en_US |
