A new method is proposed for simulating diffusion in vortex methods
for incompressible flows. The method resolves length scales up to the
spacing of the vortices. The grid-free nature of vortex methods is
fully retained and the distribution of the vortices can be irregular.
It is shown for the Stokes equations that in principle, the method can
have any order of accuracy. It also conserves circulation, linear and
angular momentum. The method is based on exchanging a conserved
quantity between arbitrary computational points. This suggests that
extensions to more general flows may be possible. For the
incompressible flows studied, circulation is exchanged between
vortices to simulate diffusion. The amounts of circulation exchanged
must satisfy a linear system of equations. Based on stability
considerations, the exchanged amounts should further be positive. A
procedure to find a solution to this problem is formulated using
linear programming techniques. To test the method, first some
two-dimensional flows due to the decay of point vortices in free space
are computed; specifically, the decay of a single point vortex and
that of a counter-rotating pair of point vortices are computed. Next,
the method is extended to handle the no-slip boundary condition on
solid walls for two-dimensional flows. To test the numerical handling
of the no-slip boundary condition, flows over impulsively rotated and
translated cylinders are computed. The method is also extended to
handle diffusion in three-dimensional incompressible flows; to test
this, the Stokes flows of a pair of vortex poles and of a vortex ring
are computed. Finally, the advantages and current limitations of our
method are discussed.
