Vortex methods (see [Leonard:80a]) are used to simulate incompressible flows at high Reynolds number. The two-dimensional inviscid vorticity equation,

is solved by discretizing the vorticity field into Lagrangian vortex particles,

where is the strength or the circulation of the particle. For an incompressible flow, the knowledge of the vorticity is sufficient to reconstruct the velocity field. Using complex notation, the induced velocity is given by

The velocity is evaluated at each particle location and the discrete
Lagrangian elements are simply advected at the local fluid velocity. In this
way, the numerical scheme approximately satisfies Kelvin and Helmholtz
theorems that govern the motion of vortex lines. The numerical
approximations have transformed the original partial differential equation
into a set of **2N** ordinary differential equations, an **N**-body problem. This class of problems
is encountered in many fields of computational physics, for example,
molecular dynamics, gravitational
interactions, plasma physics and, of course,
vortex dynamics.

Wed Mar 1 10:19:35 EST 1995