Abstract

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.