9. Discussion

In the previous chapter we demonstrated the superior performance of our method compared to other procedures by means of actual flow computations. In this chapter we will explore the theoretical reasons that give rise to this performance. We will contrast our method mainly to other vortex methods such as the Particle Strength Exchange method (subsection 1.3.3) and the related method by Fishelov (subsection 1.3.4).

Although in other studies we have found the random walk method to be robust, simple to apply, and quite reliable, in our opinion the results of the previous chapter using our new method seem to take it out of the running as a contender.

Of course, the other vortex methods do not have the independent computational points our method has. So in this chapter we will restrict ourselves to flows for which this is less important: flows around relatively simple configurations without very strong convective process. We will give theoretical reasons why our method may still be preferable even for such simple flows.

Like the particle methods, the vorticity redistribution method simulates the diffusion of a vortex by moving circulation to neighboring vortices. This similarity is rather superficial, however, since Lagrangian finite difference, finite element and spectral representations of the diffusion process would all do this. By construction, the vorticity redistribution method is closest to a finite difference method, rather than to a particle method, and reduces to one when the distribution of the vortices is uniform. For that reason, it might be considered to be a computed finite difference formula.

Finite difference, finite element, spectral, particle, Fishelov's, and our own method differ principally in the way the amounts of circulation to be moved to neighboring vortices are determined, and in the number of neighboring vortices involved. For example, the particle methods transfer circulation between vortices proportional to the local value of a ``diffusion kernel" [72,117]. In contrast, the vorticity redistribution method computes the amounts to be transferred from procedures similar to ones used to construct finite difference formulas. This allows the vorticity redistribution method to satisfy the necessary equations using only a finite number of vortices, similar to a finite difference method.

Particle methods cannot do this. For these methods to converge, the diffusion kernel must have a size $\delta$ that is asymptotically large compared to the point spacing $h$; it must be integrated correctly, [72,78]. As a result, in particle methods the diffusion of a vortex involves an unbounded number of neighboring vortices, as in a spectral method.

In practice, $\delta=O(h^p)$, $0<p<1$, and $p$ may be as small as 0.08 or 0.1 [46]. Other computations have used much smaller cores, but remeshed frequently to uniformly distributed vortices. For example, Pépin [170] uses $\delta/h = 1.8$ to compute the flow around a cylinder at Reynolds number 9,500, but reports remeshing every six or seven time steps.