6.2.4 Merging vortices

For many practical applications, high Reynolds numbers are of most interest. For such applications it would be desirable that the number of vortices within a redistribution radius remains finite in the limit of infinite Reynolds number. However, it is evident from figures 6.2 and 6.3 that the number of vortices increases without apparent bound when the Reynolds number is increased. One reason is the use of a scaled viscous time in figure 6.2; for a constant physical time the total number of vortices decays with the Reynolds number.

Yet even at a constant time, the average number of vortices in a redistribution radius still increases with the Reynolds number. The reason seems to be that fluid straining is particularly strong for this flow; there is no bound on the magnitude of the velocity at any given time when the Reynolds number increases. It is however desirable to prevent a significant increase in number of vortices under all circumstances, since it results in loss of numerical efficiency. We can achieve this by simply merging vortices which move very close to eachother together.

In the circular cylinder computations [202,203,204] presented in chapter 8, we replaced vortices of the same sign that moved very close together by a single combined vortex at their center of vorticity. In this procedure, the net circulation and the center of vorticity of the vortices are preserved. In our actual implementation, once every six time steps we searched for and combined vortices that are located within a mutual distance of $\sqrt{0.5 \nu \Delta t}$. On average this reduced the number of vortices by about $1.5\%$ each time. Using this procedure, we did not experience a significant increase in scaled vortex density with Reynolds number.

It should be emphasized that condensing nearby vortices into single vortices is not the same as the need to regenerate the mesh in particle methods. First, the only purpose here is merely to increase numerical efficiency, not to maintain accuracy. Our computation can continue without it, although at lower efficiency. Second, there is no need to produce a new ordering, or association, of the computational points; there is no repartitioning of the domain; there is no quadrature rule to update. We simply give one vortex the combined strength and location, and drop the other vortex from the further computation.

It is even possible to incorporate this condensation directly into the redistribution process itself. For a vortex located close to another vortex, we might simply try to find a solution to the redistribution equations that does not involve the vortex itself. The vortex then loses all its circulation and can be removed. The redistribution fractions could be required to be positive as before, or a less restrictive condition might be imposed to remove even more vortices. In particular, the convergence analysis for Stokes flow in chapter 5 would not be affected if the fractions were merely bounded in the $l_1$ norm and the circulation was allowed to grow by a relative amount $O(\Delta t)$. More research is needed to settle these points.

Next we discuss the ``cut-off" circulation mentioned in subsection 6.2.3.