10. Conclusions

In this thesis, we have developed a mesh-free, accurate numerical method, called the `vorticity redistribution method', for handling diffusion in vortex methods for incompressible flows.

By construction, our vorticity redistribution method might be considered to be a computed finite difference formula for vortices with disordered locations. Our intention was to design a method with the same independent computational points as the random walk method. This gives it the capability to handle complex separated flows that may be impractical using other schemes. It also allows the computational points to truly follow very strong convection processes. This may be crucial to compute flows with unsteady boundary layer separation [241]. Yet at the same time we wanted to avoid the large and awkward numerical errors in the random walk method.

Our new procedure has been tremendously successful in this quest. The results of, especially, section 8.2 provide overwhelming evidence that our method maintains the advantages of the random walk method, but with a dramatic increase in accuracy. Wang [245] complained that the large and random errors in the random walk methods (such as those between figures 8.23(a) and (b)) made the study of the effects of the true physical parameters very difficult. Our new method now allows such practical applications.

In fact, as discussed in subsection 8.2.9, the evidence suggests that our results for the impulsively translated circular cylinder at $Re=9,500$ may surpass all available finite difference results. Yet, such schemes are clearly less flexible, and some use as much as half a million mesh points where we needed 60,000 vortices. Of course, the CPU time per vortex can be expected to be much larger than the time per mesh point in a finite difference computation, especially for this simple geometry. In any case, of all computations, our results agreed closest with the preliminary spectral element results of Kruse & Fischer [120] who used half a million nodal points in half the flow region. Spectral methods are of course targeted to very different conditions than our mesh-free method. We also showed that our method works equally well in three dimensions as in two.

We further explained that it has significant further theoretical advantages over other vortex methods in addition to the fact that it is mesh-free. Our method can show a better performance for small scale phenomena; compare the discussion about a diffusing point vortex in section 9.1. This reduces the number of vortices needed. Our method uses its vortices also very efficiently since they are distributed only where there is vorticity; hence our method uses still less vortices than other methods. Our method adds new vortices automatically when the vorticity diffuses toward new locations or where straining depletes regions of vortices. Particle methods periodically restore the vortex distribution, interpolating the vorticity of the new distribution from the old. This introduces complexity as well as interpolation errors.

Other advantages of our method are that the conservation laws are satisfied exactly, and that our method can preserve the sign of the vorticity exactly even at high orders of accuracy. Another significant advantage is that there are few computational parameters in our method, and they are easy to choose. We further never needed to ``adjust" our parameters; this makes our method highly reliable.

While Fishelov's method [78] can easily achieve spectral accuracy, our method cannot. On the other hand, there is also no fundamental limitation on the order of accuracy that can be achieved by our method. However, more than first order accuracy has not yet been demonstrated in actual computations. Also accuracy beyond fourth order is not completely trivial since it requires additional consideration of the time splitting error.

The computational time required for the vorticity redistribution method is of the order of the time required for convection. In typical computations based on particle methods, on the other hand, the computational time needed for diffusion is small compared to the time needed for convection. However, at least for the case of the impulsively translated cylinder at $Re=9,500$, the particle methods use much more vortices and hence the time required for redistribution seems not really relevant for that flow. In addition, we must point out that so far we have not made any serious attempt to reduce our computational time. We certainly have no doubt that the various brute force approaches we have followed so far are unnecessarily inefficient. Since we solve two identical diffusion problems back to back, a factor 2 reduction in time would be trivial at the expense of additional storage. Interestingly, in theory the reduction in computational effort could still be arbitrarily (``infinitely") large. Future studies will have to determine how much of that is really possible.

In addition to improving the computational time, further work on the vorticity redistribution method may be continued in many ways. One is to extend the method to handle different boundary conditions; for example, free-surface and periodic boundary conditions. In section 7.3 the method was applied to three-dimensional Stokes flows; this work could be continued to flows including convection, and to flows with no-slip or other boundary conditions. Finally, one could also investigate the feasibility of extending the redistribution method to compressible flows. The fact that it is in fact a computed finite difference formula suggests a wide further range of applicability.