9.1 Resolution of short scales

The requirement of particle methods that the diffusion core size must be asymptotically large compared to the vortex spacing can be a disadvantage. It is not the vortex spacing, but the larger core size that limits the smallest scales that can be resolved during the computation.

This is especially so since the core size must be chosen before the computation can be conducted, at a time when little precise information about the flow to be computed is likely to exist. When widely differing strain rates cannot be excluded beforehand, it may be tempting to make the core sufficiently large to ensure that it will remain well resolved during the computation. A choice that optimizes both the errors in small scales and the errors in discretizing the core may not be very easy to make.

In contrast, the vorticity redistribution method proceeds without a core. The smallest scales for which the computation is meaningful are limited by the redistribution radius, which is of the order of the point spacing, not asymptotically large compared to it.

If the vorticity field itself is desired at some given time, we still need to evaluate it using an evaluation core, but this is a different problem. At the evaluation stage, all information about the solution is known, and the core can be selected based on the actual solution properties at the given time. In practice, we reduce the core size until short wave errors start to show up. In principle, it would even be possible to select a core size based on the local solution properties, but so far we have always used a spatially constant core.

Furthermore, the evaluation core does not affect the actual computation: all information about the short waves remains available for the convection algorithm to use.

In practical applications, significant short scales might be due to rapid changes in boundary conditions or due to strong straining during the vortex ejection from boundary layers that follows the unsteady separation process discovered by Van Dommelen and Shen [241] (see section 8.2). As a simple model example involving short scales we will address the case of a diffusing point vortex. This is the fundamental solution of the diffusion equation, and presents the limiting case where the entire initial vorticity distribution has such a small scale that it computationally appears to be a point. According to the exact solution for a diffusing vortex, the vorticity diffuses out over a typical distance $\sqrt{\nu t}$. The particle methods are inaccurate for times for which this diffusion distance is still small or finite compared to the kernel size $\delta$.

The method of Degond and Mas-Gallic [72] leaves the vortex largely undiffused during early times: it diffuses only a small fraction of the vortex over an area of typical size $\delta$. Instead, it should diffuse all of the vortex over the distance $\sqrt{\nu t}$. The method of Fishelov [78] initially also leaves the vortex undiffused; it simulates the diffusion by the creation of negative and positive vorticity within a region of size $\delta$.

While the smallest vortex size that the particle methods can resolve is determined by the size of the kernel, asymptotically the point spacing must be much smaller. As a result, there is a range of times for which the vortex is already large compared to the point spacing, but small or finite compared to the kernel. For those times, the particle methods give inaccurate results.

On the other hand, the vorticity redistribution method gives a valid approximation to the exact solution as soon as the size of the vortex $\ell=\sqrt{\nu t}$ becomes large compared to the typical point spacing $h_v$. We simply take the size of our smoothing function $\delta = h_v^{\alpha} \ell^{1-\alpha}$ for some $\alpha<\frac12$. According to the error bounds (5.3) and (5.11), for $\alpha$ close to zero this produces an $L_2$ relative error of almost $O(h_v/\ell)^M$, which is the best accuracy that can reasonably be expected. In this case we are using a smoothing function with a variable core size. However, this has no consequences; the redistribution process is independent of the smoothing function. The smoothing function is merely used in the final evaluation of the solution, and can be optimized for the instantaneous properties of the computed solution.

For still earlier times, after only a finite number of time-steps, the size of the delta function is of the order of the point spacing, and an accurate representation of the vorticity is not possible. This is not a shortcoming of the vorticity redistribution method. The initial data given to it cannot distinguish whether the initial condition is a true point vortex or a spike of a size smaller than the spacing of the numerical points. Thus the solution is truly indeterminate as long as the vortex distribution is of the order of the point spacing. The best that can be hoped for during these times is that the numerical solution gives the correct typical size of the vortex distribution. Since the vorticity redistribution method uses a finite scaled redistribution radius $R$, it restricts diffusion to a region of the correct order of magnitude. Moreover, except for the uncertainty in the initial data, the correct root mean square radius of the vorticity distribution is achieved.

How important the difference is for practical computations remains to be decided. As mentioned in the beginning of this section, results for the ratio of point spacing to core size vary. Some computations have used a ratio quite close to one. These computations have maintained a highly efficient vortex distribution by frequent remeshing. However, this does reintroduce concerns with regard to regeneration times, mesh generation, interpolation errors, quadrature errors, etcetera, that a truly Lagrangian computation attempts to avoid.

In any case, in practical applications we did find that our method works well at relatively low numbers of vortices. For example, in the circular cylinder computations at a Reynolds number of 9,500 in section 8.2, we found that our method with about 60,000 vortices gives results that equal or exceed all existing finite difference, spectral and vortex method data, even those using many hundred thousands of points. For the number of vortices we used there, the smallest scales, such as the boundary layer thickness, are not much larger than the typical point spacing.

We do want to point out a concern about our method when it is applied to vortices arranged according to a smoothly varying mesh distribution. Our method was not designed for such a purpose; we were interested in truly Lagrangian computations when convection has thoroughly mixed the vortices. Yet our method can be used on a uniform vortex distribution as well, since it will simply generate an explicit finite difference formula with good conservation and positivity properties. However, when our method is applied on a smoothly varying mesh of vortices, instead of a uniform one, our choice to solve the redistribution equations using a least maximum procedure is probably not the best one. The resulting redistribution weights do not always depend smoothly on the vortex positions. This will produce unnecessary short wave errors from the long wave components. Some further work could avoid this, making our method even more powerful.