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To: 9.5 Conservation laws and positivity |
It has been argued that the vorticity redistribution method introduces significant additional complexity in a vortex computation. We cannot agree with this sentiment, at least not if a fast summation algorithm is used to find the velocity field in the convection step. In its simplest implementation, followed in this paper, the redistribution method needs to do two things for each vortex: identify the neighboring vortices and solve the redistribution equations.
Neighboring vortices are already identified by the fast summation procedure used to find the velocities. In our program, that part of the fast summation process was simply repeated in the redistribution stage to account for the different neighborhood sizes in fast summation and redistribution. Thus there is no significant further complexity with regard to this requirement.
While our method also requires the solution of the redistribution equations, this does not truly add complexity to our procedure either. Solution of a linear system of equations is a standard mathematical problem, not a problem specific to the vorticity redistribution method. Ideally, the redistribution equations are merely handed to a `canned' library routine for solution. Actually, we wrote our own subroutine based on an algorithm found in literature [1].
If there is no solution to the redistribution equations, a vortex needs to be added. This too, is a very simple process. We simply try a few locations for this vortex and stick it in the largest hole we find.
It is important to note that none of the above requirements depends on what flow is being computed. The complexity of the flow does not affect them. The length scales and the strength of the convective processes do not affect them. No parameters need to be chosen based on the flow properties. In other words, these issues need to be addressed only once.
While admittedly the random walk method is even simpler to apply than our method, in our opinion particle methods are not. First, while our method extends the vortex distribution automatically from the solid surfaces into the field, a particle method faces a separate mesh generation problem: it needs to create an effective partitioning and quadrature procedure. Such problems can become difficult for complex configurations. Yet, in order to resolve the diffusion cores with the minimum number of vortices, an effective vortex distribution is highly desirable.
Further, a particle method needs to monitor its vortex distribution. It needs to formulate criteria that determine whether a given distribution, with widely varying local properties, needs to be updated globally. Or it needs to address the even more complex issue of local updating. It needs to update the vortex distribution without introducing artificial diffusion or smearing steep gradients. The best choices for the time interval between updating, the new point distribution, the transfer of vorticity between meshes, etcetera, all depend on the actual flow being computed. Optimal choices will require trial and error.
Furthermore, particle methods face the need to select a smoothing function to perform diffusion. This function must be selected a priori in order to be able to perform the computation of the diffusion process. Since at that time not much may be known about the flow to be computed, an optimal choice will not always be a simple task.
The vorticity redistribution method has it much easier since the computation of the diffusion processes does not depend on a smoothing function. A smoothing function is only used in the final evaluation of the results. At that time, much more information is available since the strength of the vortices has already been determined. It also makes it possible to optimize the size or shape of the smoothing function based on the computed properties without repeating the complete computation.
Admittedly, in an actual computation at nonzero Reynolds number, a smoothing function must still be used to find the velocity field. However, experience indicates that the choice of this smoothing function is often not very critical. For example, Milinazzo and Saffman [155] obtained meaningful random walk results with a very small smoothing function. Goodman, Hou, and Lowengrub [91] show that no smoothing function is necessary if the vortices are initially uniformly spaced.
Thus the vorticity redistribution method is simple to apply and flexible. On the other hand, additional complications can certainly arise if its performance is to be optimized. For example, while extended convection is not a problem for accuracy as in particle methods, it can certainly reduce numerical efficiency. As discussed in subsection 6.2.3, it is desirable to combine vortices that approach very closely. While there are no associations between vortices that need to be maintained, it would still have to be shown that this process does not introduce instability or inaccuracy. We note however that we have used it without difficulty for the computations in section 8.2 and elsewhere [202,203,204].
Similarly, solving the redistribution equations from scratch at every time-step seems wasteful: as pointed out in subsection 6.2.1, these equations change little from one time-step to the next. Yet, to use the solution of one time-step during the next one would clearly add complexity, such as what information to save from one time-step to the next, and how to update the old solution.
For higher order of accuracy, the conditioning of the redistribution equations needs to be considered. It would need to be determined whether it might be advantageous to recast the equations in other equivalent forms. The effects of numerical errors in the solution process would need consideration. Again, such considerations would be independent of the flow being considered.