The purpose of the vorticity redistribution method is to simulate the diffusion of each vortex during a time-step. As sketched in figure 4.1, this is done by distributing fractions of the circulation of each vortex to its neighboring vortices. The question is how to select the neighboring vortices and the fractions so that the correct diffusion is approximated. We will answer that question in the following discussion.
First, vortices will be considered to be within the neighborhood of a given
vortex if they are within a predetermined distance from that vortex.
We take this distance to be of the order of the
typical diffusion distance during a time-step.
To be precise, the typical diffusion distance will be defined as
The diffusion of a vortex will be approximated by moving fractions of its circulation towards the other vortices within this neighborhood. We will indicate the fraction moved from vortex to a vortex by , where indicates the time level. Implementation of the redistribution method is in principle merely a matter of determining fractions that approximate the correct diffusion over a time-step accurately and stably.
Yet, we choose not to identify the vortex strengths
with any particular smooth interpolated vorticity distribution.
The reason is that due to straining effects, the vortex locations
can become very irregular.
In the absence of a continuous vorticity field,
the question arises how a meaningful representation of
the diffusion process can still be achieved. Regardless
of the interpretation of that representation,
the redistribution method changes a vorticity distribution
at time level
The Fourier transform of the new vorticity distribution is
The resulting equations are the redistribution equations we were looking for.
They involve scaled relative vortex positions defined as
In terms of the scaled coordinates, the final redistribution equations are
Consistency requires that the numerical solution approximates the diffusive changes in the exact solution: the redistribution fractions must at least satisfy (4.8) through (4.10). This results in a truncation error of order . Subsequent equations, (4.11), (4.12), can be included to achieve a higher order of accuracy .
Thus in principle, the accuracy can be increased arbitrarily, although for the Navier-Stokes equations the splitting error also has to be considered. The conditioning of the above system of equations also needs to be taken into account in a practical application. The equations could be recast in terms of orthogonal polynomials such as Legendre polynomials to improve the conditioning. On the other hand, the conditioning of the system may not be very important; the requirement is not to find a particular solution for the fractions , but to satisfy the equations accurately. In the numerical results in this paper, we simply solved (4.8) through (4.10) in the form shown.
The redistribution equations are similar to the equations obtained when a Taylor series expansion of the exact solution is substituted into a finite difference formula, or to the moment conditions in the particle methods. In fact, consistency of a finite difference scheme requires the same agreement for finite wave numbers; see Strikwerda ([219] (10.1.3)), for example. For uniform point spacing and redistribution fractions, the redistribution method is equivalent to an explicit finite difference scheme. The redistribution equations do not involve the smoothing function in (4.3). This allows us to choose this function after the actual computation has already been completed.
In implementing the redistribution scheme,
it is important to realize that not all solutions to
(4.8) and following will lead to a convergent
approximation.
For example, a consistent but unstable explicit
finite difference scheme would satisfy the equations.
Some form of stability condition needs to be imposed;
following Van Dommelen [235], we will demand that all
fractions are positive:
In the next two sections we will further justify the above conditions using physical and mathematical arguments. However, the truly relevant questions are clearly whether the equations are solvable, whether they can be solved using only a finite number of neighboring points within a finite scaled distance , and whether the numerical solution approaches the exact solution with the expected rate of convergence. In the following chapters 5 and 6 we will prove that the answer to all these questions is affirmative for the linear Stokes equations. To verify that our method also works for the nonlinear Navier-Stokes equations, we will present example computations with nontrivial convection effects in chapters 7 and 8.