A High Order Method for Grid-Free Diffusion In 3D Free Space
The accurate and grid-free prediction of the diffusion of vorticity is an essential component of the Lagrangian Vortex-Boundary Element Method for viscous flow simulations.
Traditionally, diffusion has been emulated via the Random Walk Method, which is known for its low accuracy and convergence rate as well as the statistical noise it introduces to the computed flow field.
Alternatively, a number of deterministic methods for the accurate prediction of diffusion have been developed in recent years. Most of these methods are not truly grid-free as they require frequent remeshing onto a uniform background grid to improve long time solution accuracy. This diminshes the appeal of grid-free computing since remeshing is a fairly complex procedure near the boundaries, and it introduces some level of numerical diffusion into the computation. In addition, the positivity constraint precludes the formulation of arbitrarily high order schemes with most methods.
A new deterministic scheme called the Vorticity Redistribution Method (VRM) [*] alleviates the above-mentioned difficulties altogether.
VRM simulates diffusion by redistributing fractions of the circulation of each vortex element to its neighboring elements, such that the conservation properties of the system and its positivity are preserved to arbitrarily high order. VRM may be considered to be an explicit grid-free finite-difference scheme, in which the fractions (the finite-difference coefficients) are the unknowns! The significant attributes of VRM are:
2D Point Vortex
We should emphasize here that no other deterministic method for diffusion can solve this problem, since they all discretize the vorticity field (which is initially singular) rather than circulation. In VRM, the point vortex is initialized using one particle with unit circulation. New particles are inserted in subsequent time-steps, as necessary, to emulate the expansion of the vorticity field due to diffusion. Note, since VRM is an explicit scheme, the inter-particle separation is nominally set in the order of the diffusion length scale for stability reasons. (Viscosity is one for this problem).
The following plots depict
The figure on the right shows a second-order convergence rate (in inter-particle separation) for LO, and better than third-order (order 3.6) for HO. This is a significant achievement since all other grid-free methods can at best be second-order to maintain positivity.
The following animation depicts the time evolution of the scattered computational elements (vortex particles), which are color coded by their circulation values.
3D Vortex Ring
The evolution of the vortex particles in time is animated in the figure at the top of this page. Again, note that the particles are scattered with no particular order. More importantly, no special care is necessary to accommodate the "gridding" of the ever shrinking inner hole in the "donut" as the vortex ring/torus expands. That is, the same strategy that is used to diffuse an isolated point vortex is used to diffuse the torus front as it merges with itself in the inner section of the donut.
The following figure depicts a cross-sectional cut and selected vortex tubes from the diffused vortex ring at time level T = 0.2, as predicted by HO using t = 0.01. The colors represent the vorticity magnitude normalized by the maximum in the field. Note how well the field symmetry is preserved.
* S. Shankar and L. van Dommelen, "A new Diffusion Procedure for Vortex Methods," Journal of Computational Physics, Vol. 127, pp. 88-109, 1996.