The 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
A new deterministic scheme called the
VRM simulates diffusion by redistributing
fractions of the circulation of each vortex element to its neighboring
elements, such that the are positivity
to preserved. 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: arbitrarily
high order-
- Unlike other deterministic methods, VRM does not require frequent remeshing or a background grid to maintain long time accuracy, and it is free of numerical diffusion.*Truly Grid-Free* -
- VRM is capable of predicting diffusion at high accuracy and high rates of convergence, while maintaining positivity. Other methods can at best be second order accurate.*Arbitrarily High Order with Positivity* -
- The VRM formulation mathematically detects*Intelligent Particle Insertion Strategy**when*and*where*new vortex particles must be added to the computational domain to mimic the (physical) expansion of the vorticity field due to diffusion. Other methods use*ad hoc*user specified particle insertion strategies.
For this project, we have developed a (better than)
area, but
its zero support has value circulation.one
We should emphasize here that
using initialized 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).one particle
The following plots depict
The figure on the right shows a
area, and zero support
values circulation± are diffused into each
other with unit viscosity.one
The following
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. |