Vortex Methods
Acknowledgments.
The author (LFR) gratefully acknowledges the support of the National Science Foundation, Division of Mathematical
Sciences for their support of this research.
Grant DMS-9971800.
WARNING: Some of this page will seem technical to people who do not
study fluid dynamics. However, even those who do not study fluid dynamics
may enjoy parts of this page.
What is a Vortex Method?
A vortex method is a computation technique for simulation fluid flows. "Fluid
flow" most generally refers the motion of either gasses or liquids. To simulate
the fluid flow, vortex methods attempt to simulate only the evolution of
the vorticity field which is the curl of the velocity field. The reason why
some people are only interested in the vorticity of a flow is that there
are many interesting flows where the vorticity is confined to a very small
region of space even though the whole flow occupies are larger or even unbounded
area. For instance, a tornado is a very localized region of vorticity that
affects the weather all around it. A vortex method represents the vorticity
as a linear combination of localized, basis functions that freely convect
in the flow. This can be thought of as solving a problem on a moving grid.
Various properties of the vorticity equations make this an advantageous way
to solve this problem.
The BlobFlow Project... It's free!
The BlobFlow project
is aimed at developing a solid vortex code as open source for the scientific
community. The project page includes source code, some documentation,
and some examples.
My main interests:
Many people both use and study vortex methods. This page is dedicated to
my research interests because this is my way of sharing what I am working
on. Perhaps one day, there will be a Vortex Method Page which can point to
this one and others. Until then, this is a nice way for me to share information
with other researchers. Here is where I am concentrating my effort right
now.
- Corrected core spreading methods. Core spreading died a fast
death in the late 70's because it was shown to be inconsistant. I corrected
the method as part of my PhD thesis. Essentially, the method uses Gaussian
basis functions that spread at a rate governed by the viscosity of the fluid.
When blobs grow too wide, they must split so that the method remains accurate.
When one element splits into several the new configuration approximates the
field induced by the original element.
- Boundary conditions. Unlike methods that work in primitive
variables such as velocity or pressure, the boundary conditions for vorticity
are nonlocal and much more complicated.
- Convergence properties. Whenever one develops new methods,
one must prove (or at least demonstrate over a wide range of problems) that
they will converge to an exact solution. I have done some work in developing
alternative convergence formulations so that vortex methods are easier to
analyze.
- Higher order methods. Once one can describe the convergence
properties of a method, it is interesting to see if one can boost its accuracy.
I have found that this can be done to the Corrected Core Spreading Method
by using elliptical Gaussian basis functions. Again, when elements grow too
elongated or too wide, they must split into several elements.
References.
Here are some more detailed references for those interested in my work.
L. F. Rossi. High Order Vortex Methods with Deforming Elliptical
Gaussian Blobs 1: Derivation and Validation. Submitted to SIAM J.
Sci. Comput. Lost by SIAM J Sci. Comput. until 2001.
L. F. Rossi. Resurrecting Core Spreading Methods: A New Scheme That Is Both
Deterministic and Convergent. SIAM J. Sci. Comp. 17-2. 1996.
L. F. Rossi. Merging Computational Elements in Vortex Simulations. SIAM
J. Sci. Comp. 18-4. 1997.
Last modified: 24 April 2002.
rossi@math.udel.edu