Vortex methods for incompressible flow problems

I've been interested in the use of vortex methods for different problems associated with incompressible fluid motion and the in the analysis of vortex method accuracy. The work on vortex method analysis is primarily contained in the paper "On Vortex Methods". In addition, the paper also contains suggestions for computing the vortex stretching term in three dimensions and for vortex method initialization.

One application of vortex methods that I've worked on is to flows of variable density. These flows that are interesting because the interaction of density gradients and a gravitational force induce vortical motion within the fluid (e.g. this is the reason steam rises from a cup of coffee). A computational scheme to model such problems is described in "A Vortex Method for Flows with Slight Density Variations". One of the interesting aspects of this paper is the strategy of evolving the density by solving an equation for the gradients and then reconstructing the function using convolution.

Another application concerns the problem of the evolution of vortex filaments. In the early and mid 80's there was a lot of excitement over the possibility that vortex filaments could evolve in such a way that finite time singularities occur. Much of this excitement arose from the results of simulations where filaments with fixed circular core shape were evolved. (The core radius could change, but the shape was confined to being circular). In "The Vortex Ring Merger Problem at Infinite Reynolds Number" we created a simulation in which the core shapes were allowed to deform, and came to the conclusion that the filaments undergo great distortion, and moreover, a type of distortion that inhibits singularity formation.

Vortex methods can be computationally time consuming, and so I've worked on techniques for reducing their computational cost. See Fast Computational Methods for details.

In 1991, Claude Greengard and I ran an AMS-SIAM 1990 Summer Seminar on Vortex Dynamics and Vortex Methods. The proceedings of the workshop have been published as "Vortex Dynamics and Vortex Methods".

"On Vortex Methods", Christopher Anderson and Claude Greengard, SIAM J. Numer. Anal, Vol. 22, No 3, June 1985. 413-440

Abstract : We give error estimates for fully discretized two- and three-dimensional vortex methods and introduce a new way of evaluating the stretching of vorticity in three-dimensional vortex methods. The convergence theory of Beale and Majda is discussed and a simple proof of Cottet's consistency result is presented. We also describe how to obtain accurate two-dimensional vortex methods in which the initial computational points are distributed on the nodes of non-rectangular grids, and compare several three-dimensional vortex methods.


"A Vortex Method for Flows with Slight Density Variations", Christopher R. Anderson, J. of Comp. Phys., Vol. 61, No. 3, December 1985. 417-444

Abstract : We present a grid-free numerical method for solving two-dimensional, inviscid, incompressible flow problems with small density variations. The method, an extension of the vortex method, is based on a discretization of the equations written in the vorticity-stream formulation. The method is tested on an exact solution and is found to be both stable and accurate. An application to the motion of a two-dimensional line thermal is presented.


"The Vortex Ring Merger Problem at Infinite Reynolds Number", Christopher Anderson and Claude Greengard, Communications on Pure and Applied Mathematics, Vol. XLII 1123-1139 (1989).

Abstract : We present the results of a computation of the interaction of two vortex rings. The problem appears to be a good model of more general interactions of vortex tubes. We investigate the solution of the inviscid equations, a problem which to our knowledge has not previously been attempted with a fully three-dimensional vortex cores. We find that the cores flatten severely is they smash into each other. Although a tremendous increase in the magnitude of vorticity is observed, the computational evidence shows that due to the evolution of the vorticity into sheets, this does not lead to large velocities and catastrophic intensification of vorticity. In fact, the evidence suggests that the velocity remains bounded and the magnitude of the vorticity only grows linearly with time. There is also evidence of a new process for the generation small spatial scales which may be of importance in the initial range of turbulence.

"Vortex Methods and Vortex Dynamics", Proceedings of the AMS-SIAM 1990 Summer Seminar on Vortex Methods and Vortex Dynamics. Christopher R. Anderson, Claude Greengard Eds., Lectures in Applied Mathematics, AMS, 1991.