A two-dimensional fast solver for arbitrary vortex distributions

Abstract

A method which is capable of an efficient calculation of the two-dimensional stream function and velocity field produced by a large system of vortices is presented in this report. This work is based on the adaptive scheme of Carrier, Greengard, and Rokhlin with the added feature that the evaluation or target points do not have to coincide with the location of the source or vortex positions. A simple algorithm based on numerical experiments has been developed to optimize the method for cases where the number of vortices N{sub V} differs significantly from the number of target points N{sub T}. The ability to specify separate source and target fields provides an efficient means for calculating boundary conditions, trajectories of passive scalar quantities, and stream-function plots, etc. Test cases have been run to benchmark the truncation errors and CPU time savings associated with the method. For six terms in the series expansions, non-dimensional truncation errors for the magnitudes of the complex potential and velocity fields are on the order of 10{sup {minus}5} and 10{sup {minus}3} respectively. The authors found that the CPU time scales as {radical}(N{sub V}N{sub T}) for N{sub V}/N{sub T} in the range of 0.1 to 10. For {radical}(N{sub V}N{sub T}) less than 200, there is virtually no CPU time savings while for {radical}N{sub V}N{sub T} roughly equal to 20,000, the fast solver obtains solutions in about 1% of the time required for the direct solution technique depending somewhat upon the configuration of the vortex field and the target field.