Application of a Zonal Decomposition Algorithm, to Improve the Computational Operation Count of the Discrete Vortex Method Calculation. G.U. Aero Report 9711.

Abstract

The vortex method has proved a very useful tool for analysing separated, incompressible flow around two dimensional bodies. The method utilises a grid free, Lagrangian approach, to discretise the vorticity field into a series of vortex particles. These particles are then tracked in time, using the Biot-Savart law to calculate the velocity field. This calculation requires the velocity of each vortex to be found as a sum over all other particles in the flow field. A Discrete Vortex Method (DVM) has been developed at the Department of Aerospace Engineering, University of Glasgow. Currently, this vortex method uses a direct summation technique, which although relatively simple, leads to a computational operation count proportional to the square of the number of particles. In calculations that use a large number of particles, such as bluff body models, the direct summation technique becomes prohibitively expensive. A new algorithm for the velocity calculation has now been included in the DVM and is presented in this report. The procedure uses a zonal decomposition algorithm for the velocity summation. This allows the effect of groups of particles on the velocity to be calculated using a single series expansion, thus significantly reducing the operation count of the calculation. The algorithm utilises a hierarchical technique, so that the largest possible group of particles is used for each series expansion. The resulting operation count is 0(N+NlogN), and therefore offers a significant improvement over the direct summation method