Use of Routh's Correction in the Cloud-in-Cell Discrete Vortex Method

Abstract

Incompressible fluid flows comprising evolving, two-dimensional, vorticity fields arecommonly represented by collections of discrete point vortices. The vorticity in the flowis generated at the solid boundaries of the motion in the so-called “physical plane.” If thegeometry of these boundaries is not simple, it is usual to calculate the flow field, includingthe trajectories of individual vortices, not in the physical plane but in a simpler “mappingplane.” According to the rules of conformal mapping, there is a one-to-one correspondencebetweenthetwoplanes.However,intheclassical,inviscidapproachRouth’scorrection(see,for example, [2]) is required in the calculation of the trajectory of a vortex to ensure thatthe velocities are consistent between the mapping and physical planes at the location of thevortex.Thiscorrectionarisesbecauseofthenatureoftheself-potentialatthevortexlocation.The “cloud-in-cell” (CIC) discrete vortex method [1] allows the velocity field resultingfrom a large number of point vortices to be calculated more efficiently than is possible in theinviscid (gridless) approach, by the introduction of a grid. Moreover the CIC method hasalso been used in conjunction with conformal mapping (see, for example, [7]). However, itisnotclear