Application of Vortex invariants to roll up of Vortex pairs

A method developed by Betz for the rolled-up structure of vortices shed by isolated wing tips is extended to vortex pairs (two vortical regions of opposite sign), which are shed by wings of finite span. The present analysis again uses the invariants for the two-dimensional time-dependent motion of vortex systems, but the extension made here to vortex pairs depends primarily on the invariant for kinetic energy. It is found that the total energy in the flowfield can be separated into a part that governs the structure of each vortical region and a part that governs the spanwise distance between the centroids of the vortical regions. As a consequence, the rules for the rolled-up structure of vortical regions are the same as the one derived by Betz. Because the analysis does not apply a constraint that forces the circular contours of constant circulation to coincide with streamline paths, the solution is labeled first order. A method that can be used to obtain second- and higher-order approximations is described. Application of the first-order method to the vortex wake of an elliptically loaded wing indicates that a first-order solution for vortex pairs is adequate for many engineering purposes. The method derived here for single vortex pairs also justifies the superposition of axially symmetric vortical cores to simulate complex vortex wakes.