Note on governing equations for a discrete vortex method

The characteristic of a coefficient matrix is investigated which is derived from governing equations used in a discrete vortex method. The reduction of the rank of the coefficient matrix when the vortices on the body are arranged symmetrically is demonstrated. If the characteristic of the matrix is kept unchanged by the rotation and/or parallel transformation of the coordinate system, a symmetrical axis can be specified. As a result, only three cases need be considered: (1) even vortices, none on the symmetrical axis; (2) even vortices, two on the symmetrical axis; and (3) odd vortices, only one on the symmetrical axis. The singularity of the coefficient matrix for these cases is proved.