Flow Past Bluff Bodies

More than a century ago Kirchhoff solved for the velocity distribution within an elliptical patch of uniform vorticity. That solution became the basis for all further studies of elliptical vortices and has been regarded as the only known exact solution for a steady, elliptical patch of uniform vorticity. In the present paper, an exact solution for a new elliptical patch of uniform vorticity is presented. The vortex is constructed of streamlines of constant eccentricity. By specifying a velocity distribution along either of the principle axes of the vortex, continuity between differentially-spaced streamlines provides the velocity distribution throughout the vortex. Some of the unique features of the vortex are that although the vorticity is uniform throughout the vortex, the angular velocity about the center is non-uniform, unlike the Kirchhoff vortex wherein both are uniform. The point of maximum velocity occurs not at the end of the major axes as in the case of Kirchhoff\u27s vortex, but rather at the end of the minor axes, more nearly approximating the behavior of the twin vortices formed behind bluff bodies. In the present work, a non-orthogonal (non-confocal) elliptical coordinate system is employed to solve for the velocity and pressure distributions within the vortex