Solution of Two-Dimensional Vorticity Equation on a Lagrangian Mesh

A novel, vorticity-based solution methodology has been developed to compute unsteady now past bodies. Vorticity is evolved on a set of points, and the vorticity in the remainder of the field is determined by linear interpolation, Interpolation is accomplished by Delaunay triangularization of the points in the field. Triangulation of the vorticity field provides a basis to integrate the vorticity to compute the velocity. Nodal connectivity from the triangularization also provides a list of the neighboring points that are used to construct a second-order least-squares fit of the vorticity. First- and second-order spatial derivatives can then be computed based on this polynomial fit. Surface vorticity on the body is computed to satisfy the no-slip boundary condition and is introduced into the flow via diffusion. A diffusion transport velocity was derived to account for spatial movement of the vorticity due to viscous diffusion. The points are advected by the sum of the induced velocity (computed from the Biot-Savart integral) and the diffusion velocity. The remaining diffusion term includes a form of the Laplacian and is computed directly. This solution scheme was found to be stable as applied to the problem of impulsively started flow about a circular cylinder and Rat plate. Comparisons with experimental and the Blasius boundary-layer solution fur a flat plate were used to demonstrate the effectiveness of this method.