John R. Grant

Solution of Two-Dimensional Vorticity Equation on a Lagrangian Mesh

Anovel,vorticity-basedsolutionmethodologyhasbeendevelopedtocomputeunsteadye owpastbodies.Vorticity is evolved on a set of points, and the vorticity in the remainder of the e eld is determined by linear interpolation. Interpolationisaccomplished byDelaunaytriangularizationofthepointsin the e eld. Triangulationofthevorticity e eldprovidesabasistointegratethevorticitytocomputethevelocity.Nodalconnectivityfromthetriangularization also provides a list of the neighboring points that are used to construct a second-order least-squares e t of the vorticity. First- and second-order spatial derivatives can then be computed based on this polynomial e t. Surface vorticity on the body is computed to satisfy the no-slip boundary condition and is introduced into the e ow via diffusion. A diffusion transport velocity was derived to account for spatial movementof the vorticity dueto 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 e ow about a circular cylinder and e at plate. Comparisons with experimental and the Blasius boundary-layer solution for a e at plate were used to demonstrate the effectiveness of this method.