Vorticity Generation on a Flat Plate in 3D Flows

{Vortex methods, based on the splitting into Euler and Stokes operators, have been successfully adopted in numerical solutions of three-dimensional Navier-Stokes equations in free-space. Here we deal with their application to flows bounded by solid walls, discussing in particular the boundary conditions for vorticity and their approximation. In two dimensions this has been accomplished by introducing a vortex sheet at the wall, determined by the local slip-velocity, as an approximation of the vorticity source. For three-dimensional flows, we analyze in the context of the Stokes substep the integral equation for the vorticity source and its connection with the creation algorithm adopted in vortex methods. The present analysis leads to a formulation which shows the connection between the exact vorticity source at the wall and the discrete vorticity creation operator adopted in the Chorin-Marsden formula. In particular, the slip velocity at the wall is identified as an approximate solution of the integral equation for the vorticity source and the corresponding error estimate is also discussed. Besides showing the consistency of this approximation, we indicate a numerical procedure which provides a wall-generation of solenoidal vorticity. This is a crucial issue for an accurate application of vortex methods to three-dimensional flows.